Lifespan estimates for semilinear damped wave equation in a two-dimensional
exterior domain

Masahiro Ikeda***Center for Advanced Intelligence Project, RIKEN, Nihonbashi
1-chome Mitsui Building, 15th floor, 1-4-1 Nihonbashi,Chuo-ku, Tokyo,
103-0027, Japan, E-mail<EMAIL_ADDRESS>Motohiro Sobajima†††
Department of Mathematics, Faculty of Science and Technology, Tokyo University
of Science, 2641 Yamazaki, Noda-shi, Chiba, 278-8510, Japan, E-mail:
<EMAIL_ADDRESS>Koichi Taniguchi‡‡‡ Advanced Institute for Materials
Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan,
E-mail<EMAIL_ADDRESS>and Yuta Wakasugi§§§ Laboratory of
Mathematics, Graduate School of Advanced Science and Engineering, Hiroshima
University, Higashi-Hiroshima, 739-8527, Japan, E-mail:
<EMAIL_ADDRESS>
  * Abstract. Lifespan estimates for semilinear damped wave equations of the form $\partial_{t}^{2}u-\Delta u+\partial_{t}u=|u|^{p}$ in a two dimensional exterior domain endowed with the Dirichlet boundary condition are dealt with. For the critical case of the semilinear heat equation $\partial_{t}v-\Delta v=v^{2}$ with the Dirichlet boundary condition and the initial condition $v(0)=\varepsilon f$, the corresponding lifespan can be estimated from below and above by $\exp(\exp(C\varepsilon^{-1}))$ with different constants $C$. This paper clarifies that the same estimates hold even for the critical semilinear damped wave equation in the exterior of the unit ball under the restriction of radial symmetry. To achieve this result, a new technique to control $L^{1}$-type norm and a new Gagliardo–Nirenberg type estimate with logarithmic weight are introduced.

Mathematics Subject Classification (2020): Primary:35L20, Secondary:35L71.

Key words and phrases: Damped wave equations, two-dimensional exterior
problems, lifespan estimates.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Abstract version of Matsumura estimates
    2. 2.2 Dirichlet heat semigroup on the exterior domain $B^{c}$
    3. 2.3 Some functional inequalities
  3. 3 Linear decay estimates for $g\in L_{\rm rad}^{2}\cap L_{d\mu}^{1}$
    1. 3.1 A Matsumura estimate with logarithmic weight
    2. 3.2 A positively preserving property
    3. 3.3 An $L_{d\mu}^{1}$-estimate for $g\in L_{\rm rad}^{2}\cap L_{d\mu}^{1}$
  4. 4 Estimates for semilinear problem
    1. 4.1 A Gagliardo–Nirenberg type inequality with logarithmic weight
    2. 4.2 Lower bound for lifespan of semilinear problem

##  1 Introduction

In this paper, we consider the initial-boundary value problem of the
semilinear damped wave equation in the exterior of the two-dimensional closed
unit ball $B=\\{x\in\mathbb{R}^{2}\;;\;|x|\leq 1\\}$, that is,

$\begin{cases}\partial_{t}^{2}u(x,t)-\Delta
u(x,t)+\partial_{t}u(x,t)=|u(x,t)|^{p}&\text{in}\ B^{c}\times(0,T),\\\
u(x,t)=0&\text{on}\ \partial B^{c}\times(0,T),\\\
(u,\partial_{t}u)(x,0)=(0,\varepsilon g(x)),&\text{in}\ B^{c}.\end{cases}$
(1.1)

Here $p>1$ indicates the structure of the nonlinear term. The function
$g:B^{c}\to\mathbb{R}$ is given and $u:B^{c}\times[0,T)\to\mathbb{R}$ is
unknown. The constant $\varepsilon>0$ is the parameter describing the
smallness of initial data. Our interest is the behavior of solutions to (1.1)
with small initial data.

For the semilinear heat equation

$\begin{cases}\partial_{t}v(x,t)-\Delta v(x,t)=v(x,t)^{p}&\text{in}\
\mathbb{R}^{N}\times(0,T),\\\ v(x,0)=f(x)\geq 0,&\text{in}\
\mathbb{R}^{N},\end{cases}$ (1.2)

from the pioneering work [4] by Fujita, there are many papers dealing with the
existence/nonexistence of global solutions to (1.2) (see Quittner–Souplet
[17]). Nowadays, the exponent $p_{F}(N)=1+\frac{2}{N}$ is well-known as the
threshold for dividing the situation of the existence/nonexistence of
nonnegative global solutions, namely,

  * •

if $1<p\leq p_{F}(N)$, then (1.2) does not possess non-trivial global
solutions;

  * •

if $p>p_{F}(N)$, then (1.2) possesses a non-trivial global solution.

The similar phenomenon occurs also for the semilinear damped wave equation

$\begin{cases}\partial_{t}^{2}u(x,t)-\Delta
u(x,t)+\partial_{t}u(x,t)=|u(x,t)|^{p}&\text{in}\
\mathbb{R}^{N}\times(0,T),\\\
(u,\partial_{t}u)(x,0)=(u_{0}(x),u_{1}(x)),&\text{in}\
\mathbb{R}^{N}.\end{cases}$ (1.3)

The pioneering work for the problem (1.3) is the paper [12] by Matsumura via
the analysis of the profile of linear solutions in the following inequalities

$\displaystyle\|\partial_{t}^{k}\partial_{x}^{\alpha}u(t)\|_{L^{\infty}(\mathbb{R}^{N})}\leq
C(1+t)^{-\frac{N}{2}-k-\frac{|\alpha|}{2}}(\|u_{0}\|_{H^{m+1}\cap
L^{1}(\mathbb{R}^{N})}+\|u_{1}\|_{H^{m}\cap L^{1}(\mathbb{R}^{N})}),$ (1.4)
$\displaystyle\|\partial_{t}^{k}\partial_{x}^{\alpha}u(t)\|_{L^{2}(\mathbb{R}^{N})}\leq
C(1+t)^{-\frac{N}{4}-k-\frac{|\alpha|}{2}}(\|u_{0}\|_{H^{\widetilde{m}+1}\cap
L^{1}(\mathbb{R}^{N})}+\|u_{1}\|_{H^{\widetilde{m}}\cap
L^{1}(\mathbb{R}^{N})})$ (1.5)

(with $m=[\frac{N}{2}]+k+|\alpha|$ and $\widetilde{m}=k+|\alpha|-1$), which
are so-called “Matsumura estimates”. Until Todorova–Yordanov [20] and Zhang
[21], (a rough description of) the situation of the existence/nonexistence of
non-trivial (small) global solutions to (1.3) is clarified as follows:

  * •

if $1<p\leq p_{F}(N)$ and $\int_{\mathbb{R}^{N}}(u_{0}+u_{1})\,dx>0$, then
(1.3) does not possess non-trivial global solutions (a kind of smallness does
not provide global solutions);

  * •

if $p>p_{F}(N)$, then (1.2) possesses a non-trivial global solution.

After that, the precise estimates of the lifespan $T_{\varepsilon}$ (maximal
existence time) of solutions to (1.3) with the small initial data
$(\varepsilon f,\varepsilon g)$ became the subject of interest. It is firstly
discussed in Li–Zhou [11], and until Lai–Zhou [10] the sharp lifespan
estimates are clarified as the following: for sufficiently small
$\varepsilon>0$,

$\displaystyle\begin{cases}c_{p}\varepsilon^{-(\frac{1}{p-1}-\frac{N}{2})^{-1}}\leq
T_{\varepsilon}\leq
C_{p}\varepsilon^{-(\frac{1}{p-1}-\frac{N}{2})^{-1}}&\text{if}\
1<p<p_{F}(N),\\\ \exp(c_{p}\varepsilon^{-(p-1)})\leq
T_{\varepsilon}\leq\exp(C_{p}\varepsilon^{-(p-1)})&\text{if}\
p=p_{F}(N)\end{cases}$ (1.6)

for some positive constants $c_{p}$ and $C_{p}$ (independent of
$\varepsilon$); note that these estimates are completely same as the ones for
the blowup time of solutions to the semilinear heat equation (1.2) with the
initial condition $v(x,0)=\varepsilon f(x)$ (having the small parameter
$\varepsilon$).

In the case of the problem of the semilinar heat equation in an
$N$-dimensional exterior domain $\Omega$ with the Dirichlet boundary
condition:

$\begin{cases}\partial_{t}v(x,t)-\Delta v(x,t)=v(x,t)^{p}&\text{in}\
\Omega\times(0,T),\\\ v(x,t)=0&\text{on}\ \Omega\times(0,T),\\\
v(x,0)=f(x)\geq 0,&\text{in}\ \Omega,\end{cases}$ (1.7)

the situation is almost similar as in the case of the whole space when $N\geq
3$. Actually, for the linear case, Grigor’yan and Saloff-Coste [5] discussed
the asymptotic behavior of the Dirichlet heat kernel for the exterior of a
compact set in Riemannian manifolds. It is shown that the Dirichlet heat
kernel for an $N$-dimensional exterior domain $(N\geq 3)$ behaves like the one
for $\mathbb{R}^{N}$ in the far field. In contrast, the Dirichlet heat kernel
in two-dimensional exterior domains cannot be approximated by the one for
$\mathbb{R}^{2}$. This fact can be explained as the transient (recurrence)
property for $N\geq 3$ ($N=2$) of the Brownian motion in $\mathbb{R}^{N}$.
This significant difference reflects the difficulty of the case of two-
dimensional exterior problem. By using the Kaplan’s method via the Dirichlet
heat kernel found in [5], Pinsky [15] tried to draw the complete picture of
the existence/nonexistence of global solutions to (1.7), however, the critical
case $p=p_{F}(2)=2$ seems to have a gap in his proof. Later, nonexistence of
global solutions for the case $p=2$ is proved in Ikeda–Sobajima [6] via a
sharpened test function method with a distinctive shape of the (double
exponential type) lifespan estimate

$T_{\varepsilon}\leq\exp(\exp(C\varepsilon^{-1}))$ (1.8)

for the solution with the initial condition $v(x,0)=\varepsilon f(x)$. One can
prove that this is actually the sharp lifespan estimate via the supersolution-
subsolution method (explained in [17, Section 20]) with a supersolution

$U(x,t)=\alpha(t)e^{t\Delta_{\Omega}}f,\quad\alpha(t)=\varepsilon\left(1-(p-1)\varepsilon^{p-1}\int_{0}^{t}\|e^{s\Delta_{\Omega}}f\|_{L^{\infty}(\Omega)}^{p-1}\,ds\right)^{-\frac{1}{p-1}}$

with the $L^{\infty}$-estimate of the Dirichlet heat semigroup
$e^{t\Delta_{\Omega}}:$

$\|e^{t\Delta_{\Omega}}f\|_{L^{\infty}(\Omega)}\leq
Ch(t)\big{(}\|(\log|x|)f\|_{L^{1}(\Omega)}+\|f\|_{L^{\infty}(\Omega)}\big{)},\quad
t>0$ (1.9)

with the decay rate involving the logarithmic function

$h(t)=\frac{1}{(1+t)(1+\log(1+t))}.$ (1.10)

This is valid only for the case of two-dimensional (general) exterior domains.

For the semilinear damped wave equation (1.1), the existence/nonexistence of
solutions to (1.1) also has been dealt with in the literature (e.g., Ikehata
[7] for the existence and Ogawa–Takeda [13] for the nonexistence). Although
the effect of the recurrence of the Brownian motion in $\mathbb{R}^{2}$ could
appear also in the analysis of the damped wave equation, however, studies from
such a viewpoint are few. Only in [6], one can find that the lifespan estimate
of the solution $u$ to the semilinear damped wave equation (1.1) has the same
upper bound (as in (1.8)) as the case of the semilinear heat equation. In this
connection, the question about the sharpness of this lifespan estimate
naturally arises. The purpose of the present paper is to address this problem,
that is, to clarify the sharpness of the (double exponential type) lifespan
estimate for the two-dimensional exterior problem of the semilinear damped
wave equation (1.1).

To state the result, we clarify the definition of solutions to (1.1) as
follows.

######  Definition 1.1.

For general open set $\Omega$ in $\mathbb{R}^{N}$ (with a smooth boundary), we
denote $\Delta_{\Omega}$ as the Laplacian endowed with the domain
$D(\Delta_{\Omega})=H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$ (which describes the
Laplace operator $\Delta$ involving the Dirichlet boundary condition). Then we
define that $u:B^{c}\times[0,T)\to\mathbb{R}$ is a weak solution of (1.1) in
$(0,T)$ with the initial condition $(u,\partial_{t}u)(0)=(u_{0},u_{1})\in
H_{0}^{1}(B^{c})\times L^{2}(B^{c})$ if $u\in C^{1}([0,T);L^{2}(B^{c}))\cap
C([0,T);H_{0}^{1}(B^{c}))$ and

$\big{(}u(t),\partial_{t}u(t)\big{)}=e^{t\mathcal{L}_{B^{c}}}(u_{0},u_{1})+\int_{0}^{t}e^{(t-s)\mathcal{L}_{B^{c}}}(0,|u(s)|^{p})\,ds,\quad
t\in(0,T),$

where $(e^{t\mathcal{L}_{B^{c}}})_{t\geq 0}$ is the $C_{0}$-semigroup on
$\mathcal{H}=H_{0}^{1}(B^{c})\times L^{2}(B^{c})$ generated by
$\mathcal{L}_{B^{c}}(u,v)=(v,\Delta_{B^{c}}u-v)$ with the domain
$D(\mathcal{L}_{B^{c}})=D(\Delta_{B^{c}})\times H_{0}^{1}(B^{c})$.

######  Remark 1.1.

It turns out that the following representation of solution $u$ of (1.1) with
initial condition $(u,\partial_{t}u)(0)=(u_{0},u_{1})$ is also valid:

$u(t)=\partial_{t}S(t)u_{0}+S(t)(u_{0}+u_{1})+\int_{0}^{t}S(t-s)|u(s)|^{p}\,ds,$

where $S(t)g:=P_{1}e^{t\mathcal{L}_{B^{c}}}(0,g)$ with the projection
$P_{1}(u,v)=u$. Our argument in the present paper relies on this
representation.

Existence and uniqueness of solutions to (1.1) and also the blowup alternative
are well-known (see e.g., Ikehata [7]).

######  Proposition 1.1.

The following assertions hold:

  * (i)

For every $(u_{0},u_{1})\in H_{0}^{1}(B^{c})\times L^{2}(B^{c})$, there exist
a positive constant $T$ and $u\in C^{1}([0,T);L^{2}(B^{c}))\cap
C([0,T);H_{0}^{1}(B^{c}))$ such that $u$ is a unique weak solution of (1.1) in
$(0,T)$ with the initial condition $(u,\partial_{t}u)(0)=(u_{0},u_{1})$.

  * (ii)

The weak solution $u$ of (1.1) in a bounded interval $(0,T)$ cannot be able to
extend to a solution in a wider interval if and only if

$\lim_{t\to T}\Big{(}\|\partial_{t}u(t)\|_{L^{2}(B^{c})}+\|\nabla
u(t)\|_{L^{2}(B^{c})}+\|u(t)\|_{L^{2}(B^{c})}\Big{)}=+\infty.$ (1.11)

By virtue of Proposition 1.1, we can define the lifespan of solutions to
(1.1).

######  Definition 1.2.

Define the lifespan $T_{\max}(u_{0},u_{1})\in(0,\infty]$ as the maximal
existence time of weak solutions to (1.1) with the initial condition
$(u,\partial_{t}u)(0)=(u_{0},u_{1})$. Namely,

$T_{\max}(u_{0},u_{1})=\sup\big{\\{}T>0\;;\;\text{\eqref{intro:problem} has a
weak solution in $(0,T)$}\big{\\}}.$

In Ikeda–Sobajima [6], blowup of solutions to (1.1) with small initial data is
proved under the initial condition $(u,\partial_{t}u)(x,0)=(\varepsilon
f,\varepsilon g)$ with

$\int_{B^{c}}\big{(}f(x)+g(x)\big{)}\log|x|\,dx>0.$

We should point out that the weight function $\log|x|$ has been chosen as the
positive harmonic function satisfying the Dirichlet boundary condition. To
reflect the above situation and also the $L^{\infty}$-decay estimate (1.9) for
$e^{t\Delta_{B^{c}}}$ to our consideration in the present paper, we need to
introduce the following $L^{p}$-spaces with weighted measure involving
$\log|x|$.

######  Definition 1.3.

Define the measure $d\mu=(1+\log|x|)\,dx$ and for $1\leq p<\infty$,

$L_{d\mu}^{p}=\left\\{f\in
L^{p}(B^{c})\;;\;\|f\|_{L_{d\mu}^{p}}<+\infty\right\\},\quad\|f\|_{L_{d\mu}^{p}}=\left(\int_{B^{c}}|f(x)|^{p}\,d\mu\right)^{\frac{1}{p}}.$

Now we are in a position to state our result for the lower bound for the
lifespan estimate of solutions to (1.1) under the radially symmetric setting.
The following assertion is formulated in the subspace of radially symmetric
functions in $L^{2}(B^{c})$:

$L^{2}_{\rm rad}=\\{f\in L^{2}(B^{c})\;;\;\text{$f$ is radially
symmetric}\\}.$

######  Theorem 1.2.

If $g\in L_{\rm rad}^{2}\cap L_{d\mu}^{1}$, then the following assertions
hold:

  * (i)

If $1<p\leq 2$, then there exist positive constants $\varepsilon_{0}>0$ and
$c>0$ such that for $\varepsilon\in(0,\varepsilon_{0}]$,

$T_{\max}(0,\varepsilon
g)\geq\begin{cases}c\left(\dfrac{1}{\varepsilon}\log\dfrac{1}{\varepsilon}\right)^{\frac{p-1}{2-p}}&\text{if}\
1<p<2,\\\\[3.0pt] \exp(\exp(c\varepsilon^{-1}))&\text{if}\ p=2.\end{cases}$

  * (ii)

If $2<p<\infty$, then there exist positive constants $\delta$ and $C$ such
that if $\|g\|_{L^{2}(B^{c})}+\|g\|_{L_{d\mu}^{1}}\leq\delta$, then
$T_{\max}(0,g)=+\infty$ with

$\displaystyle\|u(t)\|_{L_{d\mu}^{1}}$ $\displaystyle\leq C\delta,\quad t>0,$
$\displaystyle\|\nabla u(t)\|_{L^{2}(B^{c})}$
$\displaystyle\leq\frac{C\delta}{(1+t)(1+\log(1+t))},\quad t>0,$
$\displaystyle\|\partial_{t}u(t)\|_{L^{2}(B^{c})}$
$\displaystyle\leq\frac{C\delta}{(1+t)^{\frac{3}{2}}(1+\log(1+t))},\quad t>0.$

The lower bounds of the lifespan in Theorem 1.2 (i) are derived from the
following relation:

$c_{p}\leq\varepsilon^{p-1}\int_{0}^{T_{\max}(0,\varepsilon g)}h(t)^{p-1}\,dt$

for some small constant $c_{p}$, where $h(t)$ is given in (1.10). Combining
Theorem 1.2 (i) with the upper bound of the lifespan given in Ikeda–Sobajima
[6], we obtain the following:

######  Corollary 1.3.

Let $g\in L^{2}_{\rm rad}\cap L^{1}_{d\mu}$ and let $T_{\max}$ be in
Definition 1.2. If $g\geq 0$ and $g\not\equiv 0$, then for every $1<p\leq 2$,
one has

$\displaystyle 0<\liminf_{\varepsilon\to
0}\Big{(}\varepsilon^{p-1}\int_{0}^{T_{\max}(0,\varepsilon
g)}h(t)^{p-1}\,dt\Big{)}\leq\limsup_{\varepsilon\to
0}\Big{(}\varepsilon^{p-1}\int_{0}^{T_{\max}(0,\varepsilon
g)}h(t)^{p-1}\,dt\Big{)}<+\infty.$

######  Remark 1.2.

We do not know whether the quantity

$\varepsilon^{p-1}\int_{0}^{T_{\max}(0,\varepsilon g)}h(t)^{p-1}\,dt$

converges to a constant as $\varepsilon\to 0$ or not.

The counter part about the global existence, we also have the assertion for
the total energy decay.

######  Corollary 1.4.

Let $2<p<\infty$ and let $u$ be the unique global solution of (1.1) obtained
in Theorem 1.2. Then one has the energy decay estimate for $u$:

$\int_{B^{c}}\Big{(}|\nabla
u(t)|^{2}+(\partial_{t}u(t))^{2}\Big{)}\,dx\leq\frac{C^{2}\delta^{2}}{(1+t)^{2}(1+\log(1+t))^{2}},\quad
t>0.$

######  Remark 1.3.

Ono [14] obtained the energy decay estimate

$\int_{B^{c}}\Big{(}|\nabla
u(t)|^{2}+(\partial_{t}u(t))^{2}\Big{)}\,dx\leq\frac{C_{\delta}}{(1+t)^{2-\delta}},\quad
t>0$

for the solution $u$ of linear damped wave equation with initial data
$(u,\partial_{t}u)(0)=(u_{0},u_{1})\in[H_{0}^{1}\cap
L^{1}(B^{c})]\times[L^{2}\cap L^{1}(B^{c})]$. After that, Ikehata [7] removed
the loss $\delta$ of the decay rate by assuming $|x|\log|x|(u_{0}+u_{1})\in
L^{2}(B^{c})$. The decay estimate in Corollary 1.4 (even though it is the
semilinear problem) is faster than those by a different assumption
$(u,\partial_{t}u)(0)=(0,g)$ with $g\in L^{2}_{\rm rad}\cap L^{1}_{d\mu}$.
Incidentally, one can find that the decay of total energy in Corollary 1.4 is
faster than the local energy decay proved in Dan–Shibata [2].

Now we shall describe the strategy of the present paper. Here we only focus
our attention to the two-dimensional case. If we move to the semilinear heat
equation (of course in an exterior domain), we can employ the supersolution-
subsolution method as explained before. However, it seems impossible to apply
this argument for the hyperbolic equation (1.1). Instead of this defect, in
the case of the Cauchy problem of the semilinear damped wave equation (1.3),
the positively preserving property of the linear solution map
$S_{*}(t)=P_{1}e^{t\mathcal{L}_{\mathbb{R}^{2}}}$ seems useful to control the
$L^{1}$-type norm of the solution $S_{*}(t)g$. Together with the basic energy
functional of the form $\|\partial_{t}u\|_{L^{2}(\mathbb{R}^{2})}^{2}+\|\nabla
u\|_{L^{2}(\mathbb{R}^{2})}^{2}$, this consideration enables us to control the
$L^{1}$-norm of the semilinear term $|u|^{p}$ via the Gagliardo–Nirenberg
inequality

$\|f\|_{L^{p}(\mathbb{R}^{2})}\leq C_{p}\|\nabla
f\|_{L^{2}(\mathbb{R}^{2})}^{1-\frac{1}{p}}\|f\|_{L^{1}(\mathbb{R}^{2})}^{\frac{1}{p}}\quad
f\in H^{1}(\mathbb{R}^{2})\cap L^{1}(\mathbb{R}^{2})$

(when $p=2$, this is called the Nash inequality). We emphasize that the
$L^{1}$-norm is a conserved quantity of the linear heat semigroup
$e^{t\Delta_{\mathbb{R}^{2}}}$ (for nonnegative initial data) which represents
the asymptotic behavior of solutions (1.2). Employing this procedure with the
Matsumura estimate (1.5), we can reach the sharp lower bound of lifespan of
the solution to (1.3) with the initial condition
$(u,\partial_{t}u)(0)=(0,\varepsilon g)$ with $g\in L^{2}(\mathbb{R}^{2})\cap
L^{1}(\mathbb{R}^{2})$. One of the novelty of the present paper is to
introduce this kind of strategy in the analysis of the semilinear damped wave
equation.

Let us go back to the target problem (1.1). In this case, a conserved quantity
for the Dirichlet heat semigroup $e^{t\Delta_{B_{c}}}$ can be chosen as

$I[f]=\int_{B_{c}}f(x)\log|x|\,dx,$

where the weight $\log|x|$ is chosen as a positive harmonic function
satisfying the Dirichlet boundary condition. Then it turns out that the
behavior of the above quantity in the problem (1.1) can be tracked by the
ordinary differential equation

$\displaystyle\left(\frac{d^{2}}{dt^{2}}+\frac{d}{dt}\right)\int_{B_{c}}u(x,t)\log|x|\,dx=\int_{B_{c}}|u(x,t)|^{p}\log|x|\,dx.$

This suggests that the following version of the Gagliardo–Nirenberg type
inequality seems reasonable:

$\int_{B^{c}}|f|^{p}\log|x|\,dx\leq\widetilde{C}_{p}\|\nabla
f\|_{L^{2}(B^{c})}^{p-1}\int_{B^{c}}|f|\log|x|\,dx,\quad f\in
H^{1}_{0}(B^{c})\cap L_{d\mu}^{1},$

which is not discussed so far. To proceed this strategy, the positively
preserving property of the linear solution map $S(t)$ is essential. To justify
this property, additionally we assume the radial symmetry. Moreover, to reach
the sharp lifespan estimate, we also use the corresponding Matsumura type
estimate which should have a logarithmic decay factor as in (1.9). This could
be proved via the diffusion phenomenon with the decay estimate via the
estimate for the Dirichlet heat semigroup $e^{t\Delta_{B^{c}}}$.

The present paper is organized as follows. In section 2, we collect the
important tools in this paper. More precisely, an abstract version of
Matsumura estimates, decay estimates for the Dirichlet heat semigroup
$e^{t\Delta_{B^{c}}}$ (an alternative proof is written in Appendix), the usual
Gagliardo–Nirenberg estimates and the critical Hardy inequality are listed. In
section 3, the linear solution $S(t)g$ is analysed. Here we prove a modified
Matsumura estimate with decay involving logarithmic factor, the positively
preserving property and an $L^{1}_{d\mu}$-estimate for $S(t)g$. Section 4 is
devoted to the proof of the lifespan estimate from below. A
Gagliardo–Nirenberg inequality with logarithmic weight is proved by the use of
the critical Hardy inequality at the beginning of Section 4.

##  2 Preliminaries

####

In this section we collect several important tools to analyse the target
problem (1.1).

###  2.1 Abstract version of Matsumura estimates

We first state an abstract version of Matsumura estimates which is valid for
the second order differential equation

$\begin{cases}u^{\prime\prime}(t)+Au(t)+u^{\prime}(t)=0,\quad t>0,\\\
(u,u^{\prime})(0)=(u_{0},u_{1})\end{cases}$ (2.1)

in a Hilbert space $H$. Here $A$ is a nonnegative selfadjoint operator in $H$
endowed with domain $D(A)$. Existence and uniqueness of solutions to (2.1) are
verified via the well-known Hille–Yosida theorem. Here we denote by
$S_{A}(t)g$ the solution $u\in C^{1}([0,\infty);H)\cap
C([0,\infty);D(A^{1/2}))$ of (2.1) with the initial condition
$(u,u^{\prime})(0)=(0,g)$. Note that by using $S_{A}(t)$, one can find that
the solution of (2.1) has a representation

$u(t)=\frac{d}{dt}[S_{A}(t)u_{0}]+S_{A}(t)[u_{0}+u_{1}].$

The following lemma provides the asymptotic profile of $S_{A}(t)g$ (in the
sense of the energy functional), which can be described by using the
$C_{0}$-semigroup $(e^{-tA})_{t\geq 0}$ on $H$ generated by $-A$.

######  Lemma 2.1 (Radu–Todorova–Yordanov [18]).

The following assertions hold:

  * (i)

There exists a positive constant $C_{{\rm M},1}$ such that for every $g\in H$
and $t\geq 1$,

$\displaystyle\big{\|}A^{1/2}(S_{A}(t)-e^{-tA})g\big{\|}_{H}\leq C_{{\rm
M},1}\Big{(}t^{-\frac{3}{2}}\|e^{-\frac{t}{2}A}g\|_{H}+e^{-\frac{t}{16}}\|(A^{1/2}+1)^{-1}A^{1/2}g\|_{H}\Big{)}.$

  * (ii)

There exists a positive constant $C_{{\rm M},2}$ such that for every $g\in H$
and $t\geq 1$,

$\displaystyle\left\|\frac{d}{dt}(S_{A}(t)-e^{-tA})g\right\|_{H}\leq C_{{\rm
M},2}t^{-2}\Big{(}\|g\|_{H}+e^{-\frac{t}{4}}\|g\|_{H}\Big{)}.$

######  Remark 2.1.

The assertion (ii) in Lemma 2.1 is not written explicitly in [18], but the
same strategy also provides the estimate for the derivative in $t$.

In our situation, we employ Lemma 2.1 with the (negative) Dirichlet Laplacian
$-\Delta_{B^{c}}$ in the Hilbert space $L^{2}(B^{c})$. The corresponding
linear damped wave equation is as follows:

$\begin{cases}\partial_{t}^{2}u(x,t)-\Delta
u(x,t)+\partial_{t}u(x,t)=0&\text{in}\ B^{c}\times(0,\infty),\\\
u(x,t)=0&\text{on}\ \partial B^{c}\times(0,\infty),\\\
(u,\partial_{t}u)(x,0)=(0,g(x)),&\text{in}\ B^{c}.\end{cases}$ (2.2)

The classical energy identity for the solution
$S(t)g=P_{1}e^{t\mathcal{L}_{B^{c}}}(0,g)$ (given in Remark 1.1) provides the
following basic inequality.

######  Lemma 2.2.

For every $g\in L^{2}(B^{c})$, one has

$\|\nabla
S(t)g\|_{L^{2}(B^{c})}^{2}+\|\partial_{t}S(t)g\|_{L^{2}(B^{c})}^{2}\leq\|g\|_{L^{2}(B^{c})}^{2},\quad
t\geq 0.$

The following is the result by [18] applied to the case (2.2).

######  Lemma 2.3.

There exist positive constants $C_{{\rm M},1}^{\prime}$ and $C_{{\rm
M},2}^{\prime}$ such that for every $g\in L^{2}(B^{c})$ and $t\geq 1$,

$\displaystyle\|\nabla(S(t)-e^{t\Delta_{B^{c}}})g\|_{L^{2}(B^{c})}$
$\displaystyle\leq C_{{\rm
M},1}^{\prime}t^{-\frac{3}{2}}\|g\|_{L^{2}(B^{c})},$
$\displaystyle\|\partial_{t}(S(t)-e^{t\Delta_{B^{c}}})g\|_{L^{2}(B^{c})}$
$\displaystyle\leq C_{{\rm M},2}^{\prime}t^{-2}\|g\|_{L^{2}(B^{c})}.$

###  2.2 Dirichlet heat semigroup on the exterior domain $B^{c}$

Here we state the $L^{p}$-$L^{q}$ type estimates for $e^{t\Delta_{B^{c}}}$
with logarithmic weight. Although it can be obtained via the heat kernel
estimate in [5] (as explained in Introduction), in Appendix we give an
alternative proof based on the technique of the analysis of partial
differential equations of parabolic type.

In the discussion of the present paper, the following estimates are crucial.

######  Lemma 2.4.

For every $q\in[1,2]$, there exists a positive constant $C_{{\rm H},q}>0$ such
that if $f\in L_{d\mu}^{q}$, then for every $t>0$, one has
$e^{t\Delta_{B^{c}}}f\in L^{2}(B^{c})$ with

$\|e^{t\Delta_{B^{c}}}f\|_{L^{2}(B^{c})}\leq\frac{C_{{\rm
H},q}}{t^{\frac{1}{q}-\frac{1}{2}}(1+\log(1+t))^{\frac{1}{q}}}\|f\|_{L_{d\mu}^{q}}.$

###  2.3 Some functional inequalities

In the present paper, the following form of the Gagliardo–Nirenberg
inequalities will be used (cf. Friedman [3]); note that the following
inequality with $q=2$ is also called the Nash inequality.

######  Lemma 2.5.

For every $1<q<\infty$, one has $H_{0}^{1}(B^{c})\cap L^{1}(B^{c})\subset
L^{q}(B^{c})$ and there exists a positive constant $C_{{\rm GN},q}$ such that

$\|f\|_{L^{q}(B^{c})}\leq C_{{\rm GN},q}\|\nabla
f\|_{L^{2}(B^{c})}^{1-\frac{1}{q}}\|f\|_{L^{1}(B^{c})}^{\frac{1}{q}},\quad\forall
f\in H_{0}^{1}(B^{c})\cap L^{1}(B^{c}).$

To elicit the effect from the boundary (in two dimension), we also use the
critical case of the Hardy inequality (cf. Ladyzhenskaya [9] and also
Dan–Shibata [2]). For the reader’s convenience, we give a short proof.

######  Lemma 2.6.

If $f\in H_{0}^{1}(B^{c})$, then

$\frac{1}{4}\int_{B^{c}}\frac{f^{2}}{|x|^{2}(1+\log|x|)^{2}}\,dx\leq\int_{B^{c}}|\nabla
f|^{2}\,dx.$

###### Proof.

By density, it suffices to discuss the estimate for $f\in
C_{0}^{\infty}(B^{c})$. To simplify the notation, we set $H(x)=1+\log|x|$
which is positive and harmonic in $B^{c}$. Then by using the transform
$g=H^{-\frac{1}{2}}f\in C_{0}^{\infty}(B^{c})$ and integration by parts, we
can calculate as follows:

$\displaystyle\int_{B^{c}}|\nabla f|^{2}\,dx$
$\displaystyle=\int_{B^{c}}\Big{(}|\nabla g|^{2}H+g\nabla g\cdot\nabla
H+\frac{|\nabla H|^{2}}{4H}g^{2}\Big{)}\,dx$
$\displaystyle=\int_{B^{c}}|\nabla g|^{2}H\,dx+\int_{B^{c}}\left(-\frac{\Delta
H}{2H}+\frac{|\nabla H|^{2}}{4H^{2}}\right)f^{2}\,dx.$

This gives the desired inequality. ∎

##  3 Linear decay estimates for $g\in L_{\rm rad}^{2}\cap L_{d\mu}^{1}$

####

In this section we study the decay property of the solution $u(t)=S(t)g$ of
the linear damped wave equation

$\begin{cases}\partial_{t}^{2}u(x,t)-\Delta
u(x,t)+\partial_{t}u(x,t)=0&\text{in}\ B^{c}\times(0,T),\\\
u(x,t)=0&\text{on}\ \partial B^{c}\times(0,T),\\\
(u,\partial_{t}u)(x,0)=(0,g(x)),&\text{in}\ B^{c}.\end{cases}$ (3.1)

under the additional integrability condition $g\in L_{d\mu}^{1}$. Actually,
under the condition $g\in L_{d\mu}^{1}$, we can find how a compact obstacle
affects to the behavior of solutions to the damped wave equation. The harvests
are different from the usual damped wave equation in the whole space.

###  3.1 A Matsumura estimate with logarithmic weight

Here we state the decay estimate for solutions to the damped wave equation
(3.1) for the initial condition $(u,\partial_{t}u)(0)=(0,g)$ with $g\in
L^{2}(B^{c})\cap L_{d\mu}^{1}$. We emphasize that as the effect from the
boundary, the following Matsumura type estimates have the decay rate involving
the logarithmic function. This fact can be shown via the use of the
$L_{d\mu}^{q}$-$L^{2}$ estimates for the Dirichlet heat semigroup
$e^{t\Delta_{B^{c}}}$.

######  Lemma 3.1.

For every $q\in[1,2]$, there exist positive constants $C_{{\rm
M},1,q}^{\sharp}$ and $C_{{\rm M},2,q}^{\sharp}$ such that if $g\in
L^{2}(B^{c})\cap L_{d\mu}^{q}$, then for every $t>0$,

$\displaystyle\|\nabla S(t)g\|_{L^{2}(B^{c})}$ $\displaystyle\leq C_{{\rm
M},1,q}^{\sharp}h(t)^{\frac{1}{q}}(\|g\|_{L_{d\mu}^{q}}+\|g\|_{L^{2}(B^{c})}),$
$\displaystyle\|\partial_{t}S(t)g\|_{L^{2}(B^{c})}$ $\displaystyle\leq C_{{\rm
M},2,q}^{\sharp}(1+t)^{-\frac{1}{2}}h(t)^{\frac{1}{q}}(\|g\|_{L_{d\mu}^{q}}+\|g\|_{L^{2}(B^{c})}).$

###### Proof.

The case $0<t\leq 1$ is obvious via Lemma 2.2. Let $t\geq 1$ be arbitrary.
using Lemma 2.3, we see by the notation $s=t/2$ that

$\displaystyle\|\nabla S(t)g\|_{L^{2}(B^{c})}$ $\displaystyle\leq\|\nabla
e^{t\Delta_{B^{c}}}g\|_{L^{2}(B^{c})}+\|\nabla(S(t)-e^{t\Delta_{B^{c}}})g\|_{L^{2}(B^{c})}$
$\displaystyle\leq\frac{1}{s^{\frac{1}{2}}}\|e^{s\Delta_{B^{c}}}g\|_{L^{2}(B^{c})}+\frac{C_{{\rm
M},1}^{\prime}}{t^{\frac{3}{2}}}\|g\|_{L^{2}(B^{c})}$
$\displaystyle\leq\frac{C_{{\rm
H},q}}{s^{\frac{1}{q}}(1+\log(1+s))^{\frac{1}{q}}}\|g\|_{L_{d\mu}^{q}}+\frac{C_{{\rm
M},1}^{\prime}}{t^{\frac{3}{2}}}\|g\|_{L^{2}(B^{c})}$

and also

$\displaystyle\|\partial_{t}S(t)g\|_{L^{2}(B^{c})}$
$\displaystyle\leq\|\partial_{t}e^{t\Delta_{B^{c}}}g\|_{L^{2}(B^{c})}+\|\partial_{t}(S(t)-e^{t\Delta_{B^{c}}})g\|_{L^{2}(B^{c})}$
$\displaystyle\leq\frac{1}{s}\|e^{s\Delta_{B^{c}}}g\|_{L^{2}(B^{c})}+\frac{C_{{\rm
M},2}^{\prime}}{t^{2}}\|g\|_{L^{2}(B^{c})}$ $\displaystyle\leq\frac{C_{{\rm
H},q}}{s^{\frac{1}{2}+\frac{1}{q}}(1+\log(1+s))^{\frac{1}{q}}}\|g\|_{L_{d\mu}^{q}}+\frac{C_{{\rm
M},2}^{\prime}}{t^{2}}\|g\|_{L^{2}(B^{c})}.$

These imply the desired estimates. ∎

###  3.2 A positively preserving property

In this subsection, we study a positively preserving property of the solution
map $S(t)$ for the linear damped wave equation (3.1) under the radial
symmetry.

######  Proposition 3.2.

If $g\in L_{\rm rad}^{2}$ is nonnegative, then the solution $S(t)g$ of (3.1)
is also radially symmetric and nonnegative for $t>0$.

###### Proof.

It is enough to show the assertion for the functions belonging to
$D(\Delta_{B^{c}})=H^{2}(B^{c})\cap H_{0}^{1}(B^{c})$. Indeed, since the
resolvent operator $(1-\frac{1}{n}\Delta_{B^{c}})^{-1}$ preserves the
nonnegativity and the radial symmetry (for nonnegativity see e.g, Brezis [1,
Section 9.7], the radial symmetry is the consequence of rotation), if $g\in
L_{\rm rad}^{2}$, then for each $n\in\mathbb{N}$,
$g_{n}=(1-\frac{1}{n}\Delta_{\Omega})^{-1}g\in D(\Delta_{B^{c}})$ is also
nonnegative and radially symmetric. If the assertion for $D(\Delta_{B^{c}})$
holds, then we have the nonnegativity of $S(t)g_{n}$ for $n\in\mathbb{N}$.
Noting that $g_{n}\to g$ in $L^{2}(B^{c})$ as $n\to\infty$ and recalling
$S(t)g_{n}=P_{1}e^{t\mathcal{L}_{B^{c}}}(0,g_{n})$ with the projection
$P_{1}(u,v)=u$, we see by the continuity of $e^{t\mathcal{L}_{B^{c}}}$ in
$H_{0}^{1}(B^{c})\times L^{2}(B^{c})$ that $S(t)g_{n}\to S(t)g$ in
$H_{0}^{1}(B^{c})$ as $n\to\infty$ which implies the nonnegativity and the
radial symmetry of $S(t)g$.

Now we assume $g\in D(\Delta_{B^{c}})=H^{2}(B^{c})\cap H^{1}_{0}(B^{c})$.
Then, the corresponding solution $u=S(t)g$ has the regularity

$\displaystyle u\in C([0,\infty);H^{3}(B^{c}))\cap
C^{1}([0,\infty);H^{2}(B^{c}))\cap C^{2}([0,\infty);H^{1}(B^{c}))$

with the boundary condition $u,\Delta u\in C([0,\infty);H_{0}^{1}(B^{c}))$ and
satisfies the equation of (2.2) in $H_{0}^{1}(B^{c})$. Since $g$ is radially
symmetric, so is $u(\cdot,t)=S(t)g$ (as the consequence of rotation) and
hence, noting $\Delta u=\partial_{r}^{2}u+\frac{1}{r}\partial_{r}u$ we can
find that the new function $U(r,t)=e^{t/2}r^{1/2}u(x,t)$ with $r=|x|$
satisfies the equation

$\displaystyle\partial_{t}^{2}U-\partial_{r}^{2}U=\frac{1}{4}\left(\frac{1}{r^{2}}+1\right)U\quad\text{in}\
(1,\infty)\times(0,\infty)$

with the initial condition $U(r,0)=0$ and
$\partial_{t}U(r,0)=g_{1}(r)=r^{1/2}g(r)$. Then, the above regularity of $u$
shows $U\in C^{2}([1,\infty)\times[0,\infty))$ with
$U(0,t)=\partial_{r}^{2}U(0,t)=0$, which justifies the following argument in
the classical sense.

By the reflection, we extend $g_{1}(\cdot)$ and $U(\cdot,t)$ to be odd
functions. We set $\widetilde{g}_{1}$ and $\widetilde{U}$ as

$\displaystyle\widetilde{g}_{1}(y)=\begin{cases}g_{1}(1+y)&(y>0),\\\
0&(y=0),\\\
-g_{1}(1-y)&(y<0),\end{cases}\quad\widetilde{U}(y,t)=\begin{cases}U(1+y,t)&(y>0,t>0),\\\
0&(y=0,t>0),\\\ -U(1-y,t)&(y<0,t>0),\end{cases}$

which satisfy the initial value problem

$\displaystyle\begin{cases}\partial_{t}^{2}\widetilde{U}-\partial_{y}^{2}\widetilde{U}=m\widetilde{U}&\text{in}\
\mathbb{R}\times(0,\infty),\\\
(\widetilde{U},\partial_{t}\widetilde{U})(y,0)=(0,\widetilde{g}_{1}(y))&\text{in}\
\mathbb{R}\times\\{t=0\\},\end{cases}$

where $m(y)=\frac{1}{4}\big{(}\frac{1}{(|y|+1)^{2}}+1\big{)}$. Note that
$\widetilde{g}_{1}\in C^{1}(\mathbb{R})$ and $\widetilde{U}\in
C^{2}(\mathbb{R}\times(0,\infty))$ by virtue of the behavior of the boundary
$g_{1}(1)=0$ and $U(1,t)=\partial_{r}^{2}U(1,t)=0$.

We prove the nonnegativity of $u$ by a contradiction argument similar to [16,
Chapter 4]. We assume that there exists a point $(x_{0},t_{0})\in
B^{c}\times(0,\infty)$ such that $u(x_{0},t_{0})<0$. Then clearly, setting
$y_{0}=|x_{0}|-1>0$, we have

$\widetilde{U}(y_{0},t_{0})=U(|x_{0}|,t_{0})=e^{t_{0}/2}|x_{0}|^{\frac{1}{2}}u(x_{0},t_{0})<0.$

Now we fix the parameter $\varepsilon>0$ satisfying
$\widetilde{U}(y_{0},t_{0})+\varepsilon e^{t_{0}}=0$ and put

$\displaystyle V(y,t)=\widetilde{U}(y,t)+\varepsilon
e^{t},\quad(y,t)\in\mathbb{R}\times(0,\infty).$

Then, the new function $V$ satisfies

$\displaystyle\begin{cases}\partial_{t}^{2}V-\partial_{y}^{2}V=mV+\varepsilon(1-m)e^{t}&\text{in}\
\mathbb{R}\times(0,\infty),\\\
(V,\partial_{t}V)(y,0)=(\varepsilon,\widetilde{g}_{1}(y)+\varepsilon)&\text{in}\
\mathbb{R}\times\\{t=0\\}.\end{cases}$

Then we consider the triangular region

$D_{0}=\\{(y,t)\in\mathbb{R}\times(0,\infty);t+|y-y_{0}|<t_{0}\\}.$

By $V(y,0)=\varepsilon>0$ and the continuity, $V$ is positive in
$\overline{D_{0}}$ with sufficiently small $t$. Thus, by considering a zero of
$V$ in $\overline{D_{0}\cap\\{y>0\\}}$ with the smallest time (the zero set of
$V$ in $\overline{D_{0}\cap\\{y>0\\}}$ is not empty due to
$V(y_{0},t_{0})=0$), there exists a point
$(y_{1},t_{1})\in\overline{D_{0}\cap\\{y>0\\}}$ such that

$\displaystyle V(y_{1},t_{1})=0\quad\mbox{and}\quad V>0\ \mbox{in}\
D_{1}\cap\\{y>0\\},$

where $D_{1}$ is the triangular region
$D_{1}=\\{(y,t)\in\mathbb{R}\times(0,\infty);t+|y-y_{1}|<t_{1}\\}(\subset
D_{0})$. We further define the subregion $D_{1}^{\prime}=\\{(y,t)\in
D_{1};t+|y|<t_{1}-y_{1}\\}$. Note that $D_{1}^{\prime}=\emptyset$ if
$t_{1}\leq y_{1}$ and $D_{1}\setminus D_{1}^{\prime}$ is the trapezoidal
region with the vertices
$(y_{1},t_{1}),(0,t_{1}-y_{1}),(t_{1}-y_{1},0),(y_{1}+t_{1},0)$ if
$t_{1}>y_{1}$. Applying the d’Alembert formula with the above notation, we
have

$\displaystyle 0=V(y_{1},t_{1})$
$\displaystyle=\frac{1}{2}\left(V(y_{1}-t_{1},0)+V(y_{1}+t_{1},0)\right)+\frac{1}{2}\int_{y_{1}-t_{1}}^{y_{1}+t_{1}}\partial_{t}V(y,0)\,dy$
$\displaystyle\quad+\frac{1}{2}\iint_{D_{1}}m(y)V(y,t)\,dydt+\frac{\varepsilon}{2}\iint_{D_{1}}(1-m(y))e^{t}\,dydt$
$\displaystyle=(1+t_{1})\varepsilon+\frac{1}{2}\int_{|y_{1}-t_{1}|}^{y_{1}+t_{1}}\widetilde{g}_{1}(y)\,dy+\frac{\varepsilon}{2}\iint_{D_{1}^{\prime}}m(y)e^{t}\,dydt$
$\displaystyle\quad+\frac{1}{2}\iint_{D_{1}\setminus
D_{1}^{\prime}}m(y)V(y,t)\,dydt+\frac{\varepsilon}{2}\iint_{D_{1}}(1-m(y))e^{t}\,dydt,$

where we have used that $\widetilde{g}$ and $\widetilde{U}(\cdot,t)$ are odd
functions and $m$ is an even function. Then using the conditions

$\displaystyle\widetilde{g}_{1}\geq 0\ \text{on}\ (0,\infty),\quad V>0\
\text{on}\ D_{1}\cap\\{y>0\\},\quad 0\leq m\leq\frac{1}{2}\ \text{on}\
\mathbb{R}$

and the fact $D_{1}\setminus D_{1}^{\prime}\subset D_{1}\cap\\{y>0\\}$, we
find that the right-hand side of the above identity is positive, which is
contradiction. The proof is complete. ∎

###  3.3 An $L_{d\mu}^{1}$-estimate for $g\in L_{\rm rad}^{2}\cap
L_{d\mu}^{1}$

By virtue of the positively preserving property of $S(t)$ on the radially
symmetric functions, we can discuss the validity of $L_{d\mu}^{1}$-estimates
for $S(t)g$. Basically, the following lemma is similar to the analysis of the
ordinary differential equation $y^{\prime\prime}+y^{\prime}=0$.

######  Lemma 3.3.

For every $g\in L_{\rm rad}^{2}\cap L_{d\mu}^{1}$, one has $S(t)g\in
C([0,\infty);L_{d\mu}^{1})$ with

$\|S(t)g\|_{L^{1}_{d\mu}}\leq(1-e^{-t})\|g\|_{L^{1}_{d\mu}},\quad t>0.$

In particular, $S(t)$ can be extended to the bounded operator from $L^{2}_{\rm
rad}\cap L_{d\mu}^{1}$ to itself.

###### Proof.

To shorten the notation, we use $u=S(t)g$. We divide the proof into three
steps as follows:

Step 1.

the case where $0\leq g\in L_{\rm rad}^{2}$ having bounded support,

Step 2.

the case where $0\leq g\in L_{\rm rad}^{2}\cap L_{d\mu}^{1}$ without
assumption on the support,

Step 3.

the case where $g\in L_{\rm rad}^{2}\cap L_{d\mu}^{1}$ admitting the change of
sign.

(Step 1) Let $R>1$ satisfy ${\rm supp}\,g\subset
B(0,R)=\\{x\in\mathbb{R}^{2}\;;\;|x|<R\\}$. Then by finite propagation
property, we have ${\rm supp}\,u(t)\subset B(0,R+t)$. Here we fix $\zeta\in
C^{\infty}(\mathbb{R}^{2}\times[0,\infty))$ as
$\zeta(x,t)=\zeta_{0}((R+t)^{2}-|x|^{2})$ with $\zeta_{0}\in
C^{\infty}(\mathbb{R})$ satisfying $\zeta_{0}\equiv 1$ on $[0,\infty)$ and
$\zeta_{0}\equiv 0$ on $(-\infty,-1)$. Here we note that $\zeta(\cdot,t)\in
C_{0}^{\infty}(\mathbb{R}^{N})$ and $\zeta\equiv 1$ on ${\rm supp}\,u$. By the
nonnegativity of $u(t)$ (provided by Proposition 3.2), we can see from $u\in
C([0,\infty);L^{2}(B^{c}))$ and $\zeta(x,t)(1+\log|x|)\in
C([0,\infty);L^{2}(B^{c}))$ that $u(x,t)=\zeta(x,t)u(x,t)\in
C([0,\infty);L_{d\mu}^{1})$. Moreover, Setting
$\varphi_{n}(x)=1+\log|x|-|x|^{-n}$ (for $n\in\mathbb{N}$), we have
$\varphi_{n}(x)\to 1+\log|x|$ as $n\to\infty$ and $\zeta(x,t)\varphi_{n}(x)\in
C^{2}([0,\infty);H_{0}^{1}(B^{c}))$ which is applicable to the test function
for the equation in (2.2) (verified in $H^{-1}(B^{c})$). Hereafter, all
integrals on $B^{c}$ always can be justified by using the relation
$\zeta(x,t)u(x,t)=u(x,t)$. Noting that

$\Delta\varphi_{n}=-\Delta|x|^{-n}=n\,{\rm
div}(x|x|^{-n-2})=-n^{2}|x|^{-n-2}\leq 0,$

we see from the nonnegativity of $u(t)$ that

$\displaystyle{}\langle{}\Delta
u(t),\zeta(t)\varphi_{n}{}\rangle{}_{H^{-1},H_{0}^{1}}=\int_{B^{c}}u(t)\Delta\varphi_{n}\,dx\leq
0.$

Therefore the equation in (2.2) gives

$\displaystyle\frac{d^{2}}{dt^{2}}\int_{B^{c}}u(t)\varphi_{n}\,dx+\frac{d}{dt}\int_{B^{c}}u(t)\varphi_{n}\,dx={}\langle{}\Delta
u(t),\zeta(t)\varphi_{n}{}\rangle{}_{H^{-1},H_{0}^{1}}\leq 0.$

which implies

$\int_{B^{c}}u(t)\varphi_{n}\,dx\leq(1-e^{-t})\int_{B^{c}}g\varphi_{n}\,dx.$

Letting $n\to\infty$, we obtain the desired inequality.

(Step 2) We use the cut-off approximation $g_{n}=\chi_{B^{c}\cap
B(0,n)}(x)g(x)$, where $\chi_{K}$ is the indicator function of $K$. Put
$u_{n}=S(t)g_{n}$. Then $g_{n}\to g$ in $L^{2}(B^{c})$ as $n\to\infty$ and
hence $u_{n}(t)\to u(t)$ in $H_{0}^{1}(B^{c})$ as $n\to\infty$. For each
$n\in\mathbb{N}$, we can apply the claim in Step 1 to $g_{n}$ and also
$g_{m}-g_{n}$ ($m>n$) and then $u_{n}\in C([0,\infty);L_{d\mu}^{1})$ and

$\displaystyle\|u_{n}(t)\|_{L_{d\mu}^{1}}\leq(1-e^{-t})\|g_{n}\|_{L_{d\mu}^{1}}\leq(1-e^{-t})\|g\|_{L_{d\mu}^{1}},$
$\displaystyle\|u_{m}(t)-u_{n}(t)\|_{L_{d\mu}^{1}}\leq(1-e^{-t})\|g_{m}-g_{n}\|_{L_{d\mu}^{1}}.$

Therefore $u_{n}$ is the Cauchy sequence in the Banach space
$C(I;L_{d\mu}^{1})$ endowed with the sup norm for any compact interval
$I\subset[0,\infty)$. Since $u_{n}(t)$ converses to $u(t)$ in the pointwise
sense, we can obtain $u\in C([0,\infty);L_{d\mu}^{1})$ and the desired
inequality.

(Step 3) In this case we use the decomposition

$g=g_{+}-g_{-},\quad g_{\pm}:=\max\\{\pm g,0\\}\geq 0.$

Then by the consequence of Step 2, we have $S(t)g=S(t)g_{+}-S(t)g_{-}\in
C([0,\infty);L_{d\mu}^{1})$ and

$\displaystyle\|S(t)g\|_{L_{d\mu}^{1}}$
$\displaystyle\leq\|S(t)g_{+}\|_{L_{d\mu}^{1}}+\|S(t)g_{-}\|_{L_{d\mu}^{1}}$
$\displaystyle\leq(1-e^{-t})\Big{(}\|g_{+}\|_{L_{d\mu}^{1}}+\|g_{-}\|_{L_{d\mu}^{1}}\Big{)}$
$\displaystyle=(1-e^{-t})\|g\|_{L_{d\mu}^{1}}.$

The proof is complete. ∎

##  4 Estimates for semilinear problem

In this section, we discuss the estimate from below for the lifespan of
solution

$u(t)=\varepsilon S(t)g+\int_{0}^{t}S(t-s)[|u(s)|^{p}]\,ds,\quad t\in(0,T)$

to (1.1). The argument depends on the continuity method based on the blowup
alternative (Proposition 1.1 (ii)). The quantity

$\|v\|_{X_{T}}=\sup_{0\leq t<T}\left(\|v(t)\|_{L^{1}_{d\mu}}+\frac{\|\nabla
v(t)\|_{L^{2}(B^{c})}}{h(t)}\right),\quad v\in C([0,T);H_{0}^{1}(B^{c})\cap
L_{d\mu}^{1}).$ (4.1)

plays a crucial role. We first note that the above quantity for the linear
solution $S(t)g$ with $g\in L_{\rm rad}^{2}\cap L_{d\mu}^{1}$ is finite.

######  Lemma 4.1.

There exists a positive constant $C_{1}$ such that for every $g\in L_{\rm
rad}^{2}\cap L_{d\mu}^{1}$,

$\|S(t)g\|_{X_{\infty}}\leq
C_{1}\Big{(}\|g\|_{L_{d\mu}^{1}}+\|g\|_{L^{2}(B^{c})}\Big{)}.$

###### Proof.

We can choose $C_{1}=1+C_{{\rm M},1,1}^{\sharp}$ which is a consequence of
Lemmas 3.3 and 3.1. ∎

Before treating the lifespan of $u$, we state a modified assertion about the
blowup alternative from the viewpoint of the quantity $\|u\|_{X_{T}}$.

######  Lemma 4.2.

Let $u$ be the weak solution of (1.1) in $(0,T)$ with $g$ having a compact
support. Then $T=T_{\max}(0,\varepsilon g)$ if and only if $T=+\infty$ or
$T<+\infty$ with $\lim_{t\to T}\|u\|_{X_{t}}=+\infty$.

###### Proof.

Suppose that $T_{\max}(0,\varepsilon g)=T<+\infty$. If $\lim_{t\to
T}\|u\|_{X_{t}}<+\infty$, then Lemma 2.5 with $q=2$ yields

$\|u(t)\|_{L^{2}(B^{c})}\leq C_{{\rm GN},2}\|\nabla
u(t)\|_{L^{2}(B^{c})}^{\frac{1}{2}}\|u(t)\|_{L^{1}(B^{c})}^{\frac{1}{2}}\leq
C_{{\rm GN},2}\|u\|_{X_{T}}h(t)^{\frac{1}{2}}$

and Lemmas 2.2 and 2.5 with $q=2p$ give for every $t\in(0,T)$,

$\displaystyle\|\partial_{t}u(t)\|_{L^{2}(B^{c})}$
$\displaystyle=\left\|\int_{0}^{t}\partial_{t}S(t-s)|u(s)|^{p}\,ds\right\|_{L^{2}(B^{c})}$
$\displaystyle\leq\int_{0}^{t}\|u(s)\|_{L^{2p}(B^{c})}^{p}\,ds$
$\displaystyle\leq(C_{{\rm GN},2p})^{p}\int_{0}^{t}\|\nabla
u(s)\|_{L^{2}(B^{c})}^{p-\frac{1}{2}}\|u(s)\|_{L^{1}(B^{c})}^{\frac{1}{2}}\,ds$
$\displaystyle\leq(C_{{\rm
GN},2p})^{p}\|u\|_{X_{T}}^{p}\int_{0}^{t}h(s)^{p-\frac{1}{2}}\,ds.$

These inequalities yield
$\|u\|_{H_{0}^{1}(B^{c})}+\|\partial_{t}u\|_{L^{2}(B^{c})}$ is bounded in
$(0,T)$. By Proposition 1.1, we have $T<T_{\max}(0,\varepsilon g)$ which is
contradiction.

On the contrary, suppose $T<+\infty$ with $\lim_{t\to
T}\|u\|_{X_{t}}=+\infty$. We fix $R>1$ such that ${\rm supp}\,g\in B(0,R)$. By
finite propagation property we also have ${\rm supp}\,u(t)\subset B(0,R+T)$.
We see from the Hölder inequality, that

$\displaystyle\|u(t)\|_{L_{d\mu}^{1}}$ $\displaystyle=\int_{B^{c}\cap
B(0,R+T)}|u(t)|(1+\log|x|)\,dx$ $\displaystyle\leq\left(\int_{B^{c}\cap
B(0,R+T)}(1+\log|x|)^{2}\,dx\right)^{\frac{1}{2}}\|u(t)\|_{L^{2}(B^{c})}$

and hence $\lim_{t\to T}\|u(t)\|_{H_{0}^{1}(B^{c})}=+\infty$ which means that
$T=T_{\max}(0,\varepsilon g)$. ∎

###  4.1 A Gagliardo–Nirenberg type inequality with logarithmic weight

As explained in Introduction, to estimate the $L_{d\mu}^{1}$-norm of the
solution $u$ to the problem (1.1), we have to control the quantity

$\big{\|}|u|^{p}\big{\|}_{L_{d\mu}^{1}}=\int_{B^{c}}|u(x,t)|^{p}(1+\log|x|)\,dx$

which comes from the semilinear term $|u|^{p}$. The following inequality
enables us to control such a quantity via the ingredients in $\|u\|_{X_{T}}$.
The form is close to the classical Gagliardo–Nirenberg inequality but
involving the logarithmic weight. Here we do not need to assume the radial
symmetry for functions.

######  Lemma 4.3.

For every $1<q<\infty$, one has $H^{1}_{0}(B^{c})\cap L^{1}_{d\mu}\subset
L^{q}_{d\mu}$ and there exists a positive constant $C_{{\rm GH},q}^{\sharp}$
such that

$\|f\|_{L_{d\mu}^{q}}\leq C_{{\rm GN},q}^{\sharp}\|\nabla
f\|_{L^{2}(B^{c})}^{1-\frac{1}{q}}\|f\|_{L_{d\mu}^{1}}^{\frac{1}{q}},\quad
f\in H^{1}_{0}(B^{c})\cap L^{1}_{d\mu}.$

###### Proof.

Let $f\in H^{1}_{0}(B^{c})\cap L^{1}_{d\mu}$ be fixed. Put $\eta\in
C_{0}^{\infty}(\mathbb{R})$ satisfying

$\eta(s)\begin{cases}>0&\text{if}\ s\in(1/2,2),\\\ =0&\text{if}\
s\notin(1/2,2).\end{cases}$

Note that for every $a>0$ and $\sigma\in[1,\infty)$,

$\int_{0}^{\infty}\eta\left(\frac{a}{R}\right)^{\sigma}\,dR=aK_{\sigma},\quad
K_{\sigma}=\int_{0}^{\infty}\frac{\eta(\rho)^{\sigma}}{\rho^{2}}\,d\rho<\infty.$
(4.2)

Now we define the (localized) functions $f_{R}\in H^{1}_{0}(B^{c})\cap
L^{1}(B^{c})$ for $R>0$ as

$f_{R}(x)=\eta\left(\frac{1+\log|x|}{R}\right)f(x),\quad x\in B^{c}.$

We see from the Fubini–Tonelli theorem and (4.2) that for
$1\leq\sigma<\infty$,

$\displaystyle\int_{0}^{\infty}\|f_{R}\|_{L^{\sigma}(B^{c})}^{\sigma}\,dR$
$\displaystyle=\int_{B^{c}}|f(x)|^{\sigma}\int_{0}^{\infty}\eta\left(\frac{1+\log|x|}{R}\right)^{\sigma}\,dR\,dx=K_{\sigma}\|f\|_{L_{d\mu}^{\sigma}}^{\sigma}.$
(4.3)

On the other hand, by Lemma 2.5 we have

$\displaystyle\|f_{R}\|_{L^{q}(B^{c})}^{q}\leq(C_{{\rm GN},q})^{q}\|\nabla
f_{R}\|_{L^{2}(B^{c})}^{q-1}\|f_{R}\|_{L^{1}(B^{c})}.$ (4.4)

Noting that $\frac{1}{2}R\leq 1+\log|x|\leq 2R$ on
${\operatorname{supp}}\,f_{R}$, one can compute as

$\displaystyle|\nabla f_{R}(x)|^{2}$
$\displaystyle=\left|\eta\left(\frac{1+\log|x|}{R}\right)\nabla
f(x)+\eta^{\prime}\left(\frac{1+\log|x|}{R}\right)\frac{x}{R|x|^{2}}f(x)\right|^{2}$
$\displaystyle\leq 2\|\eta\|_{L^{\infty}}^{2}|\nabla
f(x)|^{2}+8\|\eta^{\prime}\|_{L^{\infty}}^{2}\frac{|f(x)|^{2}}{|x|^{2}(1+\log|x|)^{2}}.$

Combining the above inequality with Lemma 2.6, we deduce

$\displaystyle\|\nabla
f_{R}\|_{L^{2}(B^{c})}^{2}\leq\left(2\|\eta\|_{L^{\infty}}^{2}+32\|\eta^{\prime}\|_{L^{\infty}}^{2}\right)\|\nabla
f\|_{L^{2}(B^{c})}^{2}.$ (4.5)

Consequently, using (4.4), (4.5) and (4.3) with $\sigma=q$ and also
$\sigma=1$, we obtain

$\displaystyle K_{q}\|f\|_{L_{d\mu}^{q}}^{q}$
$\displaystyle=\int_{0}^{\infty}\|f_{R}\|_{L^{q}(B^{c})}^{q}\,dR$
$\displaystyle\leq(C_{{\rm GN},q})^{q}\int_{0}^{\infty}\|\nabla
f_{R}\|_{L^{2}(B^{c})}^{q-1}\|f_{R}\|_{L^{1}(B^{c})}\,dR$
$\displaystyle\leq(C_{{\rm
GN},q})^{q}\left(2\|\eta\|_{L^{\infty}}^{2}+32\|\eta^{\prime}\|_{L^{\infty}}^{2}\right)^{\frac{q-1}{2}}\|\nabla
f\|_{L^{2}(B^{c})}^{q-1}\int_{0}^{\infty}\|f_{R}\|_{L^{1}(B^{c})}\,dR$
$\displaystyle=K_{1}(C_{{\rm
GN},q})^{q}\left(2\|\eta\|_{L^{\infty}}^{2}+32\|\eta^{\prime}\|_{L^{\infty}}^{2}\right)^{\frac{q-1}{2}}\|\nabla
f\|_{L^{2}(B^{c})}^{q-1}\|f\|_{L_{d\mu}^{1}}.$

The proof is complete. ∎

###  4.2 Lower bound for lifespan of semilinear problem

Here we discuss a priori estimate for $u$ via the quantity $\|u\|_{X_{T}}$.
The first part is the derivation of $L_{d\mu}^{1}$-estimate as a merit of the
positively preserving property of $S(t)$ for the radially symmetric functions.

######  Lemma 4.4.

Let $u$ be the weak solution of (1.1) in $(0,T)$ with $g\in L_{\rm
rad}^{2}\cap L_{d\mu}^{1}$ and let $\|u\|_{X_{T}}$ be given in (4.1). Then
there exists a positive constant $C_{2}$ (independent of $\varepsilon,g$ and
$T$) such that for every $t\in(0,T)$,

$\|u(t)\|_{L^{1}_{d\mu}}\leq\varepsilon\|g\|_{L^{1}_{d\mu}}+C_{2}\|u\|_{X_{T}}^{p}\int_{0}^{t}{h(s)}^{p-1}\,ds.$

###### Proof.

Employing Lemma 4.3 with $q=p$, we have for every $s\in(0,T)$,

$\displaystyle\|u(s)\|_{L^{p}_{d\mu}}^{p}$ $\displaystyle\leq(C_{{\rm
GN},p}^{\sharp})^{p}\|\nabla u(s)\|_{L^{2}}^{p-1}\|u(s)\|_{L^{1}_{d\mu}}$
$\displaystyle\leq(C_{{\rm GN},p}^{\sharp})^{p}\|u\|_{X_{T}}^{p}h(s)^{p-1}.$

Combining the above inequality with Lemma 3.3, we deduce that for every
$t\in(0,T)$,

$\displaystyle\|u(t)\|_{L^{1}_{d\mu}}$
$\displaystyle\leq\varepsilon\|S(t)g\|_{L_{d\mu}^{1}}+\int_{0}^{t}\|S(t-s)|u(s)|^{p}\|_{L_{d\mu}^{1}}\,ds$
$\displaystyle\leq\varepsilon\|g\|_{L_{d\mu}^{1}}+\int_{0}^{t}\|u(s)\|_{L_{d\mu}^{p}}^{p}\,ds$
$\displaystyle\leq\varepsilon\|g\|_{L_{d\mu}^{1}}+(C_{{\rm
GN},p}^{\sharp})^{p}\|u\|_{X_{T}}^{p}\int_{0}^{t}h(s)^{p-1}\,ds.$

This is the desired inequality. ∎

######  Lemma 4.5.

Let $u$ be the weak solution of 1.1 in $(0,T)$ with $g\in L_{\rm rad}^{2}\cap
L_{d\mu}^{1}$ and let $\|u\|_{X_{T}}$ be given in (4.1). Then there exists a
positive constant $C_{3}$ (independent of $\varepsilon,g$ and $T$) such that
for every $t\in(0,T)$,

$\|\nabla u(t)\|_{L^{2}(B^{c})}\leq
C_{3}h(t)\left(\varepsilon(\|g\|_{L^{1}_{d\mu}}+\|g\|_{L^{2}(B^{c})})+\|u\|_{X_{T}}^{p}\int_{0}^{t}h(s)^{p-1}\,ds\right).$

###### Proof.

Set

$J_{1}(t)=\int_{0}^{\frac{t}{2}}S(t-s)|u(s)|^{p}\,ds,\quad
J_{2}(t)=\int_{\frac{t}{2}}^{t}S(t-s)|u(s)|^{p}\,ds.$

Then it is obvious that $u(t)=\varepsilon S(t)g+J_{1}(t)+J_{2}(t)$. The
estimate for the linear part $\varepsilon S(t)g$ has already done in Lemma
4.1. For the estimate of $J_{1}(t)$, applying Lemma 3.1 with $q=1$, we see
that

$\displaystyle\|\nabla J_{1}(t)\|_{L^{2}(B^{c})}$
$\displaystyle\leq\int_{0}^{\frac{t}{2}}\|\nabla
S(t-s)|u(s)|^{p}\|_{L^{2}(B^{c})}\,ds$ $\displaystyle\leq C_{{\rm
M},1,1}^{\sharp}\int_{0}^{\frac{t}{2}}h(t-s)\Big{(}\|u(s)\|_{L^{p}_{d\mu}}^{p}+\|u(s)\|_{L^{2p}(B^{c})}^{p}\Big{)}\,ds.$

The integrands in the right-hand side of the above inequality are estimated as
follows: via Lemma 4.3 with $q=p$, one has

$\displaystyle\|u(s)\|_{L^{p}_{d\mu}}^{p}$ $\displaystyle\leq(C_{{\rm
GN},p}^{\sharp})^{p}\|\nabla
u(s)\|_{L^{2}(B^{c})}^{p-1}\|u(s)\|_{L^{1}_{d\mu}}$ $\displaystyle\leq(C_{{\rm
GN},p}^{\sharp})^{p}\|u\|_{X_{T}}^{p}h(s)^{p-1}$

and via Lemma 2.5 with $q=2p$, one has

$\displaystyle\|u(s)\|_{L^{2p}(B^{c})}^{p}$ $\displaystyle\leq(C_{{\rm
GN},2p})^{p}\|\nabla
u(s)\|_{L^{2}(B^{c})}^{p-\frac{1}{2}}\|u(s)\|_{L^{1}(B^{c})}^{\frac{1}{2}}$
$\displaystyle\leq(C_{{\rm GN},2p})^{p}\|u\|_{X_{T}}^{p}h(s)^{p-\frac{1}{2}}.$

Noting that $h(s)^{p-\frac{1}{2}}\leq h(s)^{p-1}$, we can deduce that

$\displaystyle\|\nabla J_{1}(t)\|_{L^{2}(B^{c})}$ $\displaystyle\leq C_{{\rm
M},1,1}^{\sharp}\big{(}(C_{{\rm GN},p}^{\sharp})^{p}+(C_{{\rm
GN},2p})^{p}\big{)}\|u\|_{X_{T}}^{p}\int_{0}^{\frac{t}{2}}h(t-s)h(s)^{p-1}\,ds$
$\displaystyle\leq C_{{\rm M},1,1}^{\sharp}\big{(}(C_{{\rm
GN},p}^{\sharp})^{p}+(C_{{\rm
GN},2p})^{p}\big{)}\|u\|_{X_{T}}^{p}h(t/2)\int_{0}^{\frac{t}{2}}h(s)^{p-1}\,ds.$

For the estimate of $J_{2}(t)$, applying Lemma 3.1 with $q=2$ and Lemma 4.3
with $q=2p$, we can also compute in a similar way:

$\displaystyle\|\nabla J_{2}(t)\|_{L^{2}(B^{c})}$ $\displaystyle\leq 2C_{{\rm
M},1,2}^{\sharp}\int_{\frac{t}{2}}^{t}h(t-s)^{\frac{1}{2}}\|u(s)\|_{L_{d\mu}^{2p}}^{p}\,ds$
$\displaystyle\leq 2C_{{\rm M},1,2}^{\sharp}(C_{{\rm
GN},2p}^{\sharp})^{p}\|u\|_{X_{T}}^{p}\int_{\frac{t}{2}}^{t}h(t-s)^{\frac{1}{2}}h(s)^{p-\frac{1}{2}}\,ds$
$\displaystyle\leq k_{0}C_{{\rm M},1,2}^{\sharp}(C_{{\rm
GN},2p}^{\sharp})^{p}\|u\|_{X_{T}}^{p}th(t/2)^{p}$ $\displaystyle\leq
2k_{0}C_{{\rm M},1,2}^{\sharp}(C_{{\rm
GN},2p}^{\sharp})^{p}\|u\|_{X_{T}}^{p}h(t/2)\int_{0}^{\frac{t}{2}}h(s)^{p-1}\,ds,$

where we have used the inequality

$\int_{0}^{\tau}h(s)^{\frac{1}{2}}\,ds\leq k_{0}\tau
h(\tau)^{\frac{1}{2}},\quad\tau\in(0,\infty)$

for some $k_{0}>0$. Combining these inequalities and noting $h(t/2)\leq
4h(t)$, we arrive at the desired estimate. ∎

Summarizing the above two lemmas, we conclude the following

######  Proposition 4.6.

Let $u$ be the weak solution of 1.1 in $(0,T)$ with $g\in L_{\rm rad}^{2}\cap
L_{d\mu}^{1}$ and let $\|u\|_{X_{T}}$ be given in (4.1). Then there exists a
positive constant $C_{4}$ (independent of $\varepsilon,g$ and $T$) such that
for every $t\in(0,T)$,

$\displaystyle\|u\|_{X_{t}}\leq
C_{4}\left(\varepsilon\big{(}\|g\|_{L_{d\mu}^{1}}+\|g\|_{L^{2}(B^{c})}\big{)}+\|u\|_{X_{t}}^{p}\int_{0}^{T}h(s)^{p-1}\,ds\right).$

###### Proof of Theorem 1.2.

Put $|\\!|\\!|g|\\!|\\!|=\|g\|_{L_{d\mu}^{1}}+\|g\|_{L^{2}(B^{c})}$ to shorten
the notation. We first prove the assertion in Theorem 1.2 when $g$ has a
compact support. In this case, a similar discussion with $\zeta(x,t)$ as in
Lemma 3.3, we can find that $u\in C([0,T);L_{d\mu}^{1})$ and therefore the
function $t\in(0,T_{\max}(0,\varepsilon g))\mapsto\|u\|_{X_{t}}$ is non-
decreasing and continuous. In view of Lemma 4.2, we can take

$T_{*}=\sup\\{t\in(0,T_{\max}(0,\varepsilon g)]\;;\;\|u\|_{X_{t}}\leq
2C_{4}\varepsilon|\\!|\\!|g|\\!|\\!|\\}.$

Then by Proposition 4.6 we have

$\|u\|_{X_{T_{*}}}\leq
C_{4}\varepsilon|\\!|\\!|g|\\!|\\!|\left(1+(2C_{4})^{p}|\\!|\\!|g|\\!|\\!|^{p-1}\varepsilon^{p-1}\int_{0}^{T_{*}}h(s)^{p-1}\,ds\right).$

This yields that $T_{*}$ has the lower bound $T_{*}\geq T_{\varepsilon}$ with

$T_{\varepsilon}=\sup\left\\{t\in(0,\infty]\;;\;(2C_{4})^{p}|\\!|\\!|g|\\!|\\!|^{p-1}\int_{0}^{t}h(s)^{p-1}\,ds\leq\frac{1}{\varepsilon^{p-1}}\right\\}.$
(4.6)

Since

$\int_{0}^{t}h(s)^{p-1}\,ds\begin{cases}\leq
k_{p}(1+t)^{2-p}(1+\log(1+t))^{1-p}&\text{if}\ 1<p<2,\\\
=\log(1+\log(1+t))&\text{if}\ p=2,\\\ <+\infty&\text{if}\ p>2\end{cases}$

(for some positive constants $k_{p}$), the lifespan estimates for $1<p\leq 2$
and the existence of global weak solution of (1.1) are proved.

Next we consider the case where $g\in L^{2}_{\rm rad}\cap L_{d\mu}^{1}$ does
not have compact support. In this case, we use a cut-off argument. Put
$g_{n}=g\chi_{B^{c}\cap B(0,n)}$. Note that
$|\\!|\\!|g_{n}|\\!|\\!|\leq|\\!|\\!|g|\\!|\\!|$. By the first step, we have
the respective solutions

$u_{n}\in C([0,T_{\varepsilon});H^{1}_{0}(B^{c}))\cap
C^{1}([0,T_{\varepsilon});L^{2}(B^{c}))\cap
C([0,T_{\varepsilon});L_{d\mu}^{1})$

satisfying

$\|u_{n}\|_{X_{T_{\varepsilon}}}\leq
2C_{4}\varepsilon|\\!|\\!|g_{n}|\\!|\\!|\leq
2C_{4}\varepsilon|\\!|\\!|g|\\!|\\!|;$

it should be noticed that the $T_{\varepsilon}$ can be chosen as in (4.6)
(independent of $n$). Noting that for every $1\leq q<\infty$, we see that
there exist positive constants $C_{5,q}$ and $C_{5,q}^{\prime}$ such that

$\|u_{n}(t)\|_{L_{d\mu}^{q}}\leq
C_{5,q}\varepsilon,\quad\|u_{n}(t)-u_{m}(t)\|_{L_{d\mu}^{q}}\leq
C_{5,q}^{\prime}y(t),$

where

$y(t)=\|u_{n}(t)-u_{m}(t)\|_{L_{d\mu}^{1}}+\|\nabla u_{n}(t)-\nabla
u_{m}(t)\|_{L^{2}(B^{c})}.$

Therefore we see from Lemmas 3.3 and 2.2 with the inequality
$||z_{1}|^{p}-|z_{2}|^{p}|\leq p(|z_{1}|+|z_{2}|)^{p-1}|z_{1}-z_{2}|\
(z_{1},z_{2}\in\mathbb{R})$ that

$\displaystyle\|u_{n}(t)-u_{m}(t)\|_{L_{d\mu}^{1}}$
$\displaystyle\leq\varepsilon\|g_{n}-g_{m}\|_{L_{d\mu}^{1}}+\int_{0}^{t}\big{\|}|u_{n}(s)|^{p}-|u_{m}(s)|^{p}]\big{\|}_{L_{d\mu}^{1}}\,ds$
$\displaystyle\leq\varepsilon\|g_{n}-g_{m}\|_{L_{d\mu}^{1}}+p\int_{0}^{t}\big{(}\|u_{n}(s)\|_{L_{d\mu}^{p}}+\|u_{m}(s)\|_{L_{d\mu}^{p}}\big{)}^{p-1}\|u_{n}(s)-u_{m}(s)\|_{L_{d\mu}^{p}}\,ds$
$\displaystyle\leq\varepsilon\|g_{n}-g_{m}\|_{L_{d\mu}^{1}}+p(2C_{5,p}\varepsilon)^{p-1}C_{5,p}^{\prime}\int_{0}^{t}y(s)\,ds$

and

$\displaystyle\|\nabla u_{n}(t)-\nabla u_{m}(t)\|_{L^{2}(B^{c})}$
$\displaystyle\leq\varepsilon\|g_{n}-g_{m}\|_{L^{2}(B^{c})}+\int_{0}^{t}\big{\|}|u_{n}(s)|^{p}-|u_{m}(s)|^{p}]\big{\|}_{L^{2}(B^{c})}\,ds$
$\displaystyle\leq\varepsilon\|g_{n}-g_{m}\|_{L^{2}(B^{c})}+p\int_{0}^{t}\big{(}\|u_{n}(s)\|_{L_{d\mu}^{2p}}+\|u_{m}(s)\|_{L_{d\mu}^{2p}}\big{)}^{p-1}\|u_{n}(s)-u_{m}(s)\|_{L_{d\mu}^{2p}}\,ds$
$\displaystyle\leq\varepsilon\|g_{n}-g_{m}\|_{L_{d\mu}^{1}}+p(2C_{5,2p}\varepsilon)^{p-1}C_{5,2p}^{\prime}\int_{0}^{t}y(s)\,ds.$

These imply

$y(t)\leq\varepsilon|\\!|\\!|g_{n}-g_{m}|\\!|\\!|+C_{6,p}\varepsilon^{p-1}\int_{0}^{t}y(s)\,ds,\quad
t\in[0,T_{\varepsilon})$

with
$C_{6,p}=p(2C_{5,p})^{p-1}C_{5,p}^{\prime}+p(2C_{5,2p})^{p-1}C_{5,2p}^{\prime}$.
By the Gronwall inequality we obtain

$y(t)\leq\varepsilon|\\!|\\!|g_{n}-g_{m}|\\!|\\!|e^{C_{6,p}\varepsilon^{p-1}t},\quad
t\in[0,T_{\varepsilon})$

which ensures that $\\{u_{n}\\}_{n}$ is a Cauchy sequence in the Banach space
$C(I;H_{0}^{1}(B^{c})\cap L_{d\mu}^{1})$ for any compact interval
$I\subset[0,T_{\varepsilon})$. Since the limit $u$ satisfies

$u(t)=S(t)g+\int_{0}^{t}S(t-s)|u(s)|^{p}\,ds,\quad t\in[0,T_{\varepsilon}),$

the proof of this part ($g$ with non-compact support) is complete.

It only remains to show the decay estimate for $\partial_{t}u$ when $u$ is the
global weak solution of (1.1) with $p>2$ and $|\\!|\\!|g|\\!|\\!|$ is
sufficiently small verifying $T_{\varepsilon}=\infty$. Then using Lemma 3.1,
we have

$\displaystyle\|\partial_{t}u(t)\|_{L^{2}(B^{c})}$
$\displaystyle\leq\varepsilon\|\partial_{t}S(t)g\|_{L^{2}(B^{c})}+\int_{0}^{t}\|\partial_{t}S(t-s)|u(s)|^{p}\|_{L^{2}(B^{c})}\,ds$
$\displaystyle\leq C_{{\rm
M},2,1}^{\sharp}\varepsilon(1+t)^{-\frac{1}{2}}h(t)|\\!|\\!|g|\\!|\\!|+C_{{\rm
M},2,1}^{\sharp}\int_{0}^{\frac{t}{2}}(1+t-s)^{-\frac{1}{2}}h(t-s)|\\!|\\!||u(s)|^{p}|\\!|\\!|\,ds$
$\displaystyle\quad+2C_{{\rm
M},2,2}^{\sharp}\int_{\frac{t}{2}}^{t}(1+t-s)^{-\frac{1}{2}}h(t-s)^{\frac{1}{2}}\||u(s)|^{p}\|_{L_{d\mu}^{2}}\,ds.$

The rest of the proof of the boundedness is similar to the proof of Lemma 4.5
(the difference is just the validity of $\|u\|_{X_{\infty}}\leq
2C_{4}\varepsilon|\\!|\\!|g|\\!|\\!|$). The proof is complete. ∎

## Appendix

Here we give an alternative proof of the $L^{p}$-$L^{q}$ type estimate (Lemma
2.4) involving the logarithmic weight for the Dirichlet heat semigroup
$e^{t\Delta_{B^{c}}}$, which describes the peculiarity of the two-dimensional
exterior domain. Here we shall discuss it via the classical comparison
principle for parabolic equations. Although all statements here can be shown
for general exterior domains, we only pay our attention to the typical case
$B^{c}$. A similar treatment also can be found in Sobajima [19].

######  Lemma A.1.

For every $q\in[1,\infty]$, there exists a positive constant $C_{{\rm A},1,q}$
such that if $f\in L^{q}(B^{c})$, then

$|e^{t\Delta_{B^{c}}}f(x)|\leq\frac{C_{{\rm
A},1,q}\|f\|_{L^{q}(B^{c})}}{t^{\frac{1}{q}}(1+\log(1+t))}(1+\log|x|),\quad(x,t)\in
B^{c}\times(0,\infty).$

###### Proof.

Since $e^{t\Delta_{B^{c}}}$ is a positive operator, we may assume $f\geq 0$
without loss of generality. The standard (two-dimensional)
$L^{\infty}$-$L^{q}$ estimate shows for every $(x,t)\in
B^{c}\times(0,\infty)$,

$e^{t\Delta_{B^{c}}}f(x)\leq
e^{t\Delta_{\mathbb{R}^{2}}}f(x)\leq\frac{\kappa_{q}\|f\|_{L^{q}(B^{c})}}{t^{\frac{1}{q}}},\quad\kappa_{q}=\Big{(}1-\frac{1}{q}\Big{)}^{1-\frac{1}{q}}\frac{1}{(4\pi)^{\frac{1}{q}}}$

which immediately gives for every $0<t\leq\tau=4$,

$e^{t\Delta_{B^{c}}}f(x)\leq\frac{(1+\log
5)\kappa_{q}\|f\|_{L^{q}(B^{c})}}{t^{\frac{1}{q}}(1+\log(1+t))}(1+\log|x|),\quad
x\in B^{c}.$

Therefore we focus our attention to the case $t\geq\tau=4$. Put
$\mathcal{C}_{t}=B(0,t^{1/2})(\supset B)$ and

$\mathcal{Q}_{1}=\bigcup_{t\geq\tau}(B_{c}\cap\mathcal{C}_{t})\times\\{t\\},\quad\mathcal{Q}_{2}=(B^{c}\times[\tau,\infty))\setminus\mathcal{Q}_{1}.$

We see from $1+\log(1+t)\leq 2(1+\log|x|)$ on $\mathcal{Q}_{2}$ that

$\displaystyle e^{t\Delta_{B^{c}}}f(x)$
$\displaystyle\leq\frac{\kappa_{q}\|f\|_{L^{q}(B^{c})}}{t^{\frac{1}{q}}}\times\frac{2(1+\log|x|)}{1+\log(1+t)},\quad(x,t)\in\mathcal{Q}_{2}.$

For the estimate on $\mathcal{Q}_{1}$, we employ the comparison principle. Put
for $(x,t)\in\overline{\mathcal{Q}_{1}}$,

$\Phi(x,t)=\frac{1+\log|x|^{2}}{2+\log t+\log|x|^{2}},\quad
U(x,t)=\frac{\Phi(x,t)}{t^{\frac{1}{q}}}e^{-\frac{|x|^{2}}{4t}}.$

Observing that

$\displaystyle\partial_{t}\Phi=-\frac{1+\log|x|^{2}}{t\Theta(x,t)^{2}},\quad\nabla\Phi=\frac{2(1+\log
t)x}{\Theta(x,t)^{2}|x|^{2}},\quad\Delta\Phi=-\frac{8(1+\log
t)}{\Theta(x,t)^{3}|x|^{2}}$

with $\Theta(x,t)=2+\log t+\log|x|^{2}$, we can deduce

$\displaystyle\partial_{t}U-\Delta U\geq
t^{-1-\frac{1}{q}}e^{-\frac{|x|^{2}}{4t}}\left[\frac{1+2(\log
t-\log|x|)}{\Theta^{2}}+\Big{(}1-\frac{1}{q}\Big{)}\Phi\right]\geq
0,\quad(x,t)\in\mathcal{Q}_{1}.$

Moreover, noting that $\frac{1}{2+\log t}\leq\Phi(x,t)\leq\frac{1}{2}$ on
$\mathcal{Q}_{1}$, we can find the comparison on the parabolic boundary of
$\mathcal{Q}_{1}$ as follows:

$\displaystyle\begin{cases}e^{\tau\Delta}f(x)\leq 4e^{\frac{1}{4}}(1+\log
2)\kappa_{q}\|f\|_{L^{q}(B^{c})}U(x,\tau),&x\in
B^{c}\cap\mathcal{C}_{\tau},\\\ e^{t\Delta}f(x)=0\leq
U(x,t),&x\in\partial\mathcal{C}_{t},\ t\geq\tau,\\\ e^{t\Delta}f(x)\leq
2e^{\frac{1}{4}}\kappa_{q}\|f\|_{L^{q}(B^{c})}U(x,t).&x\in\partial B^{c},\
t\geq\tau.\end{cases}$

Therefore the comparison principle shows that

$e^{t\Delta}f(x)\leq 4e^{\frac{1}{4}}(1+\log
2)\kappa_{q}\|f\|_{L^{q}(B^{c})}U(x,t),\quad(x,t)\in\mathcal{Q}_{1}.$

The proof is complete. ∎

Taking the adjoint in Lemma A.1, we have

######  Lemma A.2.

For every $q\in[1,\infty]$, there exists a positive constant $C_{{\rm A},2,q}$
such that if $f\in L_{d\mu}^{1}$, then

$\|e^{t\Delta_{B^{c}}}f\|_{L^{q}(B^{c})}\leq\frac{C_{{\rm
A},2,q}}{t^{1-\frac{1}{q}}(1+\log(1+t))}\|f\|_{L_{d\mu}^{1}},\quad t>0.$

###### Proof.

Taking $g\in L^{q^{\prime}}(B^{c})$ with $\frac{1}{q}+\frac{1}{q^{\prime}}=1$,
we have

$\displaystyle\int_{B^{c}}(e^{t\Delta_{B^{c}}}f)g\,dx=\int_{B^{c}}f(e^{t\Delta_{B^{c}}}g)\,dx\leq\frac{C_{{\rm
A},1,q^{\prime}}\|g\|_{L^{q^{\prime}}(B^{c})}}{t^{\frac{1}{q^{\prime}}}(1+\log(1+t))}\int_{B^{c}}|f|(1+\log|x|)\,dx.$

Since $g$ is arbitrary, we obtain the desired inequality. ∎

The following is $L^{p}$-$L^{q}$ type estimate with the logarithmic weight
which is essentially used in this paper.

######  Lemma A.3.

For every $p\in[1,\infty)$ and $q\in[p,\infty]$, there exists a positive
constant $C_{{\rm A},3,p,q}$ such that if $f\in L_{d\mu}^{p}$, then

$\|e^{t\Delta_{B^{c}}}f\|_{L^{q}(B^{c})}\leq\frac{C_{{\rm
A},3,p,q}}{t^{\frac{1}{p}-\frac{1}{q}}(1+\log(1+t))^{\frac{1}{p}}}\|f\|_{L_{d\mu}^{p}},\quad
t>0.$

###### Proof.

The case $p=1$ is already proved in Lemma A.2. Let $1<p<\infty$ and $t>0$ be
fixed. Then combining the $L^{\infty}$-contraction property for
$e^{t\Delta_{B^{c}}}$ written as

$\|e^{t\Delta_{B^{c}}}f\|_{L^{\infty}(B^{c})}\leq\|f\|_{L^{\infty}(B^{c})},\quad
f\in L^{\infty}(B^{c})$

with the inequality Lemma A.2 of the form

$\|e^{t\Delta_{B^{c}}}f\|_{L^{\frac{q}{p}}(B^{c})}\leq\frac{C_{{\rm
A},2,\frac{q}{p}}}{t^{1-\frac{p}{q}}(1+\log(1+t))}\|f\|_{L_{d\mu}^{1}},\quad
f\in L_{d\mu}^{1},$

we see from the Riesz–Thorin theorem with the parameter $\theta\in(0,1)$
satisfying

$\frac{\theta}{\infty}+\frac{(1-\theta)p}{q}=\frac{1}{r_{1}},\quad\frac{\theta}{\infty}+\frac{1-\theta}{1}=\frac{1}{r_{2}}$

that $e^{t\Delta_{B^{c}}}$ can be regarded as the bounded operator from
$L^{r_{2}}_{d\mu}$ to $L^{r_{1}}(B^{c})$ with

$\|e^{t\Delta_{B^{c}}}f\|_{L^{r_{1}}(B^{c})}\leq\left(\frac{C_{{\rm
A},2,\frac{q}{p}}}{t^{1-\frac{p}{q}}(1+\log(1+t))}\right)^{1-\theta}\|f\|_{L^{r_{2}}_{d\mu}},\quad
f\in L_{d\mu}^{r_{2}}.$

Choosing $\theta=1-\frac{1}{p}$, we obtain the desired inequality. ∎

### Acknowedgements

This work was supported by JSPS KAKENHI Grant Numbers 20K14346, 22H00097 and
23K03174.

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