# On an inhomogeneous coagulation model with a differential sedimentation
kernel

Iulia Cristian  Institute for Applied Mathematics, University of Bonn,
Endenicher Allee 60, 53115 Bonn, Germany<EMAIL_ADDRESS>Barbara
Niethammer  Institute for Applied Mathematics, University of Bonn, Endenicher
Allee 60, 53115 Bonn, Germany<EMAIL_ADDRESS>Juan J. L. Velázquez
Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60,
53115 Bonn, Germany<EMAIL_ADDRESS>
###### Abstract

We study an inhomogeneous coagulation equation that contains a transport term
in the spatial variable modeling the sedimentation of clusters. We prove local
existence of mass conserving solutions for a class of coagulation kernels for
which in the space homogeneous case instantaneous gelation (i.e.,
instantaneous loss of mass) occurs. Our result holds true in particular for
sum-type kernels of homogeneity greater than one, for which solutions do not
exist at all in the spatially homogeneous case. Moreover, our result covers
kernels that in addition vanish on the diagonal, which have been used to
describe the onset of rain and the behavior of air bubbles in water.

Keywords: inhomogeneous coagulation equations, mass-conserving solutions, sum-
type kernels.

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Main result
  2. 2 Proof of the main theorem
    1. 2.1 Formal approximation and discussion on the assumptions
    2. 2.2 Upper and lower bounds for the solution of the approximated model
    3. 2.3 Construction of a supersolution for a linear reformulation of the model
    4. 2.4 Proof of Theorem 1.4
  3. A Estimates for the second order derivative of $G_{L}$

##  1 Introduction

###  1.1 Background

In [25] the authors suggest a space-dependent coagulation equation to model
the onset of rain. Here spherical particles of volume $v$ move in space
vertically, for example due to gravitation, and merge when their trajectories
cross. This leads to the following inhomogeneous coagulation equation for the
density $f$ of particles of size $v$ at the point $x$:

$\displaystyle\partial_{t}f(x,v,t)+v^{\frac{2}{3}}\partial_{x}f(x,v,t)$
$\displaystyle=\frac{1}{2}\int_{(0,v)}K(v-v^{\prime},v^{\prime})f(x,v-v^{\prime},t)f(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\quad-\int_{(0,\infty)}K(v,v^{\prime})f(x,v,t)f(x,v^{\prime},t)\textup{d}v^{\prime}$
(1.1)

with a so-called differential sedimentation kernel of the form

$\displaystyle
K(v,v^{\prime})=|v^{\frac{2}{3}}-v^{\prime\frac{2}{3}}|(v^{\frac{1}{3}}+v^{\prime\frac{1}{3}})^{2}.$
(1.2)

This choice of kernel is motivated by the following consideration (see [25]):
the cross-section of interaction between two particles of radii $r$ and
$r^{\prime}$, and volume $v$ and $v^{\prime}$, respectively, that merged upon
touching is given by
$\pi(r+r^{\prime})^{2}\approx(v^{\frac{1}{3}}+v^{\prime\frac{1}{3}})^{2}$.

Additionally, the velocity is approximately $v^{\frac{2}{3}}$, which
represents the Stokes velocity of a rigid sphere with no slip boundary
condition, and the collision rate between two particles is taken to be
proportional to their relative velocities
$|v^{\frac{2}{3}}-v^{\prime\frac{2}{3}}|$.

The model is used to describe the behavior of air bubbles in water which move
due to buoyancy and it is also valid for water droplets. We refer to [20, 12,
32] for more details. In [25], slip-flow corrections for water droplets are
discussed and it is mentioned that this requires to change the power of the
volume for the velocity. More precisely, the left-hand side of (1.1) becomes
$\partial_{t}f(x,v,t)+v^{\alpha}\partial_{x}f(x,v,t)$, for some
$\alpha\in(0,1)$, and the kernel in (1.2) has the form
$K(v,v^{\prime})=|v^{\alpha}-v^{\prime\alpha}|(v^{\frac{1}{3}}+v^{\prime\frac{1}{3}})^{2}$.

The model in (1.1) with kernel (1.2) is referred to as the free merging regime
in [25], since it is assumed in its derivation that the particles merge when
their trajectories cross. When studying equations like (1.1), it is customary
to look for stationary solutions of non-zero flux (cf. [34]) of the form
$f\approx v^{d}$, for some $d\in\mathbb{R}$. One of the possible approaches is
to compute them using the so-called Zakharov transform (see [38]) and using it
we find the solution $f(x,v,t)\approx v^{-\frac{13}{6}}$. However, this
approach can be made rigorous only if the integral containing the coagulation
kernel in (1.1) is finite, which is not the case for the kernel (1.2). In
order to be able to rigorously find a solution for kernels with the same
homogeneity, the so-called forced locality regime in which only particles of
similar sizes can merge is studied in [25]. More precisely, for the forced
locality regime a cut-off in the coagulation kernel is introduced that makes
the kernel vanish outside the region where
$\frac{1}{q}<\frac{v^{\prime}}{v}<q$, for some $q>1$. With this cut-off, the
integral containing the coagulation kernel converges and thus the stationary
solution $f\approx v^{-\frac{13}{6}}$ is a valid solution.

Our main goal in this paper is to show that mass conserving solutions exist,
at least for a short time interval, for a class of inhomogeneous coagulation
equations that includes example (1.1) with (1.2). At a first glance this might
look surprising since the homogeneity of the kernel in (1.2) is greater than
one. Indeed, it is well-known that gelation (mass loss) occurs for the
standard one-dimensional coagulation equation,

$\displaystyle\partial_{t}f(v,t)=\frac{1}{2}\int_{(0,v)}K(v-v^{\prime},v^{\prime})f(v-v^{\prime},t)f(v^{\prime},t)\textup{d}v^{\prime}-\int_{(0,\infty)}K(v,v^{\prime})f(v,t)f(v^{\prime},t)\textup{d}v^{\prime},$
(1.3)

when the coagulation kernel behaves like a power law of homogeneity $\gamma>1$
(see, for example [19, 18, 26]). In particular, for sum kernels of the form

$\displaystyle K(v,v^{\prime})=v^{\gamma}+v^{\prime\gamma},$ (1.4)

with $\gamma>1$, gelation happens instantaneously. Actually, making use of
this property, one can prove that solutions which belong to $L^{1}$ for the
standard coagulation equation do not exist at all for kernels as in (1.4) (see
[2, 6, 15, 3]). In addition, it has been proven in [13] that the instantaneous
gelation phenomenon holds for Radon measure solutions of the standard
coagulation equation with sum kernels of homogeneity greater than one which
vanish on the diagonal, i.e. $K(v,v^{\prime})=0$, such as the kernel in (1.2).

Our main goal is then to prove that, in contrast to the homogeneous case,
there exist, at least for short times, mass conserving solutions to the
inhomogeneous model

$\displaystyle\partial_{t}f(x,v,t)+v^{\alpha}\partial_{x}f(x,v,t)$
$\displaystyle=\frac{1}{2}\int_{(0,v)}K(v-v^{\prime},v^{\prime})f(x,v-v^{\prime},t)f(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\qquad-\int_{(0,\infty)}K(v,v^{\prime})f(x,v,t)f(x,v^{\prime},t)\textup{d}v^{\prime},$
(1.5)

where $\alpha\in(0,1)$. Our proof holds for a rather general class of
coagulation kernels (see Assumption 1.1), in particular kernels of the form
(1.2) and (1.4). Thus, the model (1.1) provides a coagulation model in which
existence for kernels of the form (1.4) with $\gamma>1$ holds, at least for
short times.

###  1.2 Main result

#### Short time existence of mass conserving solutions for the inhomogeneous
model

Our goal is to prove short-time existence of mass conserving solutions for the
inhomogeneous model

$\displaystyle\partial_{t}f(x,v,t)+v^{\alpha}\partial_{x}f(x,v,t)$
$\displaystyle=\frac{1}{2}\int_{(0,v)}K(v-v^{\prime},v^{\prime})f(x,v-v^{\prime},t)f(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\qquad-\int_{(0,\infty)}K(v,v^{\prime})f(x,v,t)f(x,v^{\prime},t)\textup{d}v^{\prime},$
(1.6)

where

$\displaystyle\alpha\in(0,1).$ (1.7)

######  Assumption 1.1.

We assume that $K\colon[0,\infty)\times[0,\infty)\to[0,\infty)$ is a symmetric
and continuous function that satisfies

$\displaystyle 0\leq K(v,v^{\prime})\leq
K_{1}(v^{\gamma}+v^{\prime\gamma}),\textup{ with }\gamma\in[0,1+\alpha)$ (1.8)

for some constant $K_{1}>0$ and

$\displaystyle K(v-v^{\prime},v^{\prime})\leq K(v,v^{\prime}),\textup{ when
}v^{\prime}\in\big{[}0,\tfrac{v}{2}\big{]}.$ (1.9)

Condition (1.9) is a rather standard assumption in the study of coagulation
equations, see for example [27], and most of the kernels used in applications
satisfy this condition, in particular, kernels of the form
$K(v,v^{\prime})=v^{\gamma}+v^{\prime\gamma}$ or (1.2). The condition
$\gamma<1+\alpha$ in (1.8) is such that the transport term will control the
contribution coming from the coagulation term.

######  Definition 1.2.

[Mild solutions] Let $\alpha\in(0,1)$, $\gamma\in[0,1+\alpha)$ and
$m>\frac{\gamma+1}{\alpha}$. Let $T>0$ and $K$ satisfy Assumption 1.1. We say
that a non-negative function
$f\in\textup{C}([0,T]\times\mathbb{R}\times(0,\infty))$ such that

$\displaystyle\sup_{t\in[0,T]}\int_{(0,\infty)}(1+v^{\gamma})f(x,v,t)\textup{d}v$
$\displaystyle\leq\frac{C_{T}}{\operatorname{max}\Big{\\{}1,|x|^{m-\frac{\gamma+1}{\alpha}}\Big{\\}}},\textup{
for }x\in\mathbb{R},$ (1.10)

is a mild solution of equation (1.2) if

$\displaystyle f(x,v,t)-f(x-v^{\alpha}t,v,0)S[f](x,v,0,t)=$ (1.11)
$\displaystyle\frac{1}{2}\int_{0}^{t}\int_{(0,v)}S[f](x,v,s,t)K(v-v^{\prime},v^{\prime})f(x-(t-s)v^{\alpha},v-v^{\prime},s)f(x-(t-s)v^{\alpha},v^{\prime},s)\textup{d}s,$

for all $t\in[0,T]$, $v\in(0,\infty)$ and $x\in\mathbb{R}$, where

$\displaystyle
S[f](x,v,s,t):=\textup{e}^{-\int_{s}^{t}a[f](x-v^{\alpha}(t-\tau),v,\tau)\textup{d}\tau},$
(1.12)

with

$\displaystyle
a[f](x,v,t):=\int_{(0,\infty)}K(v,v^{\prime})f(x,v^{\prime},t)\textup{d}v^{\prime}.$
(1.13)

######  Definition 1.3.

We call $f\in\textup{C}([0,T]\times\mathbb{R}\times(0,\infty))$ a mass-
conserving solution of equation (1.2) if $f$ is as in Definition 1.2 and
satisfies in addition

$\displaystyle\int_{\mathbb{R}}\int_{(0,\infty)}vf(x,v,t)\textup{d}v\textup{d}x=\int_{\mathbb{R}}\int_{(0,\infty)}vf(x,v,0)\textup{d}v\textup{d}x$

for all $t\in[0,T]$.

######  Theorem 1.4 (Local existence of solutions).

Let $\alpha\in(0,1)$, $\gamma\in[0,1+\alpha)$, $m\in\mathbb{N}$ even, and
$p=\alpha m$ with
$m>\operatorname{max}\\{\frac{2\gamma+1}{\alpha},\frac{2}{\alpha}+3\\}$. Let
$K$ satisfy Assumption 1.1 and $T>0$ be sufficiently small. Let
$f_{\textup{in}}\in\textup{C}^{1}(\mathbb{R}\times(0,\infty))$ such that

$\displaystyle f_{\textup{in}}(x,v)\leq\frac{C_{0}}{1+|x|^{m}+v^{p}},$

for some $C_{0}>0$ and all $x\in\mathbb{R}$, $v\in(0,\infty)$. Then there
exists a mass-conserving solution $f$ of (1.11) as in Definition 1.3 that
satisfies

$\displaystyle f(x,v,t)\leq\frac{C}{1+|x|^{m}+v^{p}},$ (1.14)

for all $t\in[0,T]$, for some $C>0$.

######  Remark 1.5.

Theorem 1.4 is valid for coagulation kernels $K$ as in (1.2), as well as
coagulation kernels of the form $K(v,v^{\prime})=v^{\gamma}+v^{\prime\gamma}.$

######  Remark 1.6.

It is worthwhile to mention that mass conservation will follow due to the fact
that our solution will have sufficiently fast decay for large values of $|x|$
and $v$, see (1.14). For more details, see the proof of Theorem 1.4 .

When $\gamma<1$, our result could be expected according to the general theory
of existence for one-dimensional coagulation equations. This states that
solutions exist for kernels that behave like power laws of homogeneity
$\gamma<1$, see, for example, [33] for existence of solutions and [22] for
existence of self-similar profiles.

Some multi-dimensional coagulation models have been studied in the
mathematical literature, see [21, 14]. Moreover, several classes of
coagulation models for the distribution of particles with space dependence
have also been considered. In particular, models in which in addition to
coagulation there is space diffusion of the aggregating particles can be found
in [1, 8, 7, 27]. Models that contain coagulation of particles as well as
transport terms (that might include sedimentation terms) were studied in [10,
16, 17, 11, 5, 24]. In all the models mentioned above the homogeneity of the
coagulation kernel is either $\gamma<1$, case in which the solutions are
globally defined and preserve the total mass, or product-type kernels are
discussed, for which solutions preserve the mass up to a certain point in
time.

To our knowledge, the only exception that considers the case $\gamma>1$ for
space-dependent models is [23]. Indeed, existence of solutions for the
discrete version of the model in (1.1) has been established for coagulation
kernels of the form $K(i,j)=\sigma_{ij}|\nu_{i}-\nu_{j}|,$ $i,j\in\mathbb{N}$.
Here, $\nu_{i}$ is a non-negative function of volume which represents the
sedimentation velocity of the particles and $\sigma_{ij}$ can be estimated by
a function of the form $C(i^{\gamma}+j^{\gamma})$ with $\gamma<1$. Assuming
that $\nu_{i}$ scales like a power law $i^{\beta}$, we would obtain a
coagulation kernel $K(i,j)$ that behaves like a homogeneous function with
homogeneity $\beta+\gamma$ that might be larger than one. As a matter of fact,
the results in [23] hold for any choice of $\nu_{i}\geq 0$. In particular, the
solutions constructed in [23] have total mass of particles that changes
continuously in time. Nevertheless it has not been proven in [23] that the
mass of solutions is conserved. By contrast, in the present paper we construct
a theory of existence for mild solutions which conserve mass in finite time.
The class of kernels considered in [23] has a non-empty intersection with the
class of kernels considered in this paper and this intersection includes the
coagulation kernel (1.2), which is relevant in the physical literature as
explained above. However, none of two classes of kernels is included in the
other. In addition, the arguments in [23] seem to rely significantly on the
discrete character of the coagulation process and it is not clear how
difficult would be to extend these results to the continuous case. On the
other hand, the sedimentation $\nu_{i}$ can be arbitrary in [23]. This opens
the question of whether we can extend our result to powers of $\alpha$ which
are larger than one for the model in (1.2). However, a different approach of
obtaining information from the characteristics created by the transport term
in (1.2) will be needed in this case.

Moreover, kernels of the form $K(v,v^{\prime})=v^{\gamma}+v^{\prime\gamma},$
with $\gamma\in(1,1+\alpha)$, also satisfy the conditions (1.8), (1.9).
Solutions of the standard one-dimensional coagulation equation do not exist
for these types of kernels, see [2, 6, 15, 3]. This is to our knowledge the
first result of existence of mass-conserving solutions involving sum kernels
of homogeneity $\gamma>1$, regardless of whether one considers one-dimensional
or multi-dimensional coagulation models.

It is worth mentioning that our results also include the cases $\gamma=0$ and
$\gamma=1$, which are normally studied separately in the literature due to
their rich features, such as being able to predict the long-time behavior of
solutions, see [29, 30, 37, 9, 4], or prove uniqueness of self-similar
profiles, see [29, 35, 36], where the constant kernel and perturbations of the
constant kernel are discussed.

The strategy for proving existence of solutions is to consider an iterative
scheme based on a linear version of (1.2). For this equation, we are able to
find a suitable supersolution, which in turn will provide sufficiently good
moment estimates. This will give us compactness of the iterated sequence and
enable us to pass to the limit in the equation. The idea of finding
appropriate supersolutions was also used in [31], in this case for finding
self-similar solutions with fat tails. The idea of considering a linear
version of the model in order to better study properties of its solutions is
common in the study of coagulation equations, see for example [35] where this
idea is used to study uniqueness of solutions.

We present the proof of our main Theorem 1.4 in the following Section 2 with
some technical computations moved to the appendix.

##  2 Proof of the main theorem

###  2.1 Formal approximation and discussion on the assumptions

Our approach to prove existence of a solution to (1.2) is based on
constructing a suitable supersolution by approximating the coagulation term
for large particles by a transport term. To motivate this we present in this
subsection this formal approximation of (1.2). Similar computations can be
found in [25, Section 4 and Appendix 1].

Suppose now that $f$ is a solution of (1.2). Since we are interested in the
behavior for large values of $v$, we can assume, due to the fast decay of
$f(x,v,t)$, that the term $\int_{[\frac{v}{2},\infty)}K(v,v^{\prime})f(x,v,t)$
$f(x,v^{\prime},t)\textup{d}v^{\prime}$ gives a small contribution. This is
consistent with the known results in coagulation equations. We can then
approximate (1.2) via

$\displaystyle\partial_{t}f(x,v,t)+v^{\alpha}\partial_{x}f(x,v,t)$
$\displaystyle=\int_{(0,\frac{v}{2})}K(v-v^{\prime},v^{\prime})f(x,v-v^{\prime},t)f(x,v^{\prime},t)\textup{d}v^{\prime}-\int_{(0,\infty)}K(v,v^{\prime})f(x,v,t)f(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle=\int_{(0,\frac{v}{2})}[K(v-v^{\prime},v^{\prime})f(x,v-v^{\prime},t)-K(v,v^{\prime})f(x,v,t)]f(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\qquad-\int_{[\frac{v}{2},\infty)}K(v,v^{\prime})f(x,v,t)f(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\approx\int_{(0,\frac{v}{2})}[K(v-v^{\prime},v^{\prime})f(x,v-v^{\prime},t)-K(v,v^{\prime})f(x,v,t)]f(x,v^{\prime},t)\textup{d}v^{\prime}.$
(2.1)

Since our strategy relies on finding a suitable supersolution, it suffices to
find a lower bound for (2.1). This is where the assumption (1.9) is needed. We
thus use that $K(v-v^{\prime},v^{\prime})\leq K(v,v^{\prime})$ when
$v^{\prime}\in[0,\frac{v}{2}]$ and obtain that

$\displaystyle\partial_{t}$ $\displaystyle
f(x,v,t)+v^{\alpha}\partial_{x}f(x,v,t)$
$\displaystyle-\int_{(0,\frac{v}{2})}[K(v-v^{\prime},v^{\prime})f(x,v-v^{\prime},t)-K(v,v^{\prime})f(x,v,t)]f(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\geq\partial_{t}f(x,v,t)+v^{\alpha}\partial_{x}f(x,v,t)-\int_{(0,\frac{v}{2})}K(v,v^{\prime})[f(x,v-v^{\prime},t)-f(x,v,t)]f(x,v^{\prime},t)\textup{d}v^{\prime}.$
(2.2)

Assuming now that the coagulation kernel $K$ behaves like
$v^{\prime\gamma}+v^{\gamma}$ and since $v^{\prime}\in[0,\frac{v}{2}]$, we
further deduce that

$\displaystyle\partial_{t}f(x,v,t)+v^{\alpha}\partial_{x}f(x,v,t)$
$\displaystyle\approx\int_{(0,\frac{v}{2})}v^{\gamma}[f(x,v-v^{\prime},t)-f(x,v,t)]f(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\approx-v^{\gamma}\int_{(0,\frac{v}{2})}\int_{v-v^{\prime}}^{v}\partial_{w}f(x,w,t)\textup{d}wf(x,v^{\prime},t)\textup{d}v^{\prime}.$
(2.3)

Assume that $\partial_{w}f(x,w,t)$ behaves similarly for $w\in[\frac{v}{2},v]$
and thus

$\displaystyle\partial_{t}f(x,v,t)+v^{\alpha}\partial_{x}f(x,v,t)\approx-v^{\gamma}\partial_{v}f(x,v,t)\int_{(0,\frac{v}{2})}v^{\prime}f(x,v^{\prime},t)\textup{d}v^{\prime}.$
(2.4)

We denote by
$M_{1}(x,t):=\int_{(0,\infty)}v^{\prime}f(x,v^{\prime},t)\textup{d}v^{\prime}$
the first moment in $v$ of $f$. We consider only large values of $v$ so that
we can safely assume that
$\int_{(0,\frac{v}{2})}v^{\prime}f(x,v^{\prime},t)\textup{d}v^{\prime}$
contains most of the mass. In this manner, we can further approximate (2.4) by

$\displaystyle\partial_{t}f(x,v,t)+v^{\alpha}\partial_{x}f(x,v,t)\approx-v^{\gamma}\partial_{v}f(x,v,t)M_{1}(x,t).$
(2.5)

Notice that in order for our approximation to hold, the assumption (1.9) was
needed in (2.2). Otherwise, an analogous approximation of the model could be
obtained by replacing $v^{\gamma}\partial_{v}f(x,v,t)$ in (2.4) by
$\partial_{v}\big{(}v^{\gamma}f(x,v,t)\big{)}$. The approximation containing
the term $\partial_{v}\big{(}v^{\gamma}f(x,v,t)\big{)}$ is the one actually
used in [25]. However, due to (1.9), the approximation used in (2.5) suffices
in order to prove our desired result. Suppose now that $M_{1}(x,t)$ decays
sufficiently fast for large values of $x$, that is assume that

$\displaystyle M_{1}(x,t)\leq\frac{L}{1+|x|^{\overline{m}}},$ (2.6)

for some sufficiently large $\overline{m}$ and some $L>0$. Combining (2.6)
with (2.5), we obtain that $f$ should behave formally like the solution of the
equation

$\displaystyle\partial_{t}f(x,v,t)+v^{\alpha}\partial_{x}f(x,v,t)+\frac{Lv^{\gamma}\partial_{v}f(x,v,t)}{1+|x|^{\overline{m}}}=0.$
(2.7)

This motivates the analysis of the equation (2.16) below when trying to find a
supersolution for a linear version of (1.2). In order to obtain a behavior of
the form (2.6) for $M_{1}(x,t)$, we have to work with functions $f$ such that
(1.14) holds.

###  2.2 Upper and lower bounds for the solution of the approximated model

To prove short time existence of a solution to (1.2) we will set up an
iterative scheme and derive, using (2.7), a uniform supersolution for the
solutions of this scheme. More precisely, for $n\in\mathbb{N}$, we define
inductively a sequence of functions $\\{f_{n}\\}_{n\in\mathbb{N}}$ as follows:

$\displaystyle\partial_{t}f_{n+1}(x,v,t)+v^{\alpha}\partial_{x}f_{n+1}(x,v,t)$
$\displaystyle=\frac{1}{2}\int_{(0,v)}K(v-v^{\prime},v^{\prime})f_{n+1}(x,v-v^{\prime},t)f_{n}(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\qquad-\int_{(0,\infty)}K(v,v^{\prime})f_{n+1}(x,v,t)f_{n}(x,v^{\prime},t)\textup{d}v^{\prime},$
(2.8)

with $\alpha\in(0,1)$, and

$\displaystyle
f_{n+1}(x,v,0)=f_{n}(x,v,0)=f_{\textup{in}}(x,v)\leq\frac{C_{0}}{1+|x|^{m}+v^{p}},\qquad\textup{
for all }n\in\mathbb{N}.$ (2.9)

We take $f_{0}$ to be a function such that

$\displaystyle\partial_{t}f_{0}(x,v,t)+v^{\alpha}\partial_{x}f_{0}(x,v,t)=0$
(2.10)

with

$\displaystyle
f_{0}(x,v,0)=f_{\textup{in}}(x,v)\leq\frac{C_{0}}{1+|x|^{m}+v^{p}}.$ (2.11)

Notice that, if we have equality in (2.11), the solution of (2.10) is
$f_{0}(x,v,t)=\frac{C_{0}}{1+|x-v^{\alpha}t|^{m}+v^{p}}$.

######  Remark 2.1.

In principle we have to prove that the sequence in (2.2) is well-defined. A
rigorous proof would work as follows. First we approximate $K$ with a kernel
$K_{N}$ which is such that

$\displaystyle K_{N}(v,v^{\prime})=K(v,v^{\prime})\chi_{N}(v+v^{\prime}),$
(2.12)

for $N>1$ and where $\chi_{N}:[0,\infty)\rightarrow[0,1]$ is a continuous
functions such that $\chi_{N}(x)=1$, when $x\leq\frac{N}{2}$, and
$\chi_{N}(x)=0,$ when $x\geq N$. Then we can establish with a standard fixed-
point argument the existence and uniqueness of $f_{n,N}$. The key result is
then that we obtain a uniform fast decaying bound for the sequence of
solutions which is in particular independent of $N$. Then one can pass to the
limit $N\to\infty$. Since this procedure is standard once one has the bounds
on the solution, we omit the details here and work directly with $K$.

Notations and Assumptions: Let $R>0$. In the following we will denote by

$\xi_{R}\in
C([0,\infty))\,,\quad\xi_{R}\colon[0,\infty)\rightarrow[0,1]\;\mbox{ such that
}\xi_{R}(v)=1\mbox{ if }v\geq 2R\mbox{ and }\xi_{R}(v)=0\mbox{ if }v\leq R.$
(2.13)

Furthermore we assume

$\alpha\in(0,1)\,,\quad\gamma\in[0,1+\alpha)\,,\quad m\in{\varmathbb
N}\,,\quad m\mbox{ even}\,,\quad p=\alpha m\,,\quad
m>\operatorname{max}\Big{\\{}\frac{2\gamma+1}{\alpha},\frac{2}{\alpha}+3\Big{\\}}\,,$
(2.14)

and define $d$ via

$\left\\{\begin{aligned} d&=\bigg{[}\frac{2}{\alpha}\bigg{]}+1;&\textup{ if
}\bigg{[}\frac{2}{\alpha}\bigg{]}&\textup{ odd; }\\\
d&=\bigg{[}\frac{2}{\alpha}\bigg{]}+2;&\textup{ if
}\bigg{[}\frac{2}{\alpha}\bigg{]}&\textup{ even, }\end{aligned}\right.$ (2.15)

where $[\cdot]$ denotes the floor function. Note that (2.14) and (2.15) in
particular imply that $m>d+1$.

The main goal in this section will be to derive estimates for the solution of
a transport equation that approximates the coagulation equation. For $L>0$ let
$G_{L}$ be the solution of

$\displaystyle\partial_{t}G_{L}(x,v,t)+v^{\alpha}\partial_{x}G_{L}(x,v,t)+\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}G_{L}(x,v,t)$
$\displaystyle=0,$ (2.16) $\displaystyle G_{L}(x,v,0)$
$\displaystyle=\frac{C_{0}}{1+|x|^{m}+v^{p}}\,.$

We first study the properties of the backward characteristics for equation
(2.16). To this end, we define $X$ and $V$ via

$\left\\{\begin{aligned}
\partial_{t}{X}(x,v,t)&=-{V}^{\alpha},&\qquad{X}(x,v,0)&=x\,,\\\
\partial_{t}{V}(x,v,t)&=-\frac{{L}{V}^{\gamma}\xi_{R}({V})}{1+|{X}|^{m-d}},&\qquad{V}(x,v,0)&=v\,,\end{aligned}\right.$
(2.17)

where $d$ was defined in (2.15) and ${L}$ is as in (2.16).

######  Proposition 2.2 (Properties of the characteristics).

Given $L>0$ and $\delta\in\big{(}0,\frac{1}{2}\big{)}$ there exists a
sufficiently large $R>0$ such that for all $t\geq 0$ the following estimates
hold:

$\displaystyle(1-\delta)v$ $\displaystyle\leq V(x,v,t)\leq(1+\delta)v;$ (2.18)
$\displaystyle(1-\delta)v^{\alpha}t$ $\displaystyle\leq
x-X(x,v,t)\leq(1+\delta)v^{\alpha}t;$ (2.19)
$\displaystyle\frac{\alpha}{18}v^{\alpha-1}t$
$\displaystyle\leq-\partial_{v}X(x,v,t)\leq 18\alpha v^{\alpha-1}t.$ (2.20)

Moreover, if $x\not\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$, then

$\displaystyle\frac{1}{4}$
$\displaystyle\leq\partial_{v}V(x,v,t)\leq\frac{9}{4}.$ (2.21)

Otherwise, if $x\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$, then

$\displaystyle|\partial_{v}V(x,v,t)|$ $\displaystyle\leq
36L\operatorname{max}\\{1,v^{\gamma-1}t\\};$ (2.22)
$\displaystyle\partial_{v}V(x,v,t)$ $\displaystyle\leq 2.$ (2.23)

######  Corollary 2.3.

As an immediate consequence of (2.19) we obtain the following estimates for
all $t\geq 0$.

If $x\leq 0$ then

$|x|+(1-\delta)v^{\alpha}t=|x-(1-\delta)v^{\alpha}t|\leq|X(x,v,t)|\leq|x|+(1+\delta)v^{\alpha}t=|x-(1+\delta)v^{\alpha}t|\,.$
(2.24)

If $x>0$ then

$\displaystyle|x-(1-\delta)v^{\alpha}t|$
$\displaystyle\leq|X(x,v,t)|\leq|x-(1+\delta)v^{\alpha}t|\qquad\mbox{ if
}t\geq\frac{x}{(1-\delta)v^{\alpha}}\,,$ (2.25)
$\displaystyle|x-(1+\delta)v^{\alpha}t|$
$\displaystyle\leq|X(x,v,t)|\leq|x-(1-\delta)v^{\alpha}t|\qquad\mbox{ if
}t\leq\frac{x}{(1+\delta)v^{\alpha}},$ (2.26)

and

$\displaystyle|X(x,v,t)|\leq 3v^{\alpha}t\qquad\mbox{ if
}t\in\Big{(}\frac{x}{(1+\delta)v^{\alpha}},\frac{x}{(1-\delta)v^{\alpha}}\Big{)}\,.$
(2.27)

###### Proof of Proposition 2.2.

Proof of (2.18): First, we see from (2.17) that $X(x,v,t)\leq x$ and
$V(x,v,t)\leq v$ for all $t\geq 0$. Next, we define

$\displaystyle\psi(x):=\int_{-\infty}^{x}\frac{{L}\,\textup{d}\xi}{1+|\xi|^{m-d}}$
(2.28)

and $\Phi\colon(0,\infty)^{2}\to\mathbb{R}$, $(z,v)\to\Phi(z,v)$ via

$\displaystyle\Phi(\psi(X(x,v,t))-\psi(x),v):=V(x,v,t).$ (2.29)

Then it follows from (2.17) that

$\left\\{\begin{aligned}
\partial_{z}\Phi(\cdot,v)&=\Phi^{\gamma-\alpha}\xi_{R}(\Phi)\,,\qquad z>0\\\
\Phi(0,v)&=v\,.\end{aligned}\right.$ (2.30)

By integrating the ODE in (2.30) we deduce that

$\displaystyle\Phi(z,v)$ $\displaystyle=v$ $\displaystyle\qquad\mbox{ for all
}z\geq 0\,,v\in[0,R]\,,$ (2.31) $\displaystyle\Phi(z,v)$
$\displaystyle=(v^{1-(\gamma-\alpha)}+(1-(\gamma-\alpha))z)^{\frac{1}{1-(\gamma-\alpha)}}$
$\displaystyle\qquad\mbox{ for all }z\geq 0\,,v\geq 2R\,,$ (2.32)
$\displaystyle\Phi(z,v)$
$\displaystyle\leq(v^{1-(\gamma-\alpha)}+(1-(\gamma-\alpha))z)^{\frac{1}{1-(\gamma-\alpha)}}$
$\displaystyle\qquad\mbox{ for all }z\geq 0\,,\,v\geq R\,,$ (2.33)
$\displaystyle\Phi(z,v)$ $\displaystyle\geq v$ $\displaystyle\qquad\mbox{ for
all }z\geq 0\,,\,v\geq R\,.$ (2.34)

Notice also that $0\leq\psi(x)\leq CL$, for some constant $C>0$ which is
independent of $x$. It thus suffices to consider values of $z$ in the interval
$[0,CL]$. Taking in particular $R$ sufficiently large, it follows that
$(1-\delta)v\leq\Phi(z,v)\leq(1+\delta)v$ for all $z\in[0,CL]$ and $v\geq 0$.
Due to the definition of $\Phi$ in (2.29), the estimate (2.18) follows.

Proof of (2.19): Estimate (2.19) follows then from (2.18) and the relation

$\displaystyle X(x,v,t)-x=-\int_{0}^{t}V^{\alpha}(x,v,\tau)\textup{d}\tau$
(2.35)

together with the fact that $(1+\delta)^{\alpha}\leq 1+\delta$ and
$(1-\delta)^{\alpha}\geq 1-\delta$.

Proof of (2.20): In order to estimate the derivatives with respect to $v$ of
the characteristics we first prove that

$\displaystyle\frac{1}{2}\leq\partial_{v}\Phi(z,v)\leq 2.$ (2.36)

Estimate (2.36) is immediate if $v\leq R$. If $v\geq R$ we have

$\displaystyle\frac{\textup{d}}{\textup{d}z}\bigg{(}\frac{\textup{d}\Phi}{\textup{d}v}\bigg{)}=\bigg{[}(\gamma-\alpha)\Phi^{\gamma-\alpha-1}\xi_{R}(\Phi)+\Phi^{\gamma-\alpha}\xi^{\prime}_{R}(\Phi)\bigg{]}\frac{\textup{d}\Phi}{\textup{d}v}\,,\qquad\frac{\textup{d}\Phi}{\textup{d}v}(0)=1$

such that

$\displaystyle\frac{\textup{d}\Phi}{\textup{d}v}=\exp\Big{(}{\int_{0}^{z}\big{[}(\gamma-\alpha)\Phi^{\gamma-\alpha-1}\xi_{R}(\Phi)+\Phi^{\gamma-\alpha}\xi^{\prime}_{R}(\Phi)\big{]}\textup{d}s}\Big{)}\leq\exp\Big{(}CR^{\gamma-\alpha-1}\Big{)}$
(2.37)

and the upper bound in (2.36) follows since $\gamma<1+\alpha$ and if $R$ is
sufficiently large. Analogously, we obtain that

$\displaystyle\frac{\textup{d}\Phi}{\textup{d}v}\geq\exp\Big{(}-CR^{\gamma-\alpha-1}\Big{)}$

and (2.36) follows. In order to proceed, we notice that

$\displaystyle\int_{x}^{X}\frac{1}{\Phi(\psi(\xi)-\psi(x),v)^{\alpha}}\textup{d}\xi=-t.$
(2.38)

Differentiating in $v$ we obtain

$\displaystyle\frac{\partial_{v}X(x,v,t)}{\Phi(\psi(X(x,v,t))-\psi(x),v)^{\alpha}}=\alpha\int_{x}^{X}\frac{\partial_{v}\Phi(\psi(\xi)-\psi(x),v)}{\Phi(\psi(\xi)-\psi(x),v)^{\alpha+1}}\textup{d}\xi.$
(2.39)

Since $\delta\in(0,\frac{1}{2})$, (2.18) and (2.19) imply that
$\frac{v}{2}\leq V(x,v,t)\leq\frac{3v}{2}$ and that $\frac{v^{\alpha}t}{2}\leq
x-X(x,v,t)\leq\frac{3v^{\alpha}t}{2}$. From this we deduce, using (2.19), the
estimates for $\Phi$, and the fact that $\alpha<1$, that

$\displaystyle-\partial_{v}X(x,v,t)=$
$\displaystyle\alpha\int_{X}^{x}\frac{\partial_{v}\Phi(\psi(\xi)-\psi(x),v)}{\Phi(\psi(\xi)-\psi(x),v)^{\alpha+1}}\textup{d}\xi\times[\Phi(\psi(X(x,v,t))-\psi(x),v)]^{\alpha}$
$\displaystyle\leq 4\alpha
3^{\alpha}\int_{X}^{x}\frac{1}{v^{\alpha+1}}\textup{d}\xi v^{\alpha}\leq
12\alpha v^{-1}[x-X(x,v,t)]\leq 18\alpha v^{\alpha-1}t.$ (2.40)

Analogously we obtain that

$\displaystyle-\partial_{v}X\geq$
$\displaystyle\frac{\alpha}{2}\int_{X}^{x}\frac{1}{\Phi(\psi(\xi)-\psi(x),v)^{\alpha+1}}\textup{d}\xi\times[\Phi(\psi(X(x,v,t))-\psi(x),v)]^{\alpha}$
$\displaystyle\geq\frac{\alpha}{3^{\alpha+1}}\int_{X}^{x}\frac{1}{v^{\alpha+1}}\textup{d}\xi
v^{\alpha}\leq\frac{\alpha}{9}v^{-1}[x-X(x,v,t)]\geq\frac{\alpha}{18}v^{\alpha-1}t,$

which concludes the proof of (2.20).

Proof of (2.21): We now prove that (2.21) holds if
$x\not\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$. From (2.29) we
deduce that

$\displaystyle\partial_{v}V(x,v,t)=\partial_{v}\Phi(\psi(X)-\psi(x),v)+\partial_{z}\Phi(\psi(X)-\psi(x),v)\psi^{\prime}(X)\partial_{v}X.$
(2.41)

Due to (2.36) it suffices to show that

$\displaystyle|\partial_{z}\Phi(\psi(X)-\psi(x),v)\psi^{\prime}(X)\partial_{v}X|\leq\frac{1}{4}$
(2.42)

in order to conclude our proof. From (2.29), (2.30) and (2.18), we have that

$\displaystyle 0\leq\partial_{z}\Phi(z,v)\leq 2v^{\gamma-\alpha}.$

Indeed, by (2.28) and (2.18) it holds that

$\displaystyle\partial_{z}\Phi(z,v)\leq\Phi(z,v)^{\gamma-\alpha}\leq\operatorname{max}\\{2^{\alpha-\gamma},\bigg{(}\frac{3}{2}\bigg{)}^{\gamma-\alpha}\\}v^{\gamma-\alpha}\leq
2v^{\gamma-\alpha}.$

By (2.20), it thus follows that

$\displaystyle|\partial_{z}\Phi(\psi(X)-\psi(x),v)\psi^{\prime}(X)\partial_{v}X|\leq\frac{36\alpha
Lv^{\gamma-\alpha}}{1+|X(t)|^{m-d}}v^{\alpha-1}t=\frac{36\alpha
Lv^{\gamma-1}t}{1+|X(t)|^{m-d}}.$ (2.43)

We only analyze here the case when $v\geq R$, since by (2.30) we have that
$\partial_{z}\Phi(z,v)=0$ when $v\leq R$ and thus there is nothing to prove in
this case.

  1. 1)

Assume $x\leq(1-2\delta)v^{\alpha}t$.

Then by (2.25) we have
$|X(x,v,t)|\geq|x-(1-\delta)v^{\alpha}t|=(1-\delta)v^{\alpha}t-x\geq\delta
v^{\alpha}t$ and we obtain

$\displaystyle|\partial_{z}\Phi(\psi(X)-\psi(x),v)\psi^{\prime}(X)\partial_{v}X|\leq
CL\frac{v^{\gamma-1}t}{1+(\delta v^{\alpha}t)^{m-d}}\,.$

Now, if $v^{\gamma-1}t\geq\frac{1}{4LC}$, then $v^{\alpha}t\geq\frac{1}{4LC}$
since $\alpha>\gamma-1$ and we obtain

$\displaystyle|\partial_{z}\Phi(\psi(X)-\psi(x),v)\psi^{\prime}(X)\partial_{v}X|$
$\displaystyle\leq CL\frac{v^{\gamma-1}t}{1+(\delta
v^{\alpha}t)^{m-d}}\,\leq\frac{CL}{\delta}v^{\gamma-\alpha-1}(\delta
v^{\alpha}t)^{1+d-m}$ $\displaystyle\leq
C(L,\delta)v^{\gamma-\alpha-1}\leq\frac{1}{4}$

if $R>0$ is sufficiently large. If $v^{\gamma-1}t\leq\frac{1}{4LC}$, (2.43)
becomes

$\displaystyle|\partial_{z}\Phi(\psi(X)-\psi(x),v)\psi^{\prime}(X)\partial_{v}X|\leq\frac{CLv^{\gamma-1}t}{1+(\delta
v^{\alpha}t)^{m-d}}\leq\frac{1}{4}.$ (2.44)

  1. 2)

Assume $x\geq(1+2\delta)v^{\alpha}t$. Then
$|X(x,v,t)|\geq|x-(1+\delta)v^{\alpha}t|=x-(1+\delta)v^{\alpha}t\geq\delta
v^{\alpha}t$ and we can conclude as before.

Proof of (2.22) and (2.23): If
$x\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$, then from (2.43) it
follows that

$\displaystyle|\partial_{z}\Phi(\psi(X)-\psi(x),v)\psi^{\prime}(X)\partial_{v}X|\leq\frac{36Lv^{\gamma-1}t}{1+|X(t)|^{m-d}}\leq
36Lv^{\gamma-1}t$

and (2.22) follows.

In order to prove that (2.23) holds, we notice that

$\displaystyle\partial_{z}\Phi(\psi(X)-\psi(x),v)\psi^{\prime}(X)\partial_{v}X\leq
0.$

Combining this with (2.41) and (2.36), the conclusion follows. ∎

With the help of the characteristics, the solution $G_{L}$ of (2.16) can be
written as

$\displaystyle
G_{L}(x,v,t)={\frac{C_{0}}{1+|{X}(x,v,t)|^{m}+{V}(x,v,t)^{p}}}.$ (2.45)

Moreover, it holds that

$\displaystyle\partial_{v}G_{L}(x,v,t)=-{C_{0}}\frac{[m|X|^{m-2}X\partial_{v}{X}+p{V}^{p-1}\partial_{v}{V}]}{(1+|X|^{m}+{V}^{p})^{2}}\,\big{(}x,v,t\big{)}.$
(2.46)

This function $G_{L}$ will be the main building block for constructing a
uniform supersolution to the sequence $\\{f_{n}\\}_{n\in{\varmathbb N}}$ in
Subsection 2.3. We would like this supersolution to be decreasing in $v$ for
fixed $x$. Unfortunately, for $x>0$ the function $G_{L}$ is not decreasing in
$v$. The next proposition characterizes a local maximum of $G_{L}$.

######  Proposition 2.4.

$a)$ Given $L>0$ and $\delta\in\big{(}0,\frac{1}{2}\big{)}$ there exists a
sufficiently large $R>0$ such that for all $t\in[0,1]$ the following holds.
For every $x\in\mathbb{R}$ and $t\in[0,1]$ there exists at least one point
$v_{\operatorname{max}}(x,t)$ with the properties that

$\displaystyle
v_{\textup{max}}(x,t)^{\alpha}\notin\bigg{[}\frac{x}{(1+2\delta)t},\frac{x}{(1-2\delta)t}\bigg{]}$
(2.47)

and

$\displaystyle\partial_{v}G_{L}(x,v_{\operatorname{max}}(x,t),t)=0\,.$ (2.48)

Moreover, there exists a constant $K_{max}>0$, which is independent of $x$,
$t$, $\delta$, $R$, and $L$, such that the following holds:

$\displaystyle\frac{1}{K_{max}}xt^{\frac{1}{m-1}}\leq
v_{\operatorname{max}}(x,t)^{\alpha}\leq K_{max}xt^{\frac{1}{m-1}}.$ (2.49)

$b)$ Given $L>0$ and $\delta\in\big{(}0,\frac{1}{2}\big{)}$ there exists a
sufficiently large $R>0$ such that for all $t\in[0,T]$, with $T$ sufficiently
small, that is independent of $L,\delta,$ and $R$, there exists a unique point
$v_{\textup{max}}$ with the properties (2.47) and (2.48).

###### Proof.

Let
$v^{\alpha}\notin\Big{[}\frac{x}{(1+2\delta)t},\frac{x}{(1-2\delta)t}\Big{]}$
such that (2.48) holds. Notice that from (2.46), it follows that
$m|X|^{m-2}X\partial_{v}{X}+p{V}^{p-1}\partial_{v}{V}=0$. If $R$ is
sufficiently large, then the estimates from Proposition 2.2 hold. We have the
following cases.

Case 1. $x-(1+2\delta)v^{\alpha}t\geq 0$. Then, by (2.19), it holds that
$0\leq x-(1+\delta)v^{\alpha}t\leq X(x,v,t)\leq x-(1-\delta)v^{\alpha}t$ and
by (2.20), it follows that $\partial_{v}X\leq 0$. Thus, if (2.48) holds, we
have that $m|X|^{m-1}|\partial_{v}X|=pV^{p-1}\partial_{v}V$. By Proposition
2.2 and since $\delta\in(0,\frac{1}{2})$, we have that there exists a constant
$C>0$, that is independent of $\delta,L$ and $R$, such that

$\displaystyle\frac{1}{C}|x-(1+\delta)v^{\alpha}t|^{m-1}v^{\alpha-1}t\leq
m|X|^{m-1}|\partial_{v}X|=pV^{p-1}\partial_{v}V\leq Cv^{p-1}.$

This implies that $\frac{1}{C}|x-(1+\delta)v^{\alpha}t|^{m-1}t\leq
v^{\alpha(m-1)}.$ Since $m-1$ is odd, $\delta<1$, and
$x-(1+2\delta)v^{\alpha}t\geq 0$, we further have that $xt^{\frac{1}{m-1}}\leq
Cv^{\alpha}+2v^{\alpha}t^{\frac{m}{m-1}}$. Since $t\leq 1$, we then obtain
that $xt^{\frac{1}{m-1}}\leq Cv^{\alpha}$ and the lower bound in (2.49)
follows. In order to obtain the upper bound in (2.49), we use similar
computations together with the fact that $t\geq 0$ and that

$\displaystyle C|x-(1-\delta)v^{\alpha}t|^{m-1}v^{\alpha-1}t\geq
m|X|^{m-1}|\partial_{v}X|=pV^{p-1}\partial_{v}V\geq\frac{1}{C}v^{p-1}.$

Case 2. $x-(1-2\delta)v^{\alpha}t\leq 0$. Then, by (2.19), it holds that
$x-(1+\delta)v^{\alpha}t\leq X(x,v,t)\leq x-(1-\delta)v^{\alpha}t\leq 0$ and
thus
$m|X|^{m-2}X\partial_{v}{X}+p{V}^{p-1}\partial_{v}{V}=m|X|^{m-1}|\partial_{v}{X}|+p{V}^{p-1}\partial_{v}{V}>0$
in this region. Using this and (2.46), it follows that
$\partial_{v}G_{L}(x,v,t)<0$.

In Appendix A, Proposition A.1, we will prove that for a sufficiently small
$T>0$, which is independent of $L,\delta,$ and $R$, we have that

$\displaystyle\partial_{v}^{2}G_{L}(x,v,t)<0,$ (2.50)

for all $v$ that satisfy the estimate in (2.49) and all $t\in[0,T]$ which
implies the uniqueness of such a point. This concludes our proof. ∎

The following lemma will also be needed in the construction of a supersolution
in Section 2.3.

######  Proposition 2.5.

Given $L>0$ and $\delta\in\big{(}0,\frac{1}{2}\big{)}$ there exists a
sufficiently large $R>0$ such that for all $t\in[0,T]$, with $T$ sufficiently
small, which is independent of $L,\delta,$ and $R$, it holds that

$\displaystyle\partial_{x}G_{L}(x,v_{\textup{max}}(x,t),t)\leq 0,$ (2.51)

where $G_{L}$ is the solution of (2.16) and $v_{\textup{max}}(x,t)$ was
defined in (2.48).

###### Proof.

If $R$ is sufficiently large, then the estimates from Proposition 2.2 hold. We
first notice that due to (2.19) and (2.49) we have

$X(x,v_{\operatorname{max}}(x,t),t)\geq
x-2v_{\operatorname{max}}(x,t)^{\alpha}t\geq x-2K_{max}xt^{1+\frac{1}{m-1}}>0$

if $t\in[0,T]$ and $T$ is sufficiently small. Then we compute

$\displaystyle\partial_{x}G_{L}(x,v,t)=-C_{0}\frac{[m|X|^{m-2}X\partial_{x}{X}+p{V}^{p-1}\partial_{x}{V}]}{(1+|X|^{m}+{V}^{p})^{2}}(x,v,t)\,,$

and thus it suffices to prove that

$\displaystyle\Big{(}mX^{m-1}\partial_{x}{X}+p{V}^{p-1}\partial_{x}{V}\Big{)}(x,v_{\textup{max}}(x,t),t)\geq
0\,.$ (2.52)

Since $V\geq 0$ we have to prove that
$\partial_{x}{X}(x,v_{\textup{max}}(x,t),t)\geq 0$ and
$\partial_{x}{V}(x,v_{\textup{max}}(x,t),t)\geq 0.$ We start by analyzing
$\partial_{x}{X}$.

We consider the case when $v\geq 2R$, since the other cases work similarly.
Differentiating (2.38) with respect to $x$, keeping in mind that $x>0$, we
obtain

$\displaystyle\frac{\partial_{x}X(x,v,t)}{\Phi(\psi(X(x,v,t))-\psi(x),v)^{\alpha}}+\alpha\int_{x}^{X}\frac{\partial_{z}\Phi\partial_{x}\psi(x)\textup{d}\xi}{\Phi(\psi(\xi)-\psi(x),v)^{\alpha+1}}-\frac{1}{\Phi(0,v)^{\alpha}}=0.$
(2.53)

We have that $\partial_{z}\Phi\geq 0$ due to (2.30). Since $X(x,v,t)\leq x$
and by (2.29) and (2.18) we obtain

$\partial_{x}X(x,v,t)\geq\frac{\Phi(\psi(X(x,v,t))-\psi(x),v)^{\alpha}}{v^{\alpha}}\geq\frac{1}{C}>0\,.$

Thus, in order for (2.52) to hold, what is left to prove is that
$\partial_{x}{V}(x,v_{\textup{max}}(x,t),t)\geq 0.$ Integrating (2.17) over
time and differentiating with respect to $x$, we obtain

$\displaystyle{V}(x,v,t)^{-\gamma}\partial_{x}{V}(x,v,t)$
$\displaystyle=\int_{0}^{t}\frac{{L}(m-d)|X|^{m-d-2}X\partial_{x}{X}\textup{d}\xi}{(1+{X}(\xi)^{m-d})^{2}}.$
(2.54)

Since we have ${X}(x,v_{\operatorname{max}}(x,t),t)\geq 0$ and
$\partial_{x}{X}(x,v,t)\geq 0$ we obtain
$\partial_{x}{V}(x,v_{\textup{max}}(x,t),t)\geq 0.$ ∎

###  2.3 Construction of a supersolution for a linear reformulation of the
model

In this subsection we will prove that the solution $G_{L}$ of (2.16), after
suitable modifications, is a supersolution for problem (2.2).

######  Definition 2.6.

Let $\delta\in\big{(}0,\frac{1}{2}\big{)}$, $L>0$ and $R$, which depends on
$L$ and $\delta$, as in Proposition 2.2. Let $G_{L}$ be the solution of (2.16)
with given $\delta,L,$ and $R$. We define the function $H_{L}$ via

$\displaystyle H_{L}(x,v,t)=\left\\{\begin{array}[]{ll}G_{L}(x,v,t),&\mbox{ if
}\partial_{v}G_{L}(x,v,t)\leq 0\;\mbox{ or
}\;x\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t];\\\
G_{L}(x,v_{\textup{max}}(x,t),t),&\textup{otherwise},\\\ \end{array}\right.$

where $v_{\textup{max}}(x,t)$ was defined in (2.48).

By the choice of $v_{\textup{max}}$ in (2.48), if
$x\notin[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$ and $v\leq
v_{\textup{max}}(x,t)$, then we have that
$H_{L}(x,v,t)=G_{L}(x,v_{\textup{max}}(x,t),t)$. Moreover, $H_{L}$ is
decreasing in $v$ for fixed $x$ outside possibly a critical region where
$x\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$.

We first collect some properties of the function $H_{L}$, which are
independent of $\delta,L,$ and $R$.

######  Lemma 2.7.

Let $\delta\in\big{(}0,\frac{1}{2}\big{)}$, $L>0$, and $R$ as in Proposition
2.2. Let $T>0$ be sufficiently small, independent of $\delta,$ $L$, and $R$,
and $H_{L}$ as in Definition 2.6. Then there exists a constant $K_{2}>0$,
which is independent of $\delta,$ $L$, and $R$, such that the following holds
for $t\in[0,T]$.

If $x>0$ then

$\displaystyle H_{L}(x,v-v^{\prime},t)\leq K_{2}H_{L}(x,v,t)\textup{ for all
}v^{\prime}\in\bigg{(}0,\frac{v}{2}\bigg{)}$ (2.55)

If $x>0$ and for all $v$ such that
$v^{\alpha}\notin\big{[}\frac{x}{(1+2\delta)t},\frac{x}{(1-2\delta)t}\big{]}$
then there exists a sufficiently large constant $C_{l}>0$ such that

$\displaystyle-\partial_{v}H_{L}(x,v^{\prime},t)\leq-
K_{2}\partial_{v}H_{L}(x,v,t)\qquad\mbox{ for all
}v\geq\operatorname{max}\\{R,C_{l}v_{\operatorname{max}}(x,t)\\}\,,v^{\prime}\in\Big{(}\tfrac{v}{2},v\Big{)}.$
(2.56)

If $x\leq 0$ then

$\displaystyle-\partial_{v}H_{L}(x,v^{\prime},t)\leq-
K_{2}\partial_{v}H_{L}(x,v,t)\qquad\mbox{ for all }v\geq
R\,,v^{\prime}\in\big{(}\tfrac{v}{2},v\big{)}\,.$ (2.57)

###### Proof.

Proof of (2.55): We will prove that (2.55) holds by proving separately that
there exists $K_{2}>0$, which is independent of $\delta,L,$ and $R$, such
that:

If $x>0$ and for all $v$ such that
$v^{\alpha}\notin\big{[}\frac{x}{(1+2\delta)t},\frac{x}{(1-2\delta)t}\big{]}$
then

$\displaystyle H_{L}(x,v-v^{\prime},t)$ $\displaystyle\leq
K_{2}H_{L}(x,v,t)\qquad\mbox{ for all
}v^{\prime}\in\Big{(}0,\tfrac{v}{2}\Big{)}\mbox{ and }v,v-v^{\prime}\geq
v_{\textup{max}}(x,t)\,,$ (2.58) $\displaystyle
H_{L}(x,v_{\operatorname{max}}(x,t),t)$ $\displaystyle\leq
K_{2}H_{L}(x,v,t)\qquad\mbox{ for all
}v\in(v_{\operatorname{max}}(x,t),2v_{\operatorname{max}}(x,t)),$ (2.59)

If $x>0$ and for all $v$ such that
$v^{\alpha}\in\big{[}\frac{x}{(1+2\delta)t},\frac{x}{(1-2\delta)t}\big{]}$
then

$\displaystyle H_{L}(x,v-v^{\prime},t)\leq K_{2}H_{L}(x,v,t)\textup{ for all
}v^{\prime}\in\bigg{(}0,\frac{v}{2}\bigg{)}.$ (2.60)

Before beginning our proof, we make the following observation. Because the
proof of each region when
$v^{\alpha}\notin\big{[}\frac{x}{(1+2\delta)t},\frac{x}{(1-2\delta)t}\big{]}$
differs, we have to distinguish between different cases.

If $v^{\alpha}\geq\frac{x}{(1-2\delta)t}$ and $w\in[\frac{v}{2},v]$, we have
the subcases

  * 1\. a)

$x>0$, $v^{\alpha}\geq\frac{x}{(1-2\delta)t}$, and
$w^{\alpha}\geq\frac{x}{(1-2\delta)t}$. Notice that in this region
$x-(1-\delta)w^{\alpha}t\leq 0$;

  * 1\. b)

$x>0$, $v^{\alpha}\geq\frac{x}{(1-2\delta)t}$, and
$w^{\alpha}\in\bigg{[}\frac{x}{(1+2\delta)t},\frac{x}{(1-2\delta)t}\bigg{]}$;

  * 1\. c)

$x>0$, $v^{\alpha}\geq\frac{x}{(1-2\delta)t}$, and
$w^{\alpha}\leq\frac{x}{(1+2\delta)t}$. Notice that in this region
$x-(1+\delta)w^{\alpha}t\geq 0$.

The remaining case is

  * 2.

$x>0$, $v^{\alpha}\leq\frac{x}{(1+2\delta)t}$ and then
$w^{\alpha}\leq\frac{x}{(1+2\delta)t}$.

Proof of (2.58): We will prove that there exists a constant $C>0$, which is
independent of $\delta,L,$ and $R$, such that

$\displaystyle\frac{1}{C(1+|x|^{m}+v^{p})}\leq
H_{L}(x,w,t)\leq\frac{C}{1+|x|^{m}+v^{p}},$ (2.61)

for all $w\in[\frac{v}{2},v]$.

Case $1.$ $a)$ Notice that because of (2.19), we are in the region where
$X(x,w,t)\leq 0$. Because of (2.25), it follows that
$|x-(1-\delta)w^{\alpha}t|\leq|X(x,w,t)|\leq|x-(1+\delta)w^{\alpha}t|$. Thus,
due to (2.45), it suffices to show in this case that

$\displaystyle|x-(1+\delta)w^{\alpha}t|^{m}+w^{p}\leq
C\big{(}|x|^{m}+v^{p}\big{)}\leq
C\bigg{(}|x-(1-\delta)w^{\alpha}t|^{m}+w^{p}\bigg{)}.$ (2.62)

To prove (2.62) we notice that is suffices to show that
$|x-(1+\delta)w^{\alpha}t|^{m}+w^{p}\leq C\big{(}|x|^{m}+w^{p}\big{)}\leq
C\big{(}|x-(1-\delta)w^{\alpha}t|^{m}+w^{p}\big{)}$ since
$w\in[\frac{v}{2},v]$. Since $x-(1-\delta)w^{\alpha}t\leq 0$ in this case, it
holds that $|x-(1-\delta)w^{\alpha}t|\leq w^{\alpha}$. In this case, we have
that $|x|-(1-\delta)w^{\alpha}t\leq|x-(1-\delta)w^{\alpha}t|\leq w^{\alpha}$.
This implies that $|x|\leq 2w^{\alpha}$. Notice that since $x\geq 0$ and
$x-(1-\delta)w^{\alpha}t\leq 0$, it immediately follows that $0\leq x\leq
w^{\alpha}$. However, we do a more general proof as a similar estimate will be
needed in order to prove (2.57) or when $x-(1+\delta)w^{\alpha}\geq 0$ later
on.

Since $(|x|+(1+\delta)w^{\alpha}t)^{m}\leq C_{m}(|x|^{m}+t^{m}w^{p})$ and $t$
is sufficiently small, we obtain

$\displaystyle|x-(1+\delta)w^{\alpha}t|^{m}+w^{p}$
$\displaystyle\leq(|x|+(1+\delta)w^{\alpha}t)^{m}+w^{p}\leq
C_{m}(|x|^{m}+t^{m}w^{p})+w^{p}$ $\displaystyle\leq C(|x|^{m}+w^{p}).$ (2.63)

Additionally, since $|x|\leq 2w^{\alpha}$, it holds that

$\displaystyle|x-(1-\delta)w^{\alpha}t|^{m}+w^{p}\geq
w^{p}=\frac{w^{p}}{2}+\frac{w^{p}}{2}\geq\frac{1}{C}(|x|^{m}+w^{p})\,.$ (2.64)

Case $1.$ $b)$ From (2.19) and (2.27), it follows that $X(x,v,t)\leq 0$ and
$|X(x,w,t)|\leq Cw^{\alpha}t$. Since in this case
$v^{\alpha}\geq\frac{x}{(1-2\delta)t}$, from (2.62) we know that there exists
a constant $C>0$, which is independent of $\delta,L,$ and $R$, such that

$\displaystyle\frac{1}{C(1+|x|^{m}+v^{p})}\leq H_{L}(x,v,t).$

Thus, in order for (2.58) to hold in this case, we need to prove that there
exists a constant $C>0$, which is independent of $\delta,L,$ and $R$, such
that $H_{L}(x,w,t)\leq\frac{C}{1+|x|^{m}+v^{p}}$, for $w\in[\frac{v}{2},v]$.
More precisely, due to (2.45), it suffices to prove that

$\displaystyle 1+|X(x,w,t)|^{m}+V(x,w,t)^{p}\geq\frac{1}{C}(1+|x|^{m}+v^{p}).$
(2.65)

Since $t\leq 1$, we have that $x\leq(1+2\delta)w^{\alpha}t\leq 2w^{\alpha}$ in
this case. Due to (2.18), it holds that

$\displaystyle|X(x,w,t)|^{m}+V(x,w,t)^{p}\geq V(x,w,t)^{p}\geq
Cw^{p}=\frac{Cw^{p}}{2}+\frac{Cw^{p}}{2}\geq C(|x|^{m}+w^{p}).$

Case $1.$ $c)$ From (2.19), it follows that $X(x,v,t)\leq 0$ and $X(x,w,t)\geq
0$. Since in this case $v^{\alpha}\geq\frac{x}{(1-2\delta)t}$, from (2.62) we
know that there exists a constant $C>0$, independent of $\delta,L,$ and $R$,
such that $\frac{1}{C(1+|x|^{m}+v^{p})}\leq H_{L}(x,v,t)$ and we need to prove
that $H_{L}(x,w,t)\leq\frac{C}{1+|x|^{m}+v^{p}}$, for $w\in[\frac{v}{2},v]$.
More precisely, due to (2.26) and (2.45), it suffices to prove as before that
$1+|X(x,w,t)|^{m}+V(x,w,t)^{p}\geq C(1+|x|^{m}+v^{p})$. Actually, we prove
here a more general estimate that will be used in Case $2.$, namely

$\displaystyle|x-(1-\delta)w^{\alpha}t|^{m}+w^{p}\leq
C\big{(}|x|^{m}+v^{p}\big{)}\leq
C\bigg{(}|x-(1+\delta)w^{\alpha}t|^{m}+w^{p}\bigg{)}$ (2.66)

In order to prove (2.66), we distinguish between two cases:

  1. i)

$|x-(1+\delta)w^{\alpha}t|\geq w^{\alpha}$. In this case, we have that
$w^{\alpha}\leq|x-(1+\delta)w^{\alpha}t|\leq|x|+(1+\delta)w^{\alpha}t$, which
implies that $\frac{1}{2}w^{\alpha}\leq|x|$. Thus
$|x|-(1+\delta)w^{\alpha}t\geq|x|-\frac{w^{\alpha}}{4}\geq\frac{|x|}{2}>0$ and
$(|x|-(1+\delta)w^{\alpha}t)^{m}\geq\frac{|x|^{m}}{2^{m}}$. Since
$|x|-(1+\delta)w^{\alpha}t>0$ and remembering that $m$ is even, we also have
that $|x-(1+\delta)w^{\alpha}t|^{m}\geq(|x|-(1+\delta)w^{\alpha}t)^{m}$ and
thus $|x-(1+\delta)w^{\alpha}t|^{m}\geq 2^{-m}|x|^{m}$. Additionally , since
$t$ is sufficiently small and $\frac{1}{2}w^{\alpha}\leq|x|$, it holds that
$|x-(1-\delta)w^{\alpha}t|\leq|x|+w^{\alpha}t\leq 2|x|$.

  2. ii)

$|x-(1+\delta)w^{\alpha}t|\leq w^{\alpha}$. This case can be treated as in the
proof of (2.62).

Case $2.$ From (2.19), it follows that $X(x,v,t)\geq 0$ and $X(x,w,t)\geq 0$.
Thus, from (2.26) and (2.66), the conclusion follows.

Proof of (2.59): Notice that due to (2.49) and since
$v\in(v_{\textup{max}}(x,t),2v_{\textup{max}}(x,t))$, we have that
$x-(1+2\delta)v^{\alpha}t\geq 0$ if we choose $t$ to be sufficiently small.
Moreover, since $\delta<\frac{1}{2}$, $t$ can be chosen independently of
$\delta$. Since $v>v_{\textup{max}}(x,t)$, it holds by (2.66) that
$\frac{1}{C(1+|x|^{m}+v^{p})}\leq H_{L}(x,v,t)$. Thus, since
$v\in(v_{\textup{max}}(x,t),2v_{\textup{max}}(x,t))$, we further have that
$\frac{1}{C(1+|x|^{m}+v_{\textup{max}}(x,t)^{p})}\leq H_{L}(x,v,t)$. From
(2.45) and (2.26), it suffices to prove that there exists a constant $C>1$,
independent of $\delta,L,$ and $R$, such that

$\displaystyle|x|^{m}+v_{\textup{max}}(x,t)^{p}\leq
C\bigg{(}|x-(1+\delta)v_{\textup{max}}(x,t)^{\alpha}t|^{m}+v_{\textup{max}}(x,t)^{p}\bigg{)}.$

The inequality holds since due to (2.49) we can choose $t$ sufficiently small
such that
$|x-(1+\delta)v_{\textup{max}}(x,t)^{\alpha}t|=x-(1+\delta)v_{\textup{max}}(x,t)^{\alpha}t\geq\frac{x}{2}$.

Proof of (2.60): Due to (2.45) and since $v-v^{\prime}\in(\frac{v}{2},v)$ it
holds on one side that

$\displaystyle
H_{L}(x,v-v^{\prime},t)\leq\frac{C}{1+(v-v^{\prime})^{p}}\leq\frac{C}{1+v^{p}},$

for some $C>0$, independent of $\delta,L,$ and $R$. On the other side,
similarly with (2.27) and from (2.45), we have that

$\displaystyle
H_{L}(x,v,t)\geq\frac{C}{1+v^{p}t^{m}+v^{p}}\geq\frac{C}{1+v^{p}},$

for some $C>0$, independent of $\delta,L,$ and $R$. Combining the two
inequalities, we can conclude that (2.60) holds.

Proof of (2.56):

Case $1.$ $a)$ We will prove that in this case it holds that

$\displaystyle\frac{1}{C}\frac{v^{p-1}}{(1+|x|^{m}+v^{p})^{2}}\leq-\partial_{v}H_{L}(x,w,t)\leq
C\frac{v^{p-1}}{(1+|x|^{m}+v^{p})^{2}},$ (2.67)

for some $C>0$, independent of $\delta,L,$ and $R$, and for all
$w\in[\frac{v}{2},v]$. We remember we are in the case when
$x-(1+\delta)w^{\alpha}t\leq x-(1-\delta)w^{\alpha}t\leq 0$ and thus due to
(2.19) it holds that
$|X|^{m-2}(x,w,t)X(x,w,t)\partial_{v}{X}(x,w,t)(x,w,t)=|X|^{m-1}|\partial_{v}X(x,w,t)|$.
Due to (2.18)-(2.21) we thus have in this region that

$\displaystyle-\frac{1}{C}(x-(1-\delta)w^{\alpha}t)^{m-1}w^{\alpha-1}t+\frac{1}{C}(1-\delta)w^{p-1}$
$\displaystyle\leq|X|^{m-2}X\partial_{v}{X}(x,w,t)+{V}^{p-1}\partial_{v}{V}(x,w,t)$
$\displaystyle\leq-C(x-(1+\delta)w^{\alpha}t)^{m-1}w^{\alpha-1}t+C(1+\delta)w^{p-1}.$

Moreover, from (2.62), we have that

$\displaystyle|x-(1+\delta)w^{\alpha}t|^{m}+w^{p}\leq
C\big{(}|x|^{m}+v^{p}\big{)}\leq
C\bigg{(}|x-(1-\delta)w^{\alpha}t|^{m}+w^{p}\bigg{)}.$ (2.68)

In order to prove (2.67), due to (2.46), it then suffices to show in this case
that

$\displaystyle|x-(1+\delta)w^{\alpha}t|^{m-1}w^{\alpha-1}t+w^{p-1}\leq
Cv^{p-1}\leq
C\bigg{(}|x-(1-\delta)w^{\alpha}t|^{m-1}w^{\alpha-1}t+w^{p-1}\bigg{)},$ (2.69)

for all $w\in[\frac{v}{2},v]$. We thus prove (2.69). We have

$\displaystyle
w^{\alpha-1}t|x-(1+\delta)w^{\alpha}t|^{m-1}+w^{p-1}=w^{\alpha-1}\bigg{(}|x-(1+\delta)w^{\alpha}t|^{m-1}t+w^{\alpha(m-1)}\bigg{)}.$
(2.70)

Since $w\in[\frac{v}{2},v]$, it suffices to prove

$\displaystyle\tilde{J}:=|x-(1+\delta)w^{\alpha}t|^{m-1}t+w^{\alpha(m-1)}\leq
Cw^{\alpha(m-1)}$ (2.71)

and

$\displaystyle|x-(1-\delta)w^{\alpha}t|^{m-1}t+w^{\alpha(m-1)}\geq\frac{1}{C}w^{\alpha(m-1)}.$

We are in the case $v\geq C_{l}v_{\textup{max}}(x,t)$, for some sufficiently
large $C_{l}>0$ and $x>0$. We know that
$v_{\textup{max}}(x,t)^{\alpha}\geq\frac{1}{K_{max}}xt^{\frac{1}{m-1}}$, for
$K_{max}$ as in (2.49), and thus
$w^{\alpha}\geq\frac{1}{C}xt^{\frac{1}{m-1}}\geq 0$ since $w\geq\frac{v}{2}$.

We have that $|x-(1-\delta)w^{\alpha}t|^{m-1}t+w^{\alpha(m-1)}\geq
w^{\alpha(m-1)}$. On the other hand, since $0\leq xt^{\frac{1}{m-1}}\leq
Cw^{\alpha}$ and $t\leq 1$, it follows that

$\displaystyle\tilde{J}(x,w,t)\leq
C_{m}(|x|^{m-1}t+w^{\alpha(m-1)}t^{m})+w^{\alpha(m-1)}\leq Cw^{\alpha(m-1)}.$

######  Remark 2.8.

Since $v^{\alpha}\geq\frac{x}{(1-2\delta)t}$, we know from (2.67) that
$\frac{1}{C}\frac{v^{p-1}}{(1+|x|^{m}+v^{p})^{2}}\leq-\partial_{v}H_{L}(x,v,t).$
Thus, for Case $1.$ $b)$ and Case $1.$ $c)$ we need to prove that
$-\partial_{v}H_{L}(x,w,t)\leq\frac{Cv^{p-1}}{(1+|x|^{m}+v^{p})^{2}}$, for
some $C>0$, which is independent of $\delta,L,$ and $R$.

Case $1.$ $b)$ As mentioned above, we need to prove that
$-\partial_{v}H_{L}(x,w,t)\leq\frac{Cv^{p-1}}{(1+|x|^{m}+v^{p})^{2}}$ when
$w^{\alpha}\in\bigg{[}\frac{x}{(1+2\delta)t},\frac{x}{(1-2\delta)t}\bigg{]}$,
$w\in\big{[}\frac{v}{2},v\big{]}$. More precisely, due to (2.46), it suffices
to prove that

$\displaystyle 1+|X(x,w,t)|^{m}+V(x,w,t)^{p}\geq C(1+|x|^{m}+v^{p})$

and that

$\displaystyle
X(x,w,t)^{m-1}\partial_{v}X(x,w,t)+V(x,w,t)^{p-1}\partial_{v}V(x,w,t)\leq
Cv^{p-1},$

for all $w\in[\frac{v}{2},v]$ such that
$w^{\alpha}\in\big{[}\frac{x}{(1+2\delta)t},\frac{x}{(1-2\delta)t}\big{]}$.

We know from (2.65) that the first inequality holds and thus we focus on
proving the second inequality. Due to (2.18) and (2.23), it follows that

$\displaystyle
X(x,w,t)^{m-1}\partial_{v}X(x,w,t)+V(x,w,t)^{p-1}\partial_{v}V(x,w,t)\leq
X(x,w,t)^{m-1}\partial_{v}X(x,w,t)+Cw^{p-1},$

where $C>0$ is independent of $\delta,L,$ and $R$. We now analyze the term
$X(x,w,t)^{m-1}\partial_{v}X(x,w,t)$. From (2.27) and (2.20), it holds that

$\displaystyle
X(x,w,t)^{m-1}\partial_{v}X(x,w,t)\leq|X(x,w,t)|^{m-1}|\partial_{v}X(x,w,t)|\leq
Cw^{\alpha(m-1)}w^{\alpha-1}t^{m}\leq Cw^{p-1},$

for some $C>0$, which is independent of $\delta,L,$ and $R$, and we can
conclude by using that $w\in\big{[}\frac{v}{2},v\big{]}$.

Case $1.$ $c)$ As before, by Remark 2.8, we only need to prove that
$1+|X(x,w,t)|^{m}+V(x,w,t)^{p}\geq C(1+|x|^{m}+v^{p})$ and that
$X(x,w,t)^{m-1}\partial_{v}X(x,w,t)+V(x,w,t)^{p-1}\partial_{v}V(x,w,t)\leq
Cv^{p-1}$ when $w^{\alpha}\leq\frac{x}{(1+2\delta)t}$,
$w\in\big{[}\frac{v}{2},v\big{]}$.

By (2.19) and (2.20) it holds that $X(x,w,t)\geq 0$ and
$\partial_{v}X(x,w,t)\leq 0$ in this region. Moreover, due to (2.18) and
(2.21), it follows that

$\displaystyle
X(x,w,t)^{m-1}\partial_{v}X(x,w,t)+V(x,w,t)^{p-1}\partial_{v}V(x,w,t)\leq
V(x,w,t)^{p-1}\partial_{v}V(x,w,t)\leq Cw^{p-1}.$

The fact that $1+|X(x,w,t)|^{m}+V(x,w,t)^{p}\geq C(1+|x|^{m}+v^{p})$ in this
region follows from (2.66).

Case $2.$ We will prove that (2.67) holds in this case too. Due to (2.19), we
have that $X(x,v,t),$ $X(x,w,t)\geq 0$ in this region. Thus, from (2.26), it
holds that
$|x-(1+\delta)w^{\alpha}t|\leq|X(x,w,t)|\leq|x-(1-\delta)w^{\alpha}t|.$ We
know from (2.66) that $|x-(1-\delta)w^{\alpha}t|^{m}+w^{p}\leq
C\big{(}|x|^{m}+v^{p}\big{)}\leq
C\bigg{(}|x-(1+\delta)w^{\alpha}t|^{m}+w^{p}\bigg{)}$, for some constant
$C>0$, which is independent of $\delta,L,$ and $R$. Moreover, since
$X(x,w,t)\geq 0$ and $\partial_{w}X(x,w,t)\leq 0$, we have that
$X(x,w,t)\partial_{w}X(x,w,t)=-|X(x,w,t)||\partial_{w}X(x,w,t)|.$ Thus, due to
(2.46), it suffices to show in this case that there exists $C>0$, independent
of $\delta,L,$ and $R$ such that

$\displaystyle-|x-(1+\delta)w^{\alpha}t|^{m-1}w^{\alpha-1}t+w^{p-1}\leq
Cv^{p-1}\leq
C\bigg{(}-|x-(1-\delta)w^{\alpha}t|^{m-1}w^{\alpha-1}t+w^{p-1}\bigg{)},$
(2.72)

for all $w\in[\frac{v}{2},v]$.

We remember we are in the case when $x-(1+\delta)w^{\alpha}t\geq 0$. It is
clear that $-|x-(1+\delta)w^{\alpha}t|^{m-1}w^{\alpha-1}t+w^{p-1}\leq
Cv^{p-1}$. For the other inequality, due to (2.70) it suffices to prove the
statement for $-|x-(1-\delta)w^{\alpha}t|^{m-1}t+w^{\alpha(m-1)}$. Since
$x-(1-\delta)w^{\alpha}t\geq 0$ and using that
$a^{m-1}+b^{m-1}\leq(a+b)^{m-1}$, for $a,b\geq 0$, we have that
$x^{m-1}-((1-\delta)w^{\alpha}t)^{m-1}\geq(x-(1-\delta)w^{\alpha}t)^{m-1}$.
Thus, it holds that

$\displaystyle-|x-(1-\delta)w^{\alpha}t|^{m-1}t+w^{\alpha(m-1)}\geq-|x|^{m-1}t+(1-\delta)^{m-1}w^{\alpha(m-1)}t^{m}+w^{\alpha(m-1)}.$
(2.73)

Since $w\geq\frac{v}{2}\geq\frac{C_{l}}{2}v_{\textup{max}}(x,t)$ in this case
we have that
$xt^{\frac{1}{m-1}}\leq\frac{2^{\alpha}K_{max}w^{\alpha}}{C_{l}^{\alpha}}$,
for a sufficiently large constant $C_{l}>0$. (2.73) thus becomes

$\displaystyle-|x-(1-\delta)w^{\alpha}t|^{m-1}t+w^{\alpha(m-1)}\geq-\frac{2^{p-\alpha}K_{max}^{m-1}w^{\alpha(m-1)}}{C_{l}^{p-\alpha}}+w^{\alpha(m-1)}\geq\frac{w^{\alpha(m-1)}}{2},$

for $C_{l}$ sufficiently large, thus concluding our proof.

Proof of (2.57): In order to prove (2.57), it is useful to notice that if
$x\leq 0$, then (2.24) holds. We will prove that there exists $C>0$, which is
independent of $\delta,L,$ and $R$, such that

$\displaystyle\frac{1}{C}\frac{v^{\alpha-1}|x|^{m-1}t+v^{p-1}}{(1+|x|^{m}+v^{p})^{2}}\leq-\partial_{v}H_{L}(x,w,t)\leq
C\frac{v^{\alpha-1}|x|^{m-1}t+v^{p-1}}{(1+|x|^{m}+v^{p})^{2}},$ (2.74)

for all $w\in[\frac{v}{2},v]$. Using similar computations as the ones for
(2.66), we can prove that

$\displaystyle(|x|+(1+\delta)w^{\alpha}t)^{m}+w^{p}\leq
C\big{(}|x|^{m}+v^{p}\big{)}\leq
C\bigg{(}(|x|+(1-\delta)w^{\alpha}t)^{m}+w^{p}\bigg{)}.$ (2.75)

Since $X(x,w,t)\leq 0$ and $\partial_{w}X(x,w,t)\leq 0$ we have that
$X(x,w,t)\partial_{w}X(x,w,t)=|X(x,w,t)|$ $|\partial_{w}X(x,w,t)|$. Due to
(2.24) and (2.46), what is left to prove in order for (2.74) to hold is that

$\displaystyle(|x|+(1+\delta)w^{\alpha}t)^{m-1}w^{\alpha-1}t+w^{p-1}$
$\displaystyle\leq C(v^{\alpha-1}|x|^{m-1}t+v^{p-1})$ $\displaystyle\leq
C\bigg{(}(|x|+(1-\delta)w^{\alpha}t)^{m-1}w^{\alpha-1}t+w^{p-1}\bigg{)},$
(2.76)

for some $C>0$, independent of $\delta,L,$ and $R$, for all
$w\in[\frac{v}{2},v]$.

Thus, we only need to prove (2.3). For $x\leq 0$ and $w\in[\frac{v}{2},v]$, we
have that
$(|x|+(1-\delta)w^{\alpha}t)^{m-1}t+w^{\alpha(m-1)}\geq|x|^{m-1}t+w^{\alpha(m-1)}$.
Furthermore, since $t\leq 1$, it follows that

$\displaystyle(|x|+(1+\delta)w^{\alpha}t)^{m-1}t+w^{\alpha(m-1)}$
$\displaystyle\leq C_{m}|x|^{m-1}t+C_{m}w^{\alpha(m-1)}t^{m}+w^{\alpha(m-1)}$
$\displaystyle\leq C(|x|^{m-1}t+w^{\alpha(m-1)}).$

∎

We also prove some moment bounds for the function $H_{L}$, which are
independent of $\delta,L,$ and $R$.

######  Lemma 2.9 (Moment estimates).

Let $T>0$ be sufficiently small, independent of $\delta,L$, and $R$. Then
there exists $K_{3}>0$, which is independent of
$\delta\in\big{(}0,\frac{1}{2}\big{)}$, $L$, and $R$, such that for all
$t\in[0,T]$ we have

$\displaystyle M_{1,L}(x,t):=\int_{(0,\infty)}vH_{L}(x,v,t)\textup{d}v$
$\displaystyle\leq\frac{K_{3}C_{0}}{1+|x|^{m-d}}\qquad\textup{ for
}x\in\mathbb{R}\,,t\geq 0.$ (2.77)

In general, if $p>\operatorname{max}\\{2\gamma+1,2\\}$, we have that

$\displaystyle M_{n,L}(x,t):=\int_{(0,\infty)}v^{n}H_{L}(x,v,t)\textup{d}v$
$\displaystyle\leq\frac{K_{3}C_{0}}{1+|x|^{m-\frac{n+1}{\alpha}}}\qquad\textup{
for }x\in\mathbb{R}\,,t\geq 0\,,$ (2.78)

for $n\in[0,\operatorname{max}\\{2\gamma,1\\}]$.

###### Proof.

We first recall that due to (2.18) we have
$G_{L}(x,v,t)\leq\frac{C}{1+|X(x,v,t)|^{m}+v^{p}}$. We split the integral for
$M_{n,L}$ as follows:

$M_{n,L}(x,t)=\int_{0}^{v_{\operatorname{max}}(x,t)}v^{n}H_{L}(x,v,t)\,\textup{d}v+\int_{v_{\operatorname{max}}(x,t)}^{\infty}v^{n}H_{L}(x,v,t)\,\textup{d}v=:M_{n,1}(x,t)+M_{n,2}(x,t)\,,$

where $v_{\operatorname{max}}(x,t)=0$ if $x\leq 0$. From (2.18) and since
$\delta\in(0,\frac{1}{2})$, we have that $\frac{v}{2}\leq
V(x,v,t)\leq\frac{3v}{2}$. Due to (2.19) and since $\delta\in(0,\frac{1}{2})$,
it holds that $X(x,v,t)\geq x-(1+\delta)v^{\alpha}t\geq x-2v^{\alpha}t$. Then,
by (2.49), we further deduce that

$\displaystyle X(x,v_{\textup{max}}(x,t),t)\geq
x-2v_{\textup{max}}(x,t)^{\alpha}t\geq
x-2K_{max}xt^{\frac{m}{m-1}}\geq\frac{x}{2}\geq 0,$ (2.79)

for all $t\leq T$ if $T$ is sufficiently small. Thus, by Definition 2.6 and
(2.45), we obtain that

$\displaystyle M_{n,1}(x,t)=$
$\displaystyle\int_{0}^{v_{\operatorname{max}}}v^{n}H_{L}(x,v,t)\textup{d}v=\int_{0}^{v_{\operatorname{max}}}v^{n}G_{L}(x,v_{\textup{max}}(x,t),t)\textup{d}v$
$\displaystyle\leq\frac{Cv_{\operatorname{max}}^{n+1}}{1+|x|^{m}+v_{\operatorname{max}}^{p}}\leq\frac{C}{1+|x|^{m-\frac{n+1}{\alpha}}},$

for some constant $C>0$ which is independent of $\delta,L,$ and $R.$

To estimate $M_{n,2}$ we note that for all $x\in{\varmathbb R}$ we have that

$\displaystyle M_{n,2}(x,t)\leq
C\int_{0}^{\infty}\frac{v^{n}}{1+v^{p}}\,\textup{d}v\leq C$

since $p-n>1$ by assumption.

To obtain a decay for large $|x|$ we consider first $x>1$ and use
(2.25)-(2.26) to obtain

$\displaystyle M_{n,2}(x,t)$ $\displaystyle\leq
C\bigg{(}\int_{0}^{\big{(}\frac{x}{(1+2\delta)t}\big{)}^{\frac{1}{\alpha}}}\frac{v^{n}}{1+(x-(1+\delta)v^{\alpha}t)^{m}+v^{p}}\textup{d}v+\int_{\big{(}\frac{x}{(1+2\delta)t}\big{)}^{\frac{1}{\alpha}}}^{\infty}v^{n-p}\,\textup{d}v\bigg{)}.$

Then, since $\delta<\frac{1}{2}$, we have that

$\displaystyle\int_{\big{(}\frac{x}{(1+2\delta)t}\big{)}^{\frac{1}{\alpha}}}^{\infty}v^{n-p}\,\textup{d}v\leq\frac{1}{p-n-1}\Big{(}\frac{(1+2\delta)t}{x}\Big{)}^{m-\frac{n+1}{\alpha}}\leq\frac{2^{m-\frac{n+1}{\alpha}}}{p-n-1}\Big{(}\frac{t}{x}\Big{)}^{m-\frac{n+1}{\alpha}}.$

For the other term, by using the change of variables
$v=\big{(}\frac{x}{t}\big{)}^{\frac{1}{\alpha}}\xi$, we find

$\displaystyle\int_{0}^{\big{(}\frac{x}{(1+2\delta)t}\big{)}^{\frac{1}{\alpha}}}\frac{v^{n}}{1+(x-(1+\delta)v^{\alpha}t)^{m}+v^{p}}\textup{d}v$
$\displaystyle\leq\big{(}\frac{x}{t}\big{)}^{\frac{1}{\alpha}}\int_{(0,\infty)}\frac{\big{(}\frac{x}{t}\big{)}^{\frac{n}{\alpha}}\xi^{n}}{1+x^{m}|1-(1+\delta)\xi^{\alpha}|^{m}+(\frac{x}{t})^{\frac{p}{\alpha}}\xi^{p}}\textup{d}\xi$
$\displaystyle\leq\frac{1}{x^{m-\frac{1}{\alpha}}}\frac{1}{t^{\frac{1}{\alpha}}}\int_{(0,\infty)}\frac{\big{(}\frac{x}{t}\big{)}^{\frac{n}{\alpha}}\xi^{n}}{|1-(1+\delta)\xi^{\alpha}|^{m}+(\frac{1}{t})^{m}\xi^{p}}\textup{d}\xi.$

Since $t\leq 1$ and $m>1$ is even we have that the following holds

$\displaystyle
3^{m+1}[|1-(1+\delta)\xi^{\alpha}|^{m}+\frac{1}{t^{m}}\xi^{p}]\geq
1+\frac{1}{t^{m}}\xi^{p},$ (2.80)

for $\xi\geq 0$. Indeed, if $|1-(1+\delta)\xi^{\alpha}|\geq\frac{1}{2}$, then
(2.80) follows. Otherwise, if $|1-(1+\delta)\xi^{\alpha}|\leq\frac{1}{2}$,
then $\frac{1}{2}\leq(1+\delta)\xi^{\alpha}\leq\frac{3}{2}$ and since
$\delta<\frac{1}{2}$ it holds that $\xi^{\alpha}\geq\frac{1}{3}$. Thus, if we
use in addition that $t\leq 1$, we have that

$\displaystyle|1-(1+\delta)\xi^{\alpha}|^{m}+\frac{1}{t^{m}}\xi^{p}\geq\frac{1}{2t^{m}}\xi^{p}+\frac{1}{2t^{m}}\xi^{p}\geq\frac{1}{3^{m+1}}+\frac{1}{2t^{m}}\xi^{p}$

and thus (2.80) holds. It then follows that

$\displaystyle\frac{1}{x^{m-\frac{1}{\alpha}}t^{\frac{1}{\alpha}}}\int_{(0,\infty)}\frac{\big{(}\frac{x}{t}\big{)}^{\frac{n}{\alpha}}\xi^{n}}{|1-\xi^{\alpha}|^{m}+(\frac{1}{t})^{m}\xi^{p}}\textup{d}\xi$
$\displaystyle\leq\frac{C}{x^{m-\frac{1}{\alpha}}t^{\frac{1}{\alpha}}}\int_{(0,\infty)}\frac{\big{(}\frac{x}{t}\big{)}^{\frac{n}{\alpha}}\xi^{n}}{1+(\frac{1}{t})^{m}\xi^{p}}\textup{d}\xi$
$\displaystyle\leq\frac{C}{x^{m-\frac{n+1}{\alpha}}}\int_{(0,\infty)}\frac{\eta^{n}}{1+\eta^{p}}\textup{d}\eta\leq\frac{C}{x^{m-\frac{n+1}{\alpha}}},$

for some constant $C>0$ which is independent of
$\delta\in\big{(}0,\frac{1}{2}),L,$ and $R$.

In the case $x\leq 0$ we can use (2.24) to obtain the estimate similarly as
above without the need to split the integral. Estimate (2.77) follows from
(2.78) by choosing $n=1$ and the fact that $d$ in (2.15) satisfies
$d>\frac{2}{\alpha}$. ∎

######  Remark 2.10.

Let $T>0$ be sufficiently small. With similar computations as the ones used in
Lemma 2.9, we can prove that

$\displaystyle\int_{(0,\infty)}\frac{v^{n}}{1+|x|^{m}+v^{p}}\textup{d}v$
$\displaystyle\leq\frac{K_{3}}{1+|x|^{m-\frac{n+1}{\alpha}}},$ (2.81)

for all $x\in\mathbb{R},$ $t\in[0,T]$, and for
$n\in[0,\operatorname{max}\\{2\gamma,1\\}]$.

We now define the function for which we will prove is a supersolution for the
problem (2.2).

######  Definition 2.11.

Let $\delta\in\big{(}0,\frac{1}{4}\big{]}$ and $L=4K_{1}K_{2}K_{3}C_{0}$,
where $C_{0}$ is as in (2.45), $K_{1}$ is as in (1.8), $K_{2}$ is as in Lemma
2.7, and $K_{3}$ is as in Lemma 2.9. Denote by $H(x,v,t):=H_{L}(x,v,t)$, where
$H_{L}$ is as in Definition 2.6. Moreover, let $B_{t}$ to be

$\displaystyle B_{t}:=e^{\lambda t+\lambda
t^{\frac{\alpha+1-\gamma}{\alpha}}}$ (2.82)

for some $\lambda>0$. Then we define the function $G$ via

$\displaystyle G(x,v,t)=B_{t}H(x,v,t).$ (2.83)

As mentioned before, with this construction $G$ is decreasing in $v$ for fixed
$x$ outside possibly a critical region where
$x\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$. In the following we
will have to deal with this region separately.

Our key result is the following.

######  Proposition 2.12.

Let $T>0$ be sufficiently small and $\delta\in\big{(}0,\frac{1}{4}\big{]}$.
There exists a sufficiently large $\lambda>0$, which depends only on $C_{0}$
and the parameters $m,\gamma,\alpha$ such that if $f_{n}\leq G$, where the
sequence $\\{f_{n}\\}_{n\in\mathbb{N}}$ was defined in (2.2), then
$f_{n+1}\leq G$, for all $n\in{\varmathbb N}$ and all $t\in[0,T]$.

######  Remark 2.13.

Since the constants will play an important role in our proof, it is worthwhile
to mention for clarity which are the constants that the parameters in
Proposition 2.12 depend on. Let $C_{0}$ be as in (2.45), $K_{1}$ as in (1.8),
$K_{2}$ as in Lemma 2.7, and $K_{3}$ be as in Lemma 2.9. Notice that these
constants do not depend on $\delta$, $L$ or $R$ from Proposition 2.2. In order
to prove that $G$ is a supersolution, we take $L=4K_{1}K_{2}K_{3}C_{0}$. We
then take $\lambda$ to be sufficiently large depending on $C_{0},K_{1},K_{2}$,
and $K_{3}$. We then take $T$ to be such that
$\operatorname{max}\\{T,T^{\frac{\alpha+1-\gamma}{\alpha}}\\}\leq\frac{\ln(2)}{2\lambda},$
which implies that $B_{t}\leq 2$, for all $t\in[0,T]$, where $B_{t}$ was
defined in (2.82).

Before we begin with the proof of Proposition 2.12, it is worthwhile to notice
that we have some moment bounds for the function $G$ in Definition 2.11 as a
direct consequence of Lemma 2.9.

######  Lemma 2.14.

Let $T>0$ be sufficiently small with $K_{3}$ as in Lemma 2.9. Then for all
$t\in[0,T]$ we have

$\displaystyle M_{1}(x,t):=\int_{(0,\infty)}vG(x,v,t)\textup{d}v$
$\displaystyle\leq\frac{K_{3}C_{0}B_{t}}{1+|x|^{m-d}}\qquad\textup{ for
}x\in\mathbb{R},$ (2.84)

where $B_{t}$ was defined in (2.82). In general, if
$p>\operatorname{max}\\{2\gamma+1,2\\}$, we have that

$\displaystyle M_{n}(x,t):=\int_{(0,\infty)}v^{n}G(x,v,t)\textup{d}v$
$\displaystyle\leq\frac{K_{3}C_{0}B_{t}}{1+|x|^{m-\frac{n+1}{\alpha}}}\qquad\textup{
for }x\in\mathbb{R}\,,t\geq 0\,,$ (2.85)

for $n\in[0,\operatorname{max}\\{2\gamma,1\\}]$.

We now focus on proving Proposition 2.12.

###### Proof of Proposition 2.12.

We now prove that $G$ is a supersolution of the problem (2.2), that is we show

$\displaystyle\partial_{t}G(x,v,t)+v^{\alpha}\partial_{x}G(x,v,t)-\int_{0}^{\frac{v}{2}}K(v-v^{\prime},v^{\prime})G(x,v-v^{\prime},t)f_{n}(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle+\int_{0}^{\infty}K(v,v^{\prime})G(x,v,t)f_{n}(x,v^{\prime},t)\textup{d}v^{\prime}\geq
0.$ (2.86)

We prove (2.3) by showing first that

$\displaystyle\partial_{t}G(x,v,t)+v^{\alpha}\partial_{x}G(x,v,t)+\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}G(x,v,t)\geq
0,$ (2.87)

for any ${L}>0$ and where $d$ was defined in (2.15), and then that

$\displaystyle\partial_{t}G(x,v,t)+v^{\alpha}\partial_{x}G(x,v,t)-\int_{0}^{\frac{v}{2}}K(v-v^{\prime},v^{\prime})G(x,v-v^{\prime},t)f_{n}(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle+\int_{0}^{\infty}K(v,v^{\prime})G(x,v,t)f_{n}(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\geq\partial_{t}G(x,v,t)+v^{\alpha}\partial_{x}G(x,v,t)+\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}G(x,v,t),$
(2.88)

for $L>0$ as in Definition 2.11.

We now prove (2.87).

Assume $x\not\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$ and
$\partial_{v}G_{L}\leq 0$ or
$x\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$. Then $G=B_{t}G_{L}$ and
thus, using (2.16), it holds with $c_{\alpha}:=\frac{\alpha+1-\gamma}{\alpha}$
that

$\displaystyle\partial_{t}G(x,v,t)+v^{\alpha}\partial_{x}G(x,v,t)+\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}G(x,v,t)$
$\displaystyle=B_{t}\bigg{(}\partial_{t}G_{L}(x,v,t)+v^{\alpha}\partial_{x}G_{L}(x,v,t)+\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}G_{L}(x,v,t)\bigg{)}$
$\displaystyle\qquad+\lambda(1+c_{\alpha}t^{\frac{\alpha+1-\gamma}{\alpha}-1})B_{t}G_{L}\geq\lambda\textup{e}^{\lambda
t}G_{L}(x,v,t)\geq 0.$

Notice that we did not use the contribution of the term
$t^{\frac{\alpha+1-\gamma}{\alpha}}$ in the computations. This term will be
needed later in the proof.

Assume now $x\not\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$ and
$\partial_{v}G_{L}>0$ such that we have
$G(x,v,t)=B_{t}G_{L}(x,v_{\operatorname{max}}(x,t),t)$. Then

$\displaystyle\partial_{t}G(x,v,t)+v^{\alpha}\partial_{x}G(x,v,t)+\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}G(x,v,t)$
$\displaystyle=B_{t}\bigg{(}\partial_{t}G_{L}(x,v_{\textup{max}}(x,t),t)+v^{\alpha}\partial_{x}G_{L}(x,v_{\textup{max}}(x,t),t)\bigg{)}$
(2.89)
$\displaystyle\qquad+\lambda(1+c_{\alpha}t^{\frac{\alpha+1-\gamma}{\alpha}-1})B_{t}G_{L}(x,v_{\textup{max}}(x,t),t)$
$\displaystyle\geq
B_{t}\bigg{(}\partial_{t}G_{L}(x,v_{\textup{max}}(x,t),t)+v^{\alpha}\partial_{x}G_{L}(x,v_{\textup{max}}(x,t),t)\bigg{)}.$
(2.90)

By the choice of $v_{\textup{max}}(x,t)$ we have that $v\leq
v_{\textup{max}}(x,t)$ in the region where $\partial_{v}G_{L}(x,v,t)>0$.
Moreover, from Proposition 2.5, we know that
$\partial_{x}G_{L}(x,v_{\textup{max}}(x,t),t)\leq 0$. Thus, from (2.3) and
(2.16), we further obtain that

$\displaystyle\partial_{t}G(x,v,t)+v^{\alpha}\partial_{x}G(x,v,t)+\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}G(x,v,t)$
$\displaystyle\geq
B_{t}\bigg{(}\partial_{t}G_{L}(x,v_{\textup{max}}(x,t),t)+v_{\textup{max}}^{\alpha}\partial_{x}G_{L}(x,v_{\textup{max}}(x,t),t)\bigg{)}=0$

and (2.87) follows.

We are thus left to prove that (2.3) holds. We analyze the cases when
$x\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$ and
$x\not\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$ separately.

Proof of (2.3) for $x\not\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$.

We have that

$\displaystyle-\int_{0}^{\frac{v}{2}}$ $\displaystyle
K(v-v^{\prime},v^{\prime})G(x,v-v^{\prime},t)f_{n}(x,v^{\prime},t)\textup{d}v^{\prime}+\int_{0}^{\infty}K(v,v^{\prime})G(x,v,t)f_{n}(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\geq-\int_{0}^{\frac{v}{2}}K(v,v^{\prime})[G(x,v-v^{\prime},t)-G(x,v,t)]f_{n}(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\qquad\qquad+\int_{0}^{\frac{v}{2}}[K(v,v^{\prime})-K(v-v^{\prime},v^{\prime})]G(x,v-v^{\prime},t)f_{n}(x,v^{\prime},t)\textup{d}v^{\prime}.$
(2.91)

Since $G,f_{n}\geq 0$ and $K(v-v^{\prime},v^{\prime})\leq K(v,v^{\prime})$
when $v^{\prime}\in[0,\frac{v}{2})$ by (1.9), it holds that

$\displaystyle\int_{0}^{\frac{v}{2}}[K(v,v^{\prime})-K(v-v^{\prime},v^{\prime})]G(x,v-v^{\prime},t)f_{n}(x,v^{\prime},t)\textup{d}v^{\prime}\geq
0.$ (2.92)

We know that $K(v,v^{\prime})\leq K_{1}(v^{\gamma}+v^{\prime\gamma})$, where
$K_{1}$ is as in (1.8). Thus, when $v^{\prime}\leq\frac{v}{2}$, it holds that
$K(v,v^{\prime})\leq 2K_{1}v^{\gamma}$. Without loss of generality we assume
in the following that $K(v,v^{\prime})\leq v^{\gamma}$ when
$v^{\prime}\leq\frac{v}{2}$. This is in order to simplify the notation but we
allow $L$ in Definition 2.11 to depend on $K_{1}$. Additionally, by (2.83), it
holds that $\partial_{v}G\leq 0$ and thus $G(x,v-v^{\prime},t)-G(x,v,t)\geq
0$, for $v\in(0,\frac{v}{2})$. Since $f_{n}\leq G$ and with (2.92) we deduce
that

$\displaystyle-$
$\displaystyle\int_{0}^{\frac{v}{2}}v^{\gamma}[G(x,v-v^{\prime},t)-G(x,v,t)]f_{n}(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\geq-v^{\gamma}\int_{0}^{\frac{v}{2}}\big{(}G(x,v-v^{\prime},t)-G(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v^{\prime}.$

We use the following notation

$\displaystyle
I_{1}:=v^{\gamma}\int_{0}^{\frac{v}{2}}\big{(}G(x,v-v^{\prime},t)-G(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v^{\prime}=\xi_{R}(v)I_{1}+(1-\xi_{R}(v))I_{1}$

with $\xi_{R}$ as in (2.13). Assume that $H$, as in Definition 2.11, satisfies

$\displaystyle\xi_{R}(v)v^{\gamma}\int_{0}^{\frac{v}{2}}\big{(}H(x,v-v^{\prime},t)-H(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v^{\prime}\leq$
$\displaystyle-\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}H(x,v,t)$
$\displaystyle+CB_{t}H(x,v,t)$ (2.93)

and

$\displaystyle(1-\xi_{R}(v))v^{\gamma}\int_{0}^{\frac{v}{2}}\big{(}H(x,v-v^{\prime},t)-H(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v^{\prime}\leq
CB_{t}H(x,v,t),$ (2.94)

for $L=4K_{1}K_{2}K_{3}C_{0}$ as in Definition 2.11. Then

$\displaystyle\partial_{t}$ $\displaystyle
G+v^{\alpha}\partial_{x}G-v^{\gamma}\int_{0}^{\frac{v}{2}}\big{(}G(x,v-v^{\prime},t)-G(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\geq
B_{t}\Big{(}\partial_{t}H+v^{\alpha}\partial_{x}H-v^{\gamma}\int_{0}^{\frac{v}{2}}\big{(}H(x,v-v^{\prime},t)-H(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v^{\prime}\Big{)}+\lambda
e^{\lambda t+\lambda t^{\frac{\alpha+1-\gamma}{\alpha}}}H$ $\displaystyle\geq
B_{t}\Big{(}\partial_{t}H+v^{\alpha}\partial_{x}H+\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}H(x,v,t)\Big{)}$
$\displaystyle+B_{t}\big{(}-2CB_{t}+\lambda\big{)}H\geq\partial_{t}G+v^{\alpha}\partial_{x}G+\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}G$

if $\lambda\geq 2CB_{t}$, which holds true if $\lambda=4C$ and if
$\operatorname{max}\\{t,t^{\frac{\alpha+1-\gamma}{\alpha}}\\}\leq\frac{\ln(2)}{2\lambda}$.
This proves (2.3).

Proof of (2.94):. We have the following cases

Case $1.$ $a)$ $x>0$ and $v\leq v_{\textup{max}}(x,t)$: In this case
$H(x,v-v^{\prime},t)=H(x,v,t)$, for $v^{\prime}\in(0,\frac{v}{2}]$. Thus,
(2.94) holds.

Case $1.$ $b)$ $x>0$ and $v_{\textup{max}}(x,t)\leq v\leq
2v_{\textup{max}}(x,t)$: The proof of this case is the same as for Case $1.$
$b)$ when proving that (2.3) holds. We thus postpone its proof.

Case $2.$ Either $\\{x>0$ and $2v_{\textup{max}}(x,t)\leq v\\}$ or $\\{x\leq
0\\}$: Since $v\leq 2R$ and $\gamma\geq 0$, it follows, using (2.58) and Lemma
2.14, that

$\displaystyle(1-\xi_{R}(v))I_{1}\textup{e}^{-\lambda t}$
$\displaystyle\leq(2R)^{\gamma}\int_{0}^{\frac{v}{2}}\big{(}H(x,v-v^{\prime},t)-H(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\leq(2R)^{\gamma}\int_{0}^{\frac{v}{2}}H(x,v-v^{\prime},t)G(x,v^{\prime},t)\textup{d}v’$
$\displaystyle\leq
K_{2}(2R)^{\gamma}H(x,v,t)\int_{0}^{\frac{v}{2}}G(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\leq\frac{C_{0}K_{2}K_{3}(2R)^{\gamma}B_{t}H(x,v,t)}{1+|x|^{m-\frac{1}{\alpha}}}\leq
CB_{t}H(x,v,t).$ (2.95)

######  Remark 2.15.

Notice that the constant $C$ in (2.95) depends on $R$ from Proposition 2.2, on
the constant $K_{2}$ from (2.58), and on the constant $K_{3}$ from Lemma 2.14.
However, since $L$ is fixed in Definition 2.11, $R$ depends only on
$K_{1},K_{2},K_{3}$ and $C_{0}$.

Proof of (2.3): We have the following cases

Case $1.$ $a)$: $x>0$ and $v\leq v_{\textup{max}}(x,t)$: Notice that in this
case $H(x,v-v^{\prime},t)=H(x,v,t)$, for $v^{\prime}\in(0,\frac{v}{2}]$, and
$\partial_{v}H(x,v,t)=0$. Thus, (2.3) holds.

Case $1.$ $b)$: $x>0$ and $v_{\textup{max}}(x,t)\leq v\leq
2v_{\textup{max}}(x,t)$: We divide the integral on the left hand side of (2.3)
into the region $v^{\prime}\in(0,v-v_{\text{max}})$ and
$v^{\prime}\in(v-v_{\text{max}},\frac{v}{2})$, respectively. For
$v^{\prime}\in(v-v_{\text{max}},\frac{v}{2})$, using $\frac{v}{2}\leq
v_{\textup{max}}(x,t)$ and Lemma 2.14 with $n=0$, we obtain that

$\displaystyle\int_{v-v_{\textup{max}}(x,t)}^{\frac{v}{2}}\big{(}H(x,v-v^{\prime},t)-H(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\leq
H(x,v_{\textup{max}}(x,t),t)\int_{0}^{v_{\textup{max}}}G(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\leq\frac{C_{0}K_{3}B_{t}H(x,v_{\textup{max}}(x,t),t)}{1+|x|^{m-\frac{1}{\alpha}}}.$
(2.96)

We have $0\leq v^{\alpha}\leq 2v_{\textup{max}}^{\alpha}(x,t)\leq
2K_{max}xt^{\frac{1}{m-1}}$, with $K_{max}$ as in (2.49).

Moreover, using (2.59) it follows that

$\displaystyle
v^{\gamma}\int_{v-v_{\textup{max}}(x,t)}^{\frac{v}{2}}\big{(}H(x,v-v^{\prime},t)-H(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v^{\prime}\leq\frac{2^{\gamma}C_{0}K_{2}K_{3}B_{t}H(x,v,t)v_{\textup{max}}(x,t)^{\gamma}}{1+x^{m-\frac{1}{\alpha}}}.$

Since $v_{\textup{max}}(x,t)^{\alpha}\leq 2K_{max}xt^{\frac{1}{m-1}}$, with
$K_{max}$ as in (2.49), we further obtain that

$\displaystyle\frac{H(x,v,t)v_{\textup{max}}(x,t)^{\gamma}}{1+x^{m-\frac{1}{\alpha}}}\leq\frac{CH(x,v,t)x^{\frac{\gamma}{\alpha}}t^{\frac{\gamma}{\alpha(m-1)}}}{1+x^{m-\frac{1}{\alpha}}}\leq
Ct^{\frac{\gamma}{\alpha(m-1)}}H(x,v,t),$

since $m>\frac{\gamma+1}{\alpha}$. Thus

$\displaystyle
v^{\gamma}\int_{v-v_{\textup{max}}(x,t)}^{\frac{v}{2}}\big{(}H(x,v-v^{\prime},t)-H(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v^{\prime}\leq
CB_{t}t^{\frac{\gamma}{\alpha(m-1)}}H(x,v,t).$ (2.97)

We now estimate the integral

$\displaystyle
J:=\int_{0}^{v-v_{\textup{max}}(x,t)}\big{(}H(x,v-v^{\prime},t)-H(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v^{\prime}.$
(2.98)

As before, using (2.58) and Lemma 2.14, we find

$\displaystyle
J\leq\int_{0}^{v{-}v_{\textup{max}}(x,t)}H(x,v{-}v^{\prime},t)G(x,v^{\prime},t)\textup{d}v^{\prime}\leq$
$\displaystyle\int_{0}^{v-v_{\textup{max}}(x,t)}K_{2}H(x,v,t)G(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\leq\frac{C_{0}K_{2}K_{3}B_{t}H(x,v,t)}{1+x^{m-\frac{1}{\alpha}}}$

and this implies, since $v\leq 2v_{\textup{max}}(x,t)$, that

$\displaystyle
v^{\gamma}\int_{0}^{v{-}v_{\textup{max}}(x,t)}\big{(}H(x,v{-}v^{\prime},t)-H(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v$
$\displaystyle\leq
CB_{t}H(x,v,t)\frac{v_{\textup{max}}(x,t)^{\gamma}}{1+x^{m-\frac{1}{\alpha}}}$
$\displaystyle\leq
CB_{t}H(x,v,t)\frac{x^{\frac{\gamma}{\alpha}}t^{\frac{\gamma}{\alpha(m-1)}}}{1+x^{m-\frac{1}{\alpha}}}.$
(2.99)

Since $m>\frac{\gamma+1}{\alpha}$ it follows that (2.3) holds in this case.

######  Remark 2.16.

Notice that the constant $C$ in (2.97) and (2.99) depends only on the constant
$K_{2}$ from (2.58) and on the constants $C_{0},K_{3}$ from Lemma 2.14.

Case $1.$ $c)$: $x>0$ and $2v_{\operatorname{max}}(x,t)\leq v$ or $x<0$:
Notice that the computations used in Case $1.$ $b)$ hold for any $v\leq
C_{l}v_{\textup{max}}(x,t)$, for some fixed constant $C_{l}>0$. We can thus
assume without loss of generality in this case that
$v>C_{l}v_{\textup{max}}(x,t)$ and that $v\geq R$ because of the presence of
$\xi_{R}(v)$ in (2.3).

We have that $v-v^{\prime}\in(\frac{v}{2},v)$, for
$v^{\prime}\in(0,\frac{v}{2})$. It holds

$\displaystyle\int_{0}^{\frac{v}{2}}\big{(}H(x,v-v^{\prime},t)-H(x,v,t)\big{)}G(x,v^{\prime},t)\textup{d}v^{\prime}=-\int_{0}^{\frac{v}{2}}G(x,v^{\prime},t)\int_{v-v^{\prime}}^{v}\partial_{v}H(x,\tilde{v},t)\textup{d}\tilde{v}\textup{d}v^{\prime}.$

We can then use (2.57), (2.56) and Lemma 2.14 with $n=1$ in order to deduce
that

$\displaystyle-v^{\gamma}\xi_{R}(v)$
$\displaystyle\int_{0}^{\frac{v}{2}}G(x,v^{\prime},t)\int_{v-v^{\prime}}^{v}\partial_{v}H(x,\tilde{v},t)\textup{d}\tilde{v}\textup{d}v^{\prime}$
$\displaystyle\leq-
K_{2}v^{\gamma}\xi_{R}(v)\partial_{v}H(x,v,t)\int_{0}^{\frac{v}{2}}v^{\prime}G(x,v^{\prime},t)\leq-\frac{C_{0}K_{2}K_{3}B_{t}v^{\gamma}\xi_{R}(v)\partial_{v}H(x,v,t)}{1+|x|^{m-d}}$
$\displaystyle\leq-\frac{Lv^{\gamma}\xi_{R}(v)\partial_{v}H(x,v,t)}{1+|x|^{m-d}},$
(2.100)

where in the last inequality we used the definition of $L$ in Definition 2.11
and that $B_{t}\leq 2$ by Remark 2.13. Thus (2.3) holds in this case.

Proof of (2.3) for $x\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$ or,
alternatively, $v^{\alpha}t\in[\frac{x}{1+2\delta},\frac{x}{1-2\delta}].$

We will assume that $v\geq 1$ and $\gamma>1$, since the other cases are
similar but easier to treat.

In order to prove (2.3) we will first show that

$\displaystyle\int_{0}^{\frac{v}{2}}K(v,v^{\prime})f_{n}(x,v^{\prime},t)|H(x,v-v^{\prime},t)-H(x,v,t)|\textup{d}v^{\prime}\leq
CL(1+t^{\frac{\alpha+1-\gamma}{\alpha}-1})H(x,v,t)$ (2.101)

and then that

$\displaystyle\frac{v^{\gamma}}{1+|x|^{m-d}}|\partial_{v}H(x,v,t)|\leq
CLt^{\frac{\alpha+1-\gamma}{\alpha}-1}H(x,v,t),$ (2.102)

for $H$ and $L$ as in Definition 2.11. If (2.101) and (2.102) hold, then (2.3)
holds. Indeed, arguing as in (2.3), (2.92) and using $f_{n}\leq G$, we find

$\displaystyle(*):=\partial_{t}G$
$\displaystyle+v^{\alpha}\partial_{x}G-\int_{0}^{\frac{v}{2}}K(v-v^{\prime},v^{\prime})G(x,v-v^{\prime},t)f_{n}(x,v^{\prime},t)\,dv^{\prime}$
$\displaystyle\qquad+\int_{0}^{\infty}K(v,v^{\prime})G(x,v,t)f_{n}(x,v^{\prime},t)\,dv^{\prime}$
$\displaystyle\geq
B_{t}\bigg{(}\partial_{t}H+v^{\alpha}\partial_{x}H+\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}H(x,v,t)\bigg{)}$
$\displaystyle-
B_{t}\frac{{L}v^{\gamma}}{1+|x|^{m-d}}|\partial_{v}H(x,v,t)|-B_{t}v^{\gamma}\int_{0}^{\frac{v}{2}}|H(x,v-v^{\prime},t)-H(x,v,t)|f_{n}(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\qquad+\lambda(c_{\alpha}t^{\frac{\alpha+1-\gamma}{\alpha}-1}+1)B_{t}H.$

Using now (2.101) and (2.102), we can conclude that

$\displaystyle(*)$ $\displaystyle\geq
B_{t}\bigg{(}\partial_{t}H+v^{\alpha}\partial_{x}H+\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}H(x,v,t)\bigg{)}$
$\displaystyle\qquad+B_{t}\big{[}-CL-2LCt^{\frac{\alpha+1-\gamma}{\alpha}-1}+\lambda(c_{\alpha}t^{\frac{\alpha+1-\gamma}{\alpha}-1}+1)\big{]}H$
$\displaystyle\geq\partial_{t}G+v^{\alpha}\partial_{x}G+\frac{{L}v^{\gamma}}{1+|x|^{m-d}}\xi_{R}(v)\partial_{v}G(x,v,t)$

if $\lambda$ is sufficiently large.

######  Remark 2.17.

It is worthwhile to mention that $L$ depends only on $K_{1}$ from (1.8),
$K_{2}$ from Lemma 2.7, and $K_{3}$ from Lemma 2.14, see Definition 2.11.

We now prove that (2.101) holds. Let $\eta\in(0,1)$ be fixed and sufficiently
small. We want to bound the following terms.

$\displaystyle\int_{\eta
v}^{\frac{v}{2}}K(v,v^{\prime})f_{n}(x,v^{\prime},t)|H(x,v-v^{\prime},t)-H(x,v,t)|\textup{d}v^{\prime}$
$\displaystyle+\int_{0}^{\eta
v}K(v,v^{\prime})f_{n}(x,v^{\prime},t)|H(x,v-v^{\prime},t)-H(x,v,t)|\textup{d}v^{\prime}=:J_{1}+J_{2}.$

We analyze each term separately. We notice that, since $\delta<1$ and $t$ is
sufficiently small, we have that $v^{\alpha}\geq\frac{x}{(1+2\delta)t}\geq
2K_{max}xt^{\frac{1}{m-1}}\geq 2v_{\textup{max}}^{\alpha}$ in this region,
where $K_{max}$ is as in (2.49). Thus we can assume in all the following that

$\displaystyle v\geq 2v_{\textup{max}}\textup{ and }v^{\prime}\geq
v_{\textup{max}},\textup{ for all }v^{\prime}\in\Big{[}\frac{v}{2},v\Big{]}.$
(2.103)

Moreover, by (2.45), it holds that
$G(x,v,t)=\frac{C_{0}B_{t}}{1+|X|^{m}+V^{p}}\leq\frac{C_{0}B_{t}}{1+V^{p}}\leq\frac{2^{p}C_{0}B_{t}}{1+v^{p}}$.
Using (2.60) and the fact that $B_{t}\leq 2$, for $t\leq T$, we have

$\displaystyle J_{1}$ $\displaystyle\leq C\int_{\eta
v}^{\frac{v}{2}}v^{\gamma}[H(x,v-v^{\prime},t)+H(x,v,t)]G(x,v^{\prime},t)\textup{d}v^{\prime}\leq
CH(x,v,t)\int_{\eta v}^{\frac{v}{2}}\frac{v^{\gamma}}{1+v^{\prime
p}}\textup{d}v^{\prime}$ $\displaystyle\leq
C(\eta)H(x,v,t)\int_{0}^{\infty}\frac{v^{\prime\gamma}}{1+v^{\prime
p}}\textup{d}v^{\prime}\leq CH(x,v,t).$

For the second term, it holds that

$\displaystyle J_{2}\leq Cv^{\gamma}\int_{0}^{\eta
v}|H(x,v-v^{\prime},t)-H(x,v,t)|G(x,v^{\prime},t)\textup{d}v^{\prime}\,.$

We remember we are in the region where
$v^{\alpha}t\in[\frac{x}{1+2\delta},\frac{x}{1-2\delta}]$ and thus
$v^{\prime\alpha}\leq\eta^{\alpha}v^{\alpha}\leq\frac{\eta^{\alpha}x}{(1-2\delta)t}$.
Due to (2.19), it follows that

$\displaystyle X(x,v^{\prime},t)\geq x-(1+\delta)v^{\prime\alpha}t\geq
x-\frac{(1+\delta)\eta^{\alpha}x}{1-2\delta}=\frac{1-2\delta-(1+\delta)\eta^{\alpha}}{1-2\delta}x,$

for all $v^{\prime}\in[0,\eta v]$. Since $\delta\leq\frac{1}{4}$ it follows
that we can choose $\eta$ to be sufficiently small, but independent of
$\delta$, such that $1-2\delta-(1+\delta)\eta^{\alpha}\geq\frac{1}{4}$ and
thus

$\displaystyle
X(x,v^{\prime},t)\geq(1-2\delta-(1+\delta)\eta^{\alpha})x\geq\frac{x}{4},$

for all $v^{\prime}\in[0,\eta v]$. Thus, since
$v^{\alpha}t\in[\frac{x}{1+2\delta},\frac{x}{1-2\delta}]$ and
$\delta\leq\frac{1}{4}$, we have that
$X(x,v^{\prime},t)\geq\frac{v^{\alpha}t}{8}$. It follows that

$\displaystyle J_{2}$ $\displaystyle\leq CB_{t}v^{\gamma}\int_{0}^{\eta
v}\frac{\int_{v-v^{\prime}}^{v}|\partial_{\tilde{v}}H(x,\tilde{v},t)|\textup{d}\tilde{v}\textup{d}v^{\prime}}{1+(v^{\alpha}t)^{m}+v^{\prime
p}}\leq Cv^{\gamma}\int_{0}^{\eta
v}\frac{\int_{v-v^{\prime}}^{v}|\partial_{\tilde{v}}H(x,\tilde{v},t)|\textup{d}\tilde{v}\textup{d}v^{\prime}}{1+(v^{\alpha}t)^{m}+v^{\prime
p}}$

since $B_{t}\leq 2$. Let now $\tilde{v}\in[v-v^{\prime},v]$, with
$v^{\prime}\in[0,\eta v].$ We have that

$\displaystyle|\partial_{\tilde{v}}H(x,\tilde{v},t)|\leq
C_{0}\frac{|m|X|^{m-2}X\partial_{\tilde{v}}X+V^{p-1}\partial_{\tilde{v}}V|}{(1+|X|^{m}+V^{p})^{2}}(x,\tilde{v},t)\leq
C_{0}\frac{|m|X|^{m-2}X\partial_{\tilde{v}}X+V^{p-1}\partial_{\tilde{v}}V|}{(1+V^{p})^{2}}(x,\tilde{v},t).$

Assume $v^{\gamma-1}t\geq 1$ and remember we are in the case when $\gamma>1$
and $v\geq 1$. Since $v^{\prime}\leq\eta v$ it holds that
$\frac{v}{2}\leq\tilde{v}\leq v$ for $\tilde{v}\in[v-v^{\prime},v]$. In
addition, by (2.103), we have that $\tilde{v}\geq v_{\textup{max}}$. If
$\tilde{v}\geq\frac{x}{1+2\delta}$, then from (2.22), we have that

$\displaystyle|m|X|^{m-2}X\partial_{\tilde{v}}X+V^{p-1}\partial_{\tilde{v}}V|\leq
CL\Big{(}|X(x,\tilde{v},t)|^{m-1}\tilde{v}^{\alpha-1}t+\tilde{v}^{p+\gamma-2}t\Big{)}.$
(2.104)

Otherwise, if $\tilde{v}<\frac{x}{1+2\delta}$, then (2.104) still holds since
$v^{\gamma-1}t\geq 1$.

Moreover, since $\frac{v}{2}\leq\tilde{v}\leq v$ for
$\tilde{v}\in[v-v^{\prime},v]$, it holds that $|X(x,\tilde{v},t)|\leq
Cv^{\alpha}t\leq Cv^{\alpha}$ and thus
$|X(x,\tilde{v},t)|^{m-1}\tilde{v}^{\alpha-1}t\leq Cv^{p-1}t\leq
Cv^{p+\gamma-2}t$ since $\gamma>1$. Thus

$\displaystyle|\partial_{\tilde{v}}H(x,\tilde{v},t)|\leq\frac{CLv^{p+\gamma-2}t}{(1+v^{p})^{2}}$
(2.105)

and thus it holds that

$\displaystyle J_{2}\leq
CLv^{2(\gamma-1)}\frac{v^{p}t}{(1+v^{p})^{2}}\int_{0}^{\eta
v}\frac{v^{\prime}\textup{d}v^{\prime}}{1+(v^{\alpha}t)^{m}+v^{\prime p}}.$

By making the change of variables
$v^{\prime}=(1+v^{p}t^{m})^{\frac{1}{p}}\xi$, we further obtain that

$\displaystyle J_{2}\leq
CLv^{2(\gamma-1)}\frac{v^{p}}{(1+v^{p})^{2}}\frac{t}{(1+v^{p}t^{m})^{1-\frac{2}{p}}}\int_{0}^{\infty}\frac{\xi\,\textup{d}\xi}{1+\xi^{p}}\leq
CLv^{2(\gamma-1)}\frac{v^{p}}{(1+v^{p})^{2}}\frac{t}{(1+v^{p}t^{m})^{1-\frac{2}{p}}}.$

Remembering the definition of $H$ and since we have, due to (2.18), that

$\displaystyle
H(x,v,t)=\frac{C_{0}}{1+|X(x,v,t)|^{m}+V(x,v,t)^{p}}\geq\frac{C}{1+v^{p}}$
(2.106)

we deduce that

$\displaystyle
J_{2}\leq\frac{CLv^{2(\gamma-1)}t}{(1+v^{p}t^{m})^{1-\frac{2}{p}}}H(x,v,t).$
(2.107)

We now analyze the term
$\frac{v^{2(\gamma-1)}t}{(1+v^{p}t^{m})^{1-\frac{2}{p}}}$. It holds that

$\displaystyle\frac{v^{2(\gamma-1)}t}{(1+v^{p}t^{m})^{1-\frac{2}{p}}}=\frac{(v^{\alpha}t)^{\frac{2(\gamma-1)}{\alpha}}t^{1-\frac{2(\gamma-1)}{\alpha}}}{(1+(v^{\alpha}t)^{m})^{1-\frac{2}{p}}}\leq
t^{1-\frac{2(\gamma-1)}{\alpha}}\leq t^{-\frac{\gamma-1}{\alpha}}.$ (2.108)

It holds that $-\frac{\gamma-1}{\alpha}>-1$ since $\gamma<\alpha+1$ and thus
we have that $\textup{e}^{t^{1-\frac{\gamma-1}{\alpha}}}\leq C$. Then, (2.101)
follows from (2.107).

If $v^{\gamma-1}t\leq 1$, we use (2.22) and then (2.105) becomes

$\displaystyle|\partial_{\tilde{v}}H(x,\tilde{v},t)|\leq\frac{CLv^{p-1}}{(1+v^{p})^{2}}.$
(2.109)

We can conclude using the same computations as above and by noticing that
$-\frac{(\gamma-1)}{\alpha}>-1$.

Finally, we need to prove that (2.102) holds in the case when
$x\in[(1-2\delta)v^{\alpha}t,(1+2\delta)v^{\alpha}t]$. Assume first that
$v^{\gamma-1}t\geq 1$. We have that

$\displaystyle\frac{v^{\gamma}}{1+|x|^{m-d}}|\partial_{v}H(x,v,t)|\leq\frac{Cv^{\gamma}|\partial_{v}H(x,v,t)|}{1+(v^{\alpha}t)^{m-d}}.$

Making use of (2.105) and (2.106), we further obtain as before that

$\displaystyle\frac{v^{\gamma}}{1+|x|^{m-d}}|\partial_{v}H(x,v,t)|\leq
CLv^{2(\gamma-1)}\frac{v^{p}}{1+v^{p}}\frac{t}{1+(v^{\alpha}t)^{m-d}}H(x,v,t)$

and we can then use similar arguments as in (2.108). The case when
$v^{\gamma-1}t\leq 1$ can be proved similarly using (2.109) instead of
(2.105). This concludes our proof. ∎

###  2.4 Proof of Theorem 1.4

In this subsection, we finish the proof of Theorem 1.4 by establishing that
there exists a limit for the sequence $\\{f_{n}\\}_{n\in\mathbb{N}}$ defined
in (2.2) and then passing to the limit in the equation. Our proof has
analogies with the methods used to solve symmetric hyperbolic systems, see for
example [28].

We first prove some bounds that are independent of $t$ for the function $G$
defined in Definition 2.11.

######  Lemma 2.18.

Let $T>0$ be sufficiently small. Then it holds that

$\displaystyle G(x,v,t)$
$\displaystyle\leq\frac{2^{m}C_{0}B_{t}}{1+|x|^{m}+v^{p}}\leq\frac{2^{m+1}C_{0}}{1+|x|^{m}+v^{p}};$
(2.110) $\displaystyle G(x,v-v^{\prime},t)$
$\displaystyle\leq\frac{2^{m+1}K_{2}C_{0}}{1+|x|^{m}+v^{p}}\textup{ for all
}v^{\prime}\in\bigg{(}0,\frac{v}{2}\bigg{)},$ (2.111)

for all $v>0$, $x\in\mathbb{R}$, and all $t\in[0,T]$, where $B_{t}$ was
defined in (2.82), $K_{2}$ is as in Lemma 2.7, and $C_{0}$ is as in (2.45).

###### Proof.

To prove (2.110) we notice first that it holds that $B_{t}\leq 2$ if we take
$t\leq 1$ to be sufficiently small, where $B_{t}$ was defined in (2.82).

We first consider the case $x>0$ and $v\leq v_{\textup{max}}$, where
$v_{\textup{max}}(x,t)$ was defined in (2.48).

From Definition 2.11 and the fact that if $v\leq v_{\textup{max}}(x,t)$ then
we are in the region where $t\leq\frac{x}{v^{\alpha}(1+2\delta)}$, we can use
the bound in (2.26) to deduce that

$\displaystyle G(x,v,t)$
$\displaystyle\leq\frac{C_{0}}{1+|x-(1+\delta)v_{\textup{max}}(x,t)^{\alpha}t|^{m}+(1-\delta)^{p}v_{\textup{max}}(x,t)^{p}},$
(2.112)

when $v\leq v_{\textup{max}}(x,t)$, where $C_{0}$ is as in (2.45).

By (2.49), we have that $\frac{1}{K_{max}}xt^{\frac{1}{m-1}}\leq
v_{\textup{max}}(x,t)\leq K_{max}xt^{\frac{1}{m-1}}$ and thus $x\geq
x-(1+\delta)v_{\textup{max}}(x,t)^{\alpha}t\geq\frac{x}{2}>0$ when $x>0$ if
$t$ is sufficiently small. Thus, using in addition that $v\leq
v_{\textup{max}}$ and that $\delta<\frac{1}{2}$, it holds that
$(1-\delta)^{p}v_{\textup{max}}^{p}\geq\frac{v^{p}}{2^{p}}$. Thus,

$\displaystyle G(x,v,t)=G(x,v_{\textup{max}}(x,t),t)$
$\displaystyle\leq\frac{2^{m}C_{0}}{1+|x|^{m}+v^{p}}.$ (2.113)

We now treat the case when
$t\in[\frac{x}{(1+2\delta)v^{\alpha}},\frac{x}{(1-2\delta)v^{\alpha}}]$. From
(2.45) and then using the fact that $x\leq 2v^{\alpha}t\leq v^{\alpha}$, it
follows that

$\displaystyle
G(x,v,t)\leq\frac{2^{p}C_{0}}{1+v^{p}}\leq\frac{2^{p+1}C_{0}}{1+|x|^{m}+v^{p}}.$

If $v^{\alpha}t\leq\frac{x}{1+2\delta}$, from (2.26) it holds that

$\displaystyle
G(x,v,t)\leq\frac{C_{0}}{1+|x-(1+\delta)v^{\alpha}t|^{m}+(1-\delta)^{p}v^{p}}$

and (2.110) follows from (2.66).

Finally, if $v^{\alpha}t\geq\frac{x}{1-2\delta}$, from (2.25) it holds that

$\displaystyle
G(x,v,t)\leq\frac{C_{0}}{1+|x-(1-\delta)v^{\alpha}t|^{m}+(1-\delta)^{p}v^{p}}$

and (2.110) follows from (2.62).

If $x\leq 0$, from (2.24) we have that
$|X(x,v,t)|\geq|x-(1-\delta)v^{\alpha}t|=|x|+(1-\delta)v^{\alpha}t$ and the
conclusion follows.

(2.111) follows from (2.110), (2.55), and (2.75). ∎

Using these bounds, we can now prove that there exists a limit for the
sequence $\\{f_{n}\\}_{n\in\mathbb{N}}$ that was defined in (2.2).

######  Proposition 2.19.

Let $T>0$ be sufficiently small. For every $\epsilon>0$, there exists
$n_{\epsilon}\in\mathbb{N}$ such that, for every $n,m\geq n_{\epsilon}$, it
holds that
$||f_{n}-f_{m}||_{\infty}:=\sup_{t\in[0,T],x\in\mathbb{R},v\in(0,\infty)}|f_{n}(x,v,t)-f_{m}(x,v,t)|\leq\epsilon$.

###### Proof.

Step 1: (Set-up) Let $t\geq 0$, $x\in\mathbb{R}$ and $v>0$. Let
$\\{f_{n}\\}_{n\in\mathbb{N}}$ be the sequence defined in (2.2). For
$n\in\mathbb{N}$, we denote by

$\displaystyle R_{n}(x,v,t):=f_{n+1}(x,v,t)-f_{n}(x,v,t).$ (2.114)

Moreover, for two functions $f,g$, we denote by

$\displaystyle\mathbb{K}_{1}[f,g]:=\int_{0}^{\frac{v}{2}}K(v-v^{\prime},v^{\prime})f(x,v^{\prime},t)g(x,v-v^{\prime},t)\textup{d}v^{\prime},$
(2.115)

$\displaystyle\mathbb{K}_{2}[f,g]:=\int_{0}^{\infty}K(v,v^{\prime})f(x,v^{\prime},t)g(x,v,t)\textup{d}v^{\prime},$
(2.116)

and

$\displaystyle\mathbb{K}[f,g]:=\mathbb{K}_{1}[f,g]-\mathbb{K}_{2}[f,g].$
(2.117)

Using this notation, it holds that

$\displaystyle\partial_{t}R_{n}(x,v,t)+v^{\alpha}\partial_{x}R_{n}(x,v,t)$
$\displaystyle=\mathbb{K}_{1}[f_{n},f_{n+1}]-\mathbb{K}_{1}[f_{n-1},f_{n}]-\mathbb{K}_{2}[f_{n},f_{n+1}]+\mathbb{K}_{2}[f_{n-1},f_{n}]$
$\displaystyle=\mathbb{K}_{1}[f_{n},f_{n+1}]-\mathbb{K}_{1}[f_{n},f_{n}]+\mathbb{K}_{1}[f_{n},f_{n}]-\mathbb{K}_{1}[f_{n-1},f_{n}]$
$\displaystyle-\mathbb{K}_{2}[f_{n},f_{n+1}]+\mathbb{K}_{2}[f_{n},f_{n}]-\mathbb{K}_{2}[f_{n},f_{n}]+\mathbb{K}_{2}[f_{n-1},f_{n}]$
$\displaystyle=\mathbb{K}[f_{n},R_{n}]+\mathbb{K}[R_{n-1},f_{n}].$ (2.118)

######  Remark 2.20.

Notice that it suffices to analyze the term $R_{n}$ in order to obtain the
statement of Proposition 2.19. This is since we can repeat the computations in
(2.4) to obtain

$\displaystyle\partial_{t}[f_{m}-f_{n}]+v^{\alpha}\partial_{x}[f_{m}-f_{n}]=\mathbb{K}[f_{m-1},f_{m}-f_{n}]+\mathbb{K}[f_{m-1}-f_{n-1},f_{n}].$
(2.119)

Since the estimates we will prove do not depend on $n,m\in\mathbb{N}$, we can
reduce the problem to analyzing $R_{n}$ in order to simplify the notation.

Notice the following. From (2.4) and (2.9), we have that $R_{n}$ solves the
following system

$\left\\{\begin{aligned}
\partial_{t}R_{n}(x,v,t)+v^{\alpha}\partial_{x}R_{n}(x,v,t)&=\mathbb{K}_{1}[f_{n},R_{n}]+\mathbb{K}_{1}[R_{n-1},f_{n}]-\mathbb{K}_{2}[f_{n},R_{n}]-\mathbb{K}_{2}[R_{n-1},f_{n}];\\\
R_{n}(x,v,0)&=0.\end{aligned}\right.$ (2.120)

Since the system is linear in $R_{n}$, by Duhamel’s principle, it suffices to
derive estimates for

$\left\\{\begin{aligned}
\partial_{t}R^{s}_{n}(x,v,t)+v^{\alpha}\partial_{x}R^{s}_{n}(x,v,t)&=\mathbb{K}_{1}[f_{n},R^{s}_{n}]-\mathbb{K}_{2}[f_{n},R^{s}_{n}],\textup{
for }t>s;\\\
R^{s}_{n}(x,v,s)&=\mathbb{K}_{1}[R_{n-1},f_{n}](x,v,s)-\mathbb{K}_{2}[R_{n-1},f_{n}](x,v,s).\end{aligned}\right.$
(2.121)

We now prove suitable estimates for the inhomogeneous part in (2.120), which
in turn will give us suitable estimates for $R_{n}$.

Step 2: (Induction basis) It holds

$\displaystyle
R_{1}(x,v,t)\leq\frac{CT(v^{\gamma}+1)}{(1+|x|^{m}+v^{p})(1+|x|^{m-d})},$
(2.122)

for all $t\in[0,T],$ where $d$ is as in (2.15).

We will prove (2.122) after (2.123) since the estimates are similar.

Step 3: (Induction step) It holds

$\displaystyle
R_{n}(x,v,t)\leq\frac{(CT)^{n}(v^{\gamma}+1)}{(1+|x|^{m}+v^{p})(1+|x|^{m-d})},$
(2.123)

for all $t\in[0,T],$ where $d$ is as in (2.15).

We assume by induction that there exists a constant $C>0$ such that

$\displaystyle
R_{n-1}(x,v,t)\leq\frac{(CT)^{n-1}(v^{\gamma}+1)}{(1+|x|^{m}+v^{p})(1+|x|^{m-d})}.$
(2.124)

We estimate the inhomogeneous terms $\mathbb{K}_{1}[R_{n-1},f_{n}]$ and
$\mathbb{K}_{2}[R_{n-1},f_{n}]$. Assume that the following inequality holds

$\displaystyle|\mathbb{K}_{1}[R_{n-1},f_{n}]-\mathbb{K}_{2}[R_{n-1},f_{n}]|\leq\frac{C^{n}T^{n-1}(v^{\gamma}+1)}{(1+|x|^{m}+v^{p})(1+|x|^{m-d})}.$
(2.125)

From (2.121), it follows that

$\displaystyle
R_{n}^{s}(x,v,t)\leq\frac{C^{n}T^{n-1}(v^{\gamma}+1)}{(1+|x|^{m}+v^{p})(1+|x|^{m-d})},$
(2.126)

for $t\in[0,T]$, if $T$ is sufficiently small. Since from (2.120) and (2.121),
we have that

$\displaystyle R_{n}(x,v,t)=\int_{0}^{t}R_{n}^{s}(x,v,t)\textup{d}s,$

the conclusion (2.123) follows from (2.126).

We are thus left to prove that (2.125) holds. We start with
$\mathbb{K}_{2}[R_{n-1},f_{n}]$. Using Proposition 2.12, (2.110) and the fact
that $K(v,v^{\prime})\leq K_{1}(v^{\gamma}+v^{\prime\gamma})$ from (1.8), we
have that

$\displaystyle\mathbb{K}_{2}[R_{n-1},f_{n}]$
$\displaystyle=\int_{0}^{\infty}K(v,v^{\prime})R_{n-1}(x,v^{\prime},t)f_{n}(x,v,t)\textup{d}v^{\prime}\leq\frac{Cv^{\gamma}}{1+|x|^{m}+v^{p}}\int_{0}^{\infty}R_{n-1}(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle+\frac{C}{1+|x|^{m}+v^{p}}\int_{0}^{\infty}v^{\prime\gamma}R_{n-1}(x,v^{\prime},t)\textup{d}v^{\prime}.$
(2.127)

Using assumption (2.124) and then (2.81) for $n=0$ and $n=2\gamma$, we further
obtain that

$\displaystyle\int_{0}^{\infty}(1+v^{\prime\gamma})R_{n-1}(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\leq\frac{(CT)^{n-1}}{1+|x|^{m-d}}\int_{0}^{\infty}\frac{(v^{\prime\gamma}+1)^{2}}{1+|x|^{m}+v^{\prime
p}}\textup{d}v^{\prime}$
$\displaystyle\leq\frac{C^{n}T^{n-1}}{\Big{(}1+|x|^{m-d}\Big{)}\Big{(}1+|x|^{m-\frac{2\gamma+1}{\alpha}}\Big{)}}\leq\frac{C^{n}T^{n-1}}{1+|x|^{m-d}}.$
(2.128)

Plugging (2.128) into (2.127), we deduce that

$\displaystyle\mathbb{K}_{2}[R_{n-1},f_{n}]\leq\frac{C^{n}T^{n-1}(v^{\gamma}+1)}{(1+|x|^{m}+v^{p})(1+|x|^{m-d})}.$

Similarly, we can bound the term $\mathbb{K}_{1}[R_{n-1},f_{n}]$. More
precisely, since $v^{\prime}\in(0,\frac{v}{2})$, we make use of Proposition
2.12 and (2.111) and, as before, it holds that

$\displaystyle\mathbb{K}_{1}[R_{n-1},f_{n}]$
$\displaystyle=\int_{0}^{\frac{v}{2}}K(v-v^{\prime},v^{\prime})R_{n-1}(x,v^{\prime},t)f_{n}(x,v-v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\leq\frac{Cv^{\gamma}}{1+|x|^{m}+v^{p}}\int_{0}^{\infty}R_{n-1}(x,v^{\prime},t)\textup{d}v^{\prime}$
$\displaystyle\leq\frac{C^{n}T^{n-1}(v^{\gamma}+1)}{(1+|x|^{m}+v^{p})(1+|x|^{m-d})}.$

This concludes the proof of (2.123).

Step 4: (Proof of (2.122)) Using the definition of $f_{0}$ in (2.10), we
obtain that

$\displaystyle\partial_{t}R_{1}(x,v,t)+v^{\alpha}\partial_{x}R_{1}(x,v,t)=\mathbb{K}[f_{0},R_{1}]+\mathbb{K}[f_{0},f_{0}].$

Following the steps of the proof of (2.123), we obtain that

$\displaystyle\mathbb{K}[f_{0},f_{0}]\leq\frac{C(1+v^{\gamma})}{(1+|x|^{m}+v^{p})(1+|x|^{m-d})}$

and the conclusion follows by Duhamel’s principle as before.

Step 5: (Conclusion) We combine Remark 2.20 with (2.122) and (2.123) and
choose the time $T$ in (2.123) to be sufficiently small, such that the right-
hand side of (2.123) tends to zero as $n\rightarrow\infty$.

∎

We are now able to conclude the proof of Theorem 1.4.

###### Proof of Theorem 1.4.

We first prove that if $t\leq T$ and $T$ is sufficiently small, it holds that

$\displaystyle f_{0}(x,v,t)\leq\frac{2C_{0}}{1+|x|^{m}+v^{p}}.$ (2.129)

From (2.10) and (2.11) it follows that
$f_{0}(x,v,t)\leq\frac{C_{0}}{1+|x-v^{\alpha}t|^{m}+v^{p}}.$ We first consider
$x>0$. If $v^{\alpha}t\leq x\bigg{(}1-\frac{1}{\sqrt[m]{2}}\bigg{)}$ then
$x-v^{\alpha}t\geq\frac{x}{\sqrt[m]{2}}$ and thus

$\displaystyle\frac{1}{1+|x-v^{\alpha}t|^{m}+v^{p}}\leq\frac{1}{1+\frac{x^{m}}{2}+v^{p}}\leq\frac{2}{1+x^{m}+v^{p}}.$

If $v^{\alpha}t\geq x\bigg{(}1-\frac{1}{\sqrt[m]{2}}\bigg{)}$ then $t\neq 0$
and since $t\leq 1$ is sufficiently small we have that $v^{p}\geq 2x^{m}$ and
thus

$\displaystyle\frac{1}{1+|x-v^{\alpha}t|^{m}+v^{p}}\leq\frac{1}{1+v^{p}}=\frac{1}{1+\frac{v^{p}}{2}+\frac{v^{p}}{2}}\leq\frac{1}{1+x^{m}+\frac{v^{p}}{2}}\leq\frac{2}{1+x^{m}+v^{p}}.$

If $x\leq 0$, then, since $m$ is even, we have
$(x-v^{\alpha}t)^{m}=(|x|+v^{\alpha}t)^{m}\geq x^{m}$ and the conclusion
follows. Then, by (2.129) and (2.81), we obtain that

$\displaystyle\int_{(0,\infty)}vf_{0}(x,v,t)\textup{d}v$
$\displaystyle\leq\frac{2K_{3}C_{0}}{1+|x|^{m-\frac{2}{\alpha}}}.$ (2.130)

On the other hand, by Lemma 2.14, it follows that we can find a constant
$K_{3}>0$ such that

$\displaystyle\int_{(0,\infty)}vG(x,v,t)\textup{d}v$
$\displaystyle\leq\frac{K_{3}C_{0}B_{t}}{1+|x|^{m-d}},$ (2.131)

where $d$ was defined in (2.15), $B_{t}$ was defined in (2.82), and $G$ was
defined in Definition 2.11. Moreover, we can choose $t\leq 1$ to be
sufficiently small (as in Remark 2.13) such that $B_{t}\leq 2$ and thus it
holds that
$\int_{(0,\infty)}vG(x,v,t)\textup{d}v\leq\frac{2K_{3}C_{0}}{1+|x|^{m-d}}$.

We use induction in order to prove that $f_{n}\leq G$ for all
$n\in\mathbb{N}$. By Proposition 2.12, the induction step holds true. For the
induction basis, we need to prove that (2.3) holds true for $n=0$. This is
done with the same estimates as in Proposition 2.12 by using (2.130). Thus, if
we take $L$ in (2.16) to be as in Definition 2.11, namely
$L=4K_{1}K_{2}K_{3}C_{0}$, where $K_{1}$ is as in (1.8), $K_{2}$ is as in
Lemma 2.7, and $K_{3}$ is as in (2.131), we can conclude that

$\displaystyle f_{n}(x,v,t)\leq G(x,v,t),$ (2.132)

for all $n\in\mathbb{N}$.

From (2.132) and (2.110), it follows that there exists some $C>0$ such that

$\displaystyle f_{n}(x,v,t)\leq\frac{C}{1+|x|^{m}+v^{p}},$ (2.133)

for all $n\in\mathbb{N}$, $t\in[0,T]$, $x\in\mathbb{R}$, and $v\in(0,\infty)$.

By Proposition 2.19 we have that the exists a limit of the sequence
$\\{f_{n}\\}_{n\in\mathbb{N}}$. It remains to show that the limit of the
sequence $\\{f_{n}\\}_{n\in\mathbb{N}}$ satisfies equation (1.11). With the
bound on $\\{f_{n}\\}$ in (2.133) and the bound on the kernel (1.8) it is
completely standard to pass to the limit in the equation. Mass-conservation of
$f_{n}$ follows by testing with $v$ in (2.2) and then integrating in $v$ and
$x$. Mass-conservation of $f$ then follows by passing to the limit as
$n\rightarrow\infty$ in
$\int_{\mathbb{R}}\int_{(0,\infty)}vf_{n}(x,v,t)\textup{d}v\textup{d}x.$ We
omit the details here.

∎

##  Appendix A Estimates for the second order derivative of $G_{L}$

We will prove in this appendix that (2.50) holds true.

######  Proposition A.1.

Given $L>0$ and $\delta\in\big{(}0,\frac{1}{2}\big{)}$ there exists a
sufficiently large $R>0$ such that for all $t\in[0,T]$, with $T$ sufficiently
small, which is independent of $L,\delta,$ and $R$, it holds that

$\displaystyle\partial_{v}^{2}G_{L}(x,v,t)<0$ (A.1)

if

$\displaystyle\frac{1}{K_{max}}xt^{\frac{1}{m-1}}\leq v^{\alpha}\leq
K_{max}xt^{\frac{1}{m-1}},$ (A.2)

where $K_{max}$ is as in (2.49).

###### Proof.

Let $Q(x,v,t)=1+|X|^{m}+V^{p}$. Since $G_{L}=\frac{1}{Q}$, we have that

$\displaystyle\partial_{v}Q(x,v,t)$
$\displaystyle=m|X|^{m-2}X\partial_{v}X+pV^{p-1}\partial_{v}V,$ (A.3)
$\displaystyle\partial^{2}_{v}Q(x,v,t)$
$\displaystyle=m(m-1)|X|^{m-2}|\partial_{v}X|^{2}+p(p-1)V^{p-2}(\partial_{v}V)^{2}+m|X|^{m-2}X\partial^{2}_{v}X+pV^{p-1}\partial^{2}_{v}V,$
(A.4)

and

$\displaystyle\partial_{v}^{2}G_{L}=2\frac{|\partial_{v}Q|^{2}}{Q^{3}}-\frac{\partial_{v}^{2}Q}{Q^{2}}=\frac{2|\partial_{v}Q|^{2}-Q\partial_{v}^{2}Q}{Q^{3}}.$
(A.5)

It is worthwhile to notice that if $v$ is as in (A.2), then
$x\geq(1+2\delta)v^{\alpha}t$ if $t$ is sufficiently small and thus (2.21)
holds. We analyze first the term $2|\partial_{v}Q|^{2}$ in (A.5). Since
$v^{\alpha}\leq K_{max}xt^{\frac{1}{m-1}}$, we have that $0\leq
x-(1+\delta)v^{\alpha}t\leq X\leq x-(1-\delta)v^{\alpha}t\leq x$ and $x>0$.
Using Proposition 2.2 and (A.2) we deduce that

$\displaystyle|\partial_{v}Q|\leq
C_{1}x^{m-1}v^{\alpha-1}t+C_{2}v^{p-1}\leq\frac{Cx^{m}}{v}t^{\frac{m}{m-1}},$

for some constants $C_{1},C_{2},C>0$, and thus

$\displaystyle
2|\partial_{v}Q(x,v,t)|^{2}\leq\frac{Cx^{2m}}{v^{2}}t^{\frac{2m}{m-1}}.$ (A.6)

We then analyze the terms in (A.4). We have $|X|^{m-2}|\partial_{v}X|^{2}\geq
0$ and from Proposition 2.2 we know

$\displaystyle V^{p-2}(\partial_{v}V)^{2}\geq\frac{1}{C}v^{p-2}.$ (A.7)

We are going to show that if $t$ is sufficiently small and $v$ as in (A.2) we
have

$\displaystyle\partial^{2}_{v}X(x,v,t)\geq 0$ (A.8)

and that for any $\epsilon>0$ and $v$ as in (A.2) we have

$\displaystyle|V^{p-1}\partial^{2}_{v}V|$ $\displaystyle\leq\epsilon v^{p-2}$
(A.9)

if $R$ is sufficiently large and $t$ is sufficiently small. Combining then
(A.7)-(A.9) with (A.2) and noticing that
$Q\geq|X|^{m}\geq(x-(1+\delta)v^{\alpha}t)^{m}\geq\frac{x^{m}}{2^{m}}$ if $t$
is sufficiently small, it follows that

$\displaystyle
Q\partial_{v}^{2}Q\geq\frac{x^{m}}{C}v^{p-2}\geq\frac{x^{2m}}{Cv^{2}}t^{\frac{m}{m-1}}.$
(A.10)

(A.5), (A.6) and (A.10) imply that we can find a sufficiently small
$T\in(0,1)$ such that (A.1) holds for all $t\leq T$.

In all the following computations, we will assume for simplicity that $v\geq
2R$, with $R$ as in Proposition 2.2. The rest of the cases can be proved using
similar computations.

Auxiliary result: Given $L>0$ and $\delta\in\big{(}0,\frac{1}{2}\big{)}$ there
exists a sufficiently large $R>0$ such that for all $t\in[0,T]$, with $T$
sufficiently small, which is independent of $L,\delta,$ and $R$, it holds that

$\displaystyle|\partial^{2}_{v}X(x,v,t)|\leq Cv^{\alpha-2}t$ (A.11)

if $v$ is as in (A.2).

In order to show (A.11) we first prove that there exists $C>0$ such that

$\displaystyle|\partial^{2}_{v}\Phi(z,v)|\leq Cv^{-1},\textup{ for }v\geq 2R,$
(A.12)

where $\Phi$ is as (2.30). We differentiate two times with respect to $v$ in
(2.32) in order to deduce that

$\displaystyle\partial^{2}_{v}\Phi(z,v)$
$\displaystyle=(\gamma{-}\alpha)\bigg{(}v^{1-(\gamma-\alpha)}+(1-(\gamma{-}\alpha))z\bigg{)}^{\frac{2(\gamma-\alpha)-1}{1-(\gamma-\alpha)}}v^{-2(\gamma-\alpha)}$
$\displaystyle\qquad-(\gamma{-}\alpha)(v^{1-(\gamma-\alpha)}+(1-(\gamma{-}\alpha))z)^{\frac{\gamma-\alpha}{1-(\gamma{-}\alpha)}}v^{-(\gamma-\alpha)-1}.$

Since $0\leq z\leq C$ and we are in the case when $v\geq 2R$, we can choose
$R>0$ sufficiently large such that $v^{1-(\gamma-\alpha)}\leq
v^{1-(\gamma-\alpha)}+(1-(\gamma-\alpha))z\leq 2v^{1-(\gamma-\alpha)}$ and
thus $|\partial^{2}_{v}\Phi(z,v)|\leq Cv^{-1}$.

In order to prove (A.11) we differentiate (2.38) twice with respect to $v$ and
obtain

$\displaystyle\frac{\partial^{2}_{v}X(x,v,t)}{\Phi(\psi(X(x,v,t))-\psi(x),v)^{\alpha}}-\alpha\frac{\partial_{v}X(x,v,t)[\partial_{v}\Phi+\partial_{z}\Phi\psi^{\prime}(X)\partial_{v}X]}{\Phi(\psi(X(x,v,t))-\psi(x),v)^{\alpha+1}}$
$\displaystyle=\alpha\int_{x}^{X}\frac{\partial^{2}_{v}\Phi\textup{d}\xi}{\Phi(\psi(\xi)-\psi(x),v)^{\alpha+1}}+\alpha(\alpha+1)\int_{X}^{x}\frac{(\partial_{v}\Phi)^{2}\textup{d}\xi}{\Phi(\psi(\xi)-\psi(x),v)^{\alpha+2}}$
$\displaystyle\qquad+\frac{\alpha\partial_{v}\Phi}{\Phi(\psi(X)-\psi(x),v)^{\alpha+1}}\partial_{v}X.$
(A.13)

From (A.12), (2.19) and (2.18) we deduce that

$\displaystyle\bigg{|}\int_{x}^{X}\frac{\partial^{2}_{v}\Phi\textup{d}\xi}{\Phi(\psi(\xi)-\psi(x),v)^{\alpha+1}}\bigg{|}\leq
Cv^{-\alpha-2}[x-X]\leq Cv^{-2}t.$ (A.14)

Furthermore, Proposition 2.2 implies that

$\displaystyle\frac{|\partial_{v}\Phi\partial_{v}X|}{\Phi(\psi(X)-\psi(x),v)^{\alpha+1}}$
$\displaystyle\leq Cv^{-2}t\,,$ (A.15)
$\displaystyle\int_{X}^{x}\frac{(\partial_{v}\Phi)^{2}\textup{d}\xi}{\Phi(\psi(\xi)-\psi(x),v)^{\alpha+2}}\leq
Cv^{-\alpha-2}[x-X]$ $\displaystyle\leq Cv^{-2}t$ (A.16)

and that $\frac{1}{C}v^{\alpha}\leq\Phi(\psi(X)-\psi(x),v)^{\alpha}\leq
Cv^{\alpha}.$

We claim that if $v$ satisfies (A.2) it holds

$\displaystyle
0\leq\frac{\partial_{z}\Phi\psi^{\prime}(X)|\partial_{v}X|^{2}}{\Phi(\psi(X)-\psi(x),v)^{\alpha+1}}\leq
Cv^{-2}t.$ (A.17)

Indeed, Proposition 2.2 implies that $|\partial_{v}X|\leq Cv^{\alpha-1}t$ and
$\frac{1}{\Phi(\psi(X)-\psi(x),v)^{\alpha+1}}\leq Cv^{-\alpha-1}$.

We use (A.2) again to deduce that we are in the region where
$x-(1+2\delta)v^{\alpha}t\geq 0$ for sufficiently small $t$. From the estimate
(2.42) we obtain that

$\displaystyle 0\leq\partial_{z}\Phi\psi^{\prime}(X)|\partial_{v}X|\leq C.$
(A.18)

Thus, (A.17) follows.

Combining the estimates (A.14)-(A.17) and then making use of (A), we obtain
(A.11).

Proof of (A.9):

We only look at the case when $\gamma>1$. The case $\gamma\in[0,1]$ can be
proved in a similar manner. As before, we assume in the following for
simplicity that the constant $L$ in (2.17) is $L=1$. If
$\gamma\in(1,1+\alpha)$, then by integrating in (2.17) it follows that

$\displaystyle
V(x,v,t)^{1-\gamma}-v^{1-\gamma}=(\gamma-1)\int_{0}^{t}\frac{\textup{d}\xi}{1+|X(x,v,\xi)|^{m-d}}$
(A.19)

such that

$\displaystyle
V(x,v,t)=\frac{v}{\bigg{(}1+(\gamma-1)v^{\gamma-1}\int_{0}^{t}\frac{\textup{d}\xi}{1+|X|^{m-d}}\bigg{)}^{\frac{1}{\gamma-1}}}.$
(A.20)

Differentiating in $v$, we obtain that

$\displaystyle\partial_{v}V(x,v,t)$
$\displaystyle=\frac{1}{\bigg{(}1+(\gamma-1)v^{\gamma-1}\int_{0}^{t}\frac{\textup{d}\xi}{1+|X|^{m-d}}\bigg{)}^{\frac{1}{\gamma-1}}}-\frac{(\gamma-1)vv^{\gamma-2}\int_{0}^{t}\frac{\textup{d}\xi}{1+|X|^{m-d}}}{\bigg{(}1+(\gamma-1)v^{\gamma-1}\int_{0}^{t}\frac{\textup{d}\xi}{1+|X|^{m-d}}\bigg{)}^{\frac{\gamma}{\gamma-1}}}$
(A.21)
$\displaystyle+\frac{vv^{\gamma-1}\int_{0}^{t}\frac{(m-d)|X|^{m-d-2}X\partial_{v}X\textup{d}\xi}{(1+|X|^{m-d})^{2}}}{\bigg{(}1+(\gamma-1)v^{\gamma-1}\int_{0}^{t}\frac{\textup{d}\xi}{1+|X|^{m-d}}\bigg{)}^{\frac{\gamma}{\gamma-1}}}=:\frac{T_{1}}{T_{2}^{\frac{\gamma}{\gamma-1}}},$

where

$\displaystyle
T_{1}:=1+v^{\gamma}\int_{0}^{t}\frac{(m-d)|X|^{m-d-2}X\partial_{v}X\textup{d}\xi}{(1+|X|^{m-d})^{2}}$

and

$\displaystyle
T_{2}:=1+(\gamma-1)v^{\gamma-1}\int_{0}^{t}\frac{\textup{d}\xi}{1+|X|^{m-d}}.$

Differentiating in $v$, we obtain that

$\displaystyle\partial^{2}_{v}V(x,v,t)=\frac{\partial_{v}T_{1}}{T_{2}^{\frac{\gamma}{\gamma-1}}}-\frac{\gamma}{\gamma-1}\frac{T_{1}\partial_{v}T_{2}}{T_{2}^{\frac{2\gamma-1}{\gamma-1}}}.$

It holds that

$\displaystyle\partial_{v}T_{1}$ $\displaystyle=\gamma
v^{\gamma-1}\int_{0}^{t}\frac{(m-d)|X|^{m-d-2}X\partial_{v}X\textup{d}\xi}{(1+|X|^{m-d})^{2}}-2v^{\gamma}\int_{0}^{t}\frac{\big{|}(m-d)|X|^{m-d-2}X\partial_{v}X\big{|}^{2}\textup{d}\xi}{(1+|X|^{m-d})^{3}}$
$\displaystyle\qquad+v^{\gamma}\int_{0}^{t}\frac{(m-d)|X|^{m-d-2}X\partial^{2}_{v}X\textup{d}\xi}{(1+|X|^{m-d})^{2}}+v^{\gamma}\int_{0}^{t}\frac{(m-d)(m-d-1)|X|^{m-d-2}|\partial_{v}X|^{2}\textup{d}\xi}{(1+|X|^{m-d})^{2}}.$

Moreover, we have that

$\displaystyle\partial_{v}T_{2}=(\gamma-1)^{2}v^{\gamma-2}\int_{0}^{t}\frac{\textup{d}\xi}{1+|X|^{m-d}}-(\gamma-1)v^{\gamma-1}\int_{0}^{t}\frac{(m-d)|X|^{m-d-2}X\partial_{v}X\textup{d}\xi}{(1+|X|^{m-d})^{2}}.$

We first prove the following estimates:

$\displaystyle 0\leq
v^{\gamma-1}\int_{0}^{t}\frac{\textup{d}\xi}{1+|X(x,v,\xi)|^{m-d}}\leq\frac{1}{2}$
(A.22)

and

$\displaystyle
v^{\gamma}\bigg{|}\int_{0}^{t}\frac{(m-d)|X(x,v,\xi)|^{m-d-2}X(x,v,\xi)\partial_{v}X(x,v,\xi)\textup{d}\xi}{(1+|X(x,v,\xi)|^{m-d})^{2}}\bigg{|}\leq\frac{1}{2}.$
(A.23)

We first prove that (A.22) holds. For sufficiently small $t$, we are in the
case when $x\geq(1+2\delta)v^{\alpha}t$ since $v$ is as in (A.2). We use
(2.19) and the fact that
$x\geq(1+2\delta)v^{\alpha}t\geq(1+2\delta)v^{\alpha}s$ for all $s\in[0,t]$.
Thus, $|X(s)|\geq|x-(1+\delta)v^{\alpha}s|$ when
$x\geq(1+2\delta)v^{\alpha}t$, for all $s\in[0,t]$. It follows that

$\displaystyle
v^{\gamma-1}\int_{0}^{t}\frac{\textup{d}\xi}{1+|X(x,v,\xi)|^{m-d}}$
$\displaystyle\leq
Cv^{\gamma-1}\int_{0}^{t}\frac{\textup{d}\xi}{1+|x-(1+\delta)v^{\alpha}\xi|^{m-d}}$
$\displaystyle\leq
Cv^{\gamma-\alpha-1}\int_{0}^{v^{\alpha}t}\frac{\textup{d}z}{1+|x-(1+\delta)z|^{m-d}}\leq
Cv^{\gamma-\alpha-1},$ (A.24)

which, since $v\geq R$ and $\gamma<\alpha+1$, implies (A.22) if $R$ is
sufficiently large.

We now prove (A.23). As before, we consider the case when
$x\geq(1+2\delta)v^{\alpha}t$ since we can choose $t$ sufficiently small. It
follows that

$\displaystyle
v^{\gamma}\bigg{|}\int_{0}^{t}\frac{(m-d)|X|^{m-d-2}X\partial_{v}X\textup{d}\xi}{(1+|X|^{m-d})^{2}}\bigg{|}$
$\displaystyle\leq
Cv^{\gamma-1}\int_{0}^{t}\frac{|x-(1-\delta)v^{\alpha}\xi|^{m-d-1}v^{\alpha}\xi\textup{d}\xi}{(1+|x-(1+\delta)v^{\alpha}\xi|^{m-d})^{2}}.$

If $v$ is as in (A.2), then $x-(1+\delta)z\geq x-2z\geq\frac{x}{2}$ for
sufficiently small $t$, for $z\in[0,v^{\alpha}t]$. By making the change of
variables $z=v^{\alpha}\xi$ and using the fact that $x-(1-\delta)z\leq x$ we
further obtain that

$\displaystyle
v^{\gamma-1}\int_{0}^{t}\frac{|x-(1-\delta)v^{\alpha}\xi|^{m-d-1}v^{\alpha}\xi\textup{d}\xi}{(1+|x-(1+\delta)v^{\alpha}\xi|^{m-d})^{2}}$
$\displaystyle\leq
Cv^{\gamma-\alpha-1}\int_{0}^{v^{\alpha}t}\frac{|x-(1-\delta)z|^{m-d-1}z\textup{d}z}{(1+|x-(1+\delta)z|^{m-d})^{2}}$
$\displaystyle\leq
Cv^{\gamma-\alpha-1}\int_{0}^{\frac{x}{1+2\delta}}\frac{|x|^{m-d}\textup{d}z}{(1+|x|^{m-d})^{2}}\leq\frac{Cv^{\gamma-\alpha-1}|x|^{m-d+1}}{(1+|x|^{m-d})^{2}}$
$\displaystyle\leq Cv^{\gamma-\alpha-1},$

and thus (A.23) follows if we take $R$ sufficiently large since $v\geq R$ and
$\gamma<\alpha+1$.

From (A.22) and (A.23), it holds that $T_{1}\in[\frac{1}{2},1]$ and
$T_{2}\in[\frac{1}{2},\frac{3}{2}]$. From this we deduce that

$\displaystyle|\partial^{2}_{v}V(x,v,t)|\leq
C\big{(}|\partial_{v}T_{1}|+|\partial_{v}T_{2}|\big{)}\,.$

We analyze each term of $\partial_{v}T_{1}$ separately. Following the
computations for (A.23), we deduce that

$\displaystyle
v^{\gamma-1}\bigg{|}\int_{0}^{t}\frac{(m-d)|X|^{m-d-2}X\partial_{v}X\textup{d}\xi}{(1+|X|^{m-d})^{2}}\bigg{|}\leq
Cv^{\gamma-\alpha-1}v^{-1}.$ (A.25)

Similarly, by making the change of variables $z=v^{\alpha}\xi$, it follows
that

$\displaystyle
v^{\gamma}\int_{0}^{t}\frac{\big{|}|X|^{m-d-2}X\partial_{v}X\big{|}^{2}\textup{d}\xi}{(1+|X|^{m-d})^{3}}$
$\displaystyle\leq
Cv^{\gamma}\int_{0}^{t}\frac{\big{(}|x-(1-\delta)v^{\alpha}\xi|^{m-d-1}v^{\alpha-1}\xi\big{)}^{2}\textup{d}\xi}{(1+|x-(1+\delta)v^{\alpha}\xi|^{m-d})^{3}}$
$\displaystyle\leq
Cv^{\gamma-\alpha-2}\int_{0}^{v^{\alpha}t}\frac{\big{(}|x-(1-\delta)z|^{m-d-1}z\big{)}^{2}\textup{d}z}{(1+|x-(1+\delta)z|^{m-d})^{3}}\leq
Cv^{\gamma-\alpha-1}v^{-1}.$ (A.26)

Moreover, (A.11) implies that

$\displaystyle
v^{\gamma}\bigg{|}\int_{0}^{t}\frac{(m-d)|X|^{m-d-2}X\partial^{2}_{v}X\textup{d}\xi}{(1+|X|^{m-d})^{2}}\bigg{|}$
$\displaystyle\leq
Cv^{\alpha-2}v^{\gamma}\int_{0}^{t}\frac{(m-d)|X|^{m-d-1}\xi\textup{d}\xi}{(1+|X|^{m-d})^{2}}$
$\displaystyle\leq
Cv^{-\alpha-2}v^{\gamma}\int_{0}^{v^{\alpha}t}\frac{|x-(1-\delta)z|^{m-d-1}z\textup{d}z}{(1+|x-(1+\delta)z|^{m-d})^{2}}$
$\displaystyle\leq Cv^{\gamma-\alpha-1}v^{-1}.$ (A.27)

We now analyze the last term in $\partial_{v}T_{1}$. We have that

$\displaystyle
v^{\gamma}\int_{0}^{t}\frac{|X|^{m-d-2}|\partial_{v}X|^{2}\textup{d}\xi}{(1+|X|^{m-d})^{2}}\leq
Cv^{\gamma-\alpha-2}\int_{0}^{v^{\alpha}t}\frac{|x-(1-\delta)z|^{m-d-2}z^{2}\textup{d}z}{(1+|x-(1+\delta)z|^{m-d})^{2}}\leq
Cv^{\gamma-\alpha-1}v^{-1}.$ (A.28)

We continue by finding upper bounds for each term in $\partial_{v}T_{2}$. From
(A), it holds that

$\displaystyle v^{\gamma-2}\int_{0}^{t}\frac{\textup{d}\xi}{1+|X|^{m-d}}\leq
Cv^{\gamma-\alpha-1}v^{-1},$ (A.29)

which offers the right estimate in order to prove (A.9). Notice that the last
term in $\partial_{v}T_{2}$, which is
$v^{\gamma-1}\int_{0}^{t}\frac{|X|^{m-d-2}X\partial_{v}X\textup{d}\xi}{(1+|X|^{m-d})^{2}}$,
has already been analyzed in (A.25).

Combining (A.25)-(A.29) together with the fact that $v\geq R$ and that
$\gamma<\alpha+1$, we obtain that

$\displaystyle|v^{p-1}\partial_{v}^{2}V(x,v,t)|\leq
Cv^{\gamma-\alpha-2}v^{p-1}\leq\epsilon v^{p-2},$

if $v\geq R$ is sufficiently large. This concludes the proof of (A.9).

Proof of (A.8):

From (2.17) it follows that

$\displaystyle\partial_{v}^{2}X(x,v,t)=\alpha(1-\alpha)\int_{0}^{t}V^{\alpha-2}(\partial_{v}V)^{2}\textup{d}s-\alpha\int_{0}^{t}V^{\alpha-1}\partial_{v}^{2}V\textup{d}s.$
(A.30)

Now (A.9) implies that the second term on the right hand side can be absorbed
into the first one and thus (A.8) follows. ∎

### Acknowledgements

The authors gratefully acknowledge the financial support of the collaborative
research centre The mathematics of emerging effects (CRC 1060, Project-ID
211504053) and of the Bonn International Graduate School of Mathematics at the
Hausdorff Center for Mathematics (EXC 2047/1, Project-ID 390685813) funded
through the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

#### Statements and Declarations

Conflict of interest The authors declare that they have no conflict of
interest.

Data availability Data sharing not applicable to this article as no datasets
were generated or analysed during the current study.

## References

  * [1] M. Aizenman and T. A. Bak  “Convergence to equilibrium in a system of reacting polymers”  In _Communications in Mathematical Physics_ 65, 1979, pp. 203–230
  * [2] J. M. Ball and J. Carr  “The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation”  In _Journal of Statistical Physics_ 61, 1990, pp. 203–234
  * [3] J. Banasiak, W. Lamb and P. Laurençot  “Analytic Methods for Coagulation-Fragmentation Models, Volume II”  Boca Raton: CRC Press, 2019
  * [4] M. Bonacini, B. Niethammer and J. J. L. Velázquez  “Self-similar solutions to coagulation equations with time-dependent tails: The case of homogeneity one”  In _Archive for Rational Mechanincs and Analysis_ 233, 2019, pp. 1–43
  * [5] A. V. Burobin  “Existence and uniqueness of a solution of a Cauchy problem for the inhomogeneous three-dimensional coagulation equation”  In _Differential Equations_ 19, 1983, pp. 1568–1579
  * [6] J. Carr and F. P. Costa  “Instantaneous gelation in coagulation dynamics”  In _Zeitschrift für angewandte Mathematik und Physik ZAMP_ 43.6, 1992, pp. 974–983
  * [7] J. A. Carrillo, L. Desvillettes and K. Fellner  “Exponential decay towards equilibrium for the inhomogeneous Aizenman-Bak model”  In _Communications in Mathematical Physics_ 278, 2008, pp. 433–451
  * [8] J. A. Cañizo, L. Desvillettes and K. Fellner  “Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion”  In _Annales de l’Institut Henri Poincaré (C), Analyse non linéaire_ 27, 2010, pp. 639–654
  * [9] J. A. Cañizo, S. Mischler and C. Mouhot  “Rate of convergence to self-similarity for Smoluchowski’s coagulation equation with constant coefficients”  In _SIAM Journal on Mathematical Analysis_ 41.6, 2010, pp. 2283–2314
  * [10] D. Chae and P. Dubovskiǐ  “Existence and uniqueness for spatially inhomogeneous coagulation - condensation equation with unbounded kernels”  In _Journal of Integral Equations and Applications_ 9.3, 1997, pp. 219–236
  * [11] D. Chae and P. Dubovskiǐ  “Existence and uniqueness for spatially inhomogeneous coagulation equation with sources and effluxes”  In _Zeitschrift für angewandte Mathematik und Physik_ 47, 1995, pp. 580–594
  * [12] R. Clift, J. R. Grace and M. E. Weber  “Bubbles, Drops, and Particles”  Academic Press, 1978
  * [13] I. Cristian, B. Niethammer and J. J. L. Velázquez  “A note on instantaneous gelation for coagulation kernels vanishing on the diagonal”  In _Preprint_ , 2024
  * [14] I. Cristian and J. J. L. Velázquez  “Coagulation equations for non-spherical clusters”  In _Preprint arXiv: 2209.00644_ , 2023
  * [15] P. G. J. Dongen  “On the possible occurrence of instantaneous gelation in Smoluchowski’s coagulation equation”  In _Journal of Physics A: Mathematical and General_ 20.7, 1987, pp. 1889
  * [16] P. Dubovskiǐ  “An iterative method for solving the coagulation equation with spatially inhomogeneous velocity fields”  In _Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki_ 30.11, 1990, pp. 1755–1757
  * [17] P. Dubovskiǐ  “Solutions of a spatially inhomogeneous coagulation equation with particle fractionation taken into account”  In _Differential Equations_ 26.3, 1990, pp. 380–384
  * [18] M. Escobedo, P. Laurençot, S. Mischler and B. Perthame  “Gelation and mass conservation in coagulation-fragmentation models”  In _Journal of Differential Equations_ 195.1, 2003, pp. 143–174
  * [19] M. Escobedo, S. Mischler and B. Perthame  “Gelation in coagulation and fragmentation models”  In _Communications in Mathematical Physics_ 231, 2002, pp. 157–188
  * [20] G. Falkovich, M. G. Stepanov and M. Vucelja  “Rain initiation time in turbulent warm clouds”  In _Journal of Applied Meteorology and Climatology_ 45.4, 2006, pp. 591–599
  * [21] M. A. Ferreira, J. Lukkarinen, A. Nota and J. J. L. Velázquez  “Localization in stationary non-equilibrium solutions for multicomponent coagulation systems”  In _Communications in Mathematical Physics_ 388.1, 2021, pp. 479–506
  * [22] N. Fournier and P. Laurençot  “Existence of self-similar solutions to Smoluchowski’s coagulation equation”  In _Communications in Mathematical Physics_ 256, 2005, pp. 589–609
  * [23] V. A. Galkin  “Generalized solution of the Smoluchowski kinetic equation for spatially inhomogeneous systems”  In _Doklady Akademii Nauk SSSR_ 293.1, 1987, pp. 74–77
  * [24] V. A. Galkin  “The Smoluchowski equation of the kinetic theory of coagulation for spatially non-uniform systems”  In _Doklady Akademii Nauk SSSR_ 285, 1985, pp. 1087–1091
  * [25] P. Horvai, S. V. Nazarenko and T. H. M. Stein  “Coalescence of particles by differential sedimentation”  In _Journal of Statistical Physics_ 130, 2007, pp. 1177–1195
  * [26] P. Laurençot  “On a class of continuous coagulation-fragmentation equations”  In _Journal of Differential Equations_ 167.2, 2000, pp. 245–274
  * [27] P. Laurençot and S. Mischler  “The continuous coagulation-fragmentation equations with diffusion”  In _Archive for Rational Mechanincs and Analysis_ 162, 2002, pp. 45–99
  * [28] A. Majda  “Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables”  Springer New York, NY, 2012
  * [29] G. Menon and R. L. Pego  “Approach to self-similarity in Smoluchowski’s coagulation equation”  In _Communications on Pure and Applied Mathematics_ 57.9, 2004, pp. 1197–1232
  * [30] G. Menon and R. L. Pego  “Dynamical scaling in Smoluchowski’s coagulation equations: uniform convergence”  In _SIAM Review_ 48.4, 2006, pp. 745–768
  * [31] B. Niethammer and J. J. L. Velázquez  “Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with locally bounded kernels”  In _Commununications in Mathematical Physics_ 318, 2013, pp. 505–532
  * [32] H. R. Pruppacher and J. D. Klett  “Microphysics of Clouds and Precipitation”  Springer Dordrecht, 1997
  * [33] I. W. Stewart  “A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels”  In _Mathematical Methods in the Applied Sciences_ 11.5, 1989, pp. 627–648
  * [34] H. Tanaka, S. Inaba and K. Nakazawa  “Steady-state size distribution for the self-similar collision cascade”  In _Icarus_ 123.2, 1996, pp. 450–455
  * [35] S. Throm  “Stability and uniqueness of self-similar profiles in $L^{1}$ spaces for perturbations of the constant kernel in Smoluchowski’s coagulation equation”  In _Communications in Mathematical Physics_ 383, 2021, pp. 1361 –1407
  * [36] S. Throm  “Uniqueness of fat-tailed self-similar profiles to Smoluchowski’s coagulation equation for a perturbation of the constant kernel”  In _Memoirs of the American Mathematical Society_ 271.1328, 2021
  * [37] H. Tran and T.-S. Van  “Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel”  In _Communications on Pure and Applied Mathematics_ 75.6, 2019, pp. 1292–1331
  * [38] V. E. Zakharov and N. N. Filonenko  “Weak turbulence of capillary waves”  In _Journal of Applied Mechanics and Technical Physics_ 8.5, 1967, pp. 37–40