# Large-time behavior of unbounded solutions of the viscous Hamilton-Jacobi
equation: quadratic and subquadratic cases

Alexander Quaas and Andrei Rodríguez-Paredes

###### Abstract.

We determine the large-time behavior of unbounded solutions for the so-called
viscous Hamilton Jacobi equation, $u_{t}-\Delta u+|Du|^{m}=f(x)$, in the
quadratic and subquadratic cases (i.e., for $1<m\leq 2$), with a particular
focus on allowing arbitrary growth at infinity for $f$ and the prescribed
initial data. The lack of a comparison principle for the associated ergodic
problem is overcome by proving that a generalized simplicity holds for sub-
and supersolutions of the ergodic problem. Moreover, as the uniqueness of
solutions of the parabolic problem remains open in the current setting, our
result on large-time holds for any solution, even if multiple solutions exist.

Keywords: Hamilton-Jacobi equations, viscous Hamilton-Jacobi equation,
unbounded solutions, large-time behavior, ergodic behavior, viscosity
solutions.

MSC (2010): 35B40, 35B51, 35D40, 35K15, 35K55.

##  1\. Introduction

The following work deals with the parabolic viscous Hamilton-Jacobi Equation,

(1) $\displaystyle u_{t}-\Delta u+|Du|^{m}=f(x)$
$\displaystyle{}\quad\textrm{in }\mathbb{R}^{N}\times(0,+\infty),$ (2)
$\displaystyle u(x,0)=u_{0}(x)$ $\displaystyle{}\quad\textrm{in
}\mathbb{R}^{N},$

in the the subquadratic and quadratic cases, i.e. when $1<m\leq 2$. The
functions $f,u_{0}$ are continuous and assumed bounded from below, but not
from above. Thus, in this setting, solutions to (1)-(2) are in general
unbounded. Further assumptions on $f,u_{0}$ are stated as needed.

Our main interest is determining the large-time behavior of solutions to
(1)-(2) for data—$u_{0}$ and $f$—as general as possible, particularly with the
most general growth conditions at infinity.

We recall that the problem of large-time behavior for (1) set on a bounded
domain (along with the necessary boundary data) is addressed in Tabet [24] for
the superquadratic case, then by this same author in collaboration with Barles
and Porretta in [7] for the subquadratic case. While some key ideas from these
works remain useful in the present setting, there is considerable difficulty
in obtaining similar results for unbounded solutions over an unbounded domain.

There are several works that address the large-time behavior of solutions of
(1) set in the whole space in the context of weak solutions, but by necessity
these work within the class of bounded solutions; we refer the reader to [9,
10, 12, 14, 16, 20] and the works cited therein, though this list is by no
means exhaustive.

For unbounded solutions over the whole space, Ichihara [17] obtains results on
large-time behavior for a class of equations which includes the model (1), for
all $m>1$, by combining probabilistic and PDE techniques. Additionally, these
results make use of precise growth assumptions for both $f$ and $u_{0}$:
roughly, it is assumed in [17] that $f(x)\approx|x|^{m^{*}}$, where
$m^{*}=\frac{m}{m-1}$ (the conjugate exponent of $m$) and that $u_{0}$ has at
most polynomial growth.

The results of [17] show that the large-time behavior of solutions of (1) is
determined by the associated ergodic problem,

(EP) $\lambda-\Delta\phi+|D\phi|^{m}=f(x)\quad\textrm{in }\mathbb{R}^{N},$

also studied therein, where both $\lambda$ and $\phi$ are unknown. More
precisely, it is shown in [17] that the solution of (1) behaves like
$\phi(x)+\lambda t$ as $t\to+\infty$, where $(\lambda,\phi)$ is the unique
solution pair of (EP) within the class of bounded-from-below functions with at
most polynomial growth (here, $\phi$ is unique up to an additive constant).
This result if commonly referred to as ergodic large-time behavior, and is in
accordance with the bounded domain case.

Given the importance of the ergodic problem, we comment with some detail on
some of the previous results in the unbounded setting. Also, we will sometimes
write “(EP)λ” to stress the dependence of the problem on the constant
$\lambda$; in particular, when a given value for $\lambda$ is considered.

The results of [17] were extended by Barles and Meireles [6] in terms of the
generality allowed for the data, though only the model equation (1) is
addressed. In particular, in the subquadratic case, it is shown that
uniqueness of solutions for (EP) holds if $f(x)\approx|x|^{\alpha}$—where
$\alpha>0$ is now arbitrary—and $Df$ behaves accordingly (see [6], Theorem
3.10 and Proposition 3.7).

In contrast, in a recent work by Arapostathis, Biswas and Caffarelli [1],
uniqueness of solutions for (EP) is obtained assuming only a technical
hypothesis which relates $f$ and $Df$, but still allows $f$ to have arbitrary
growth—see (H2) below. This is accomplished by means of the uniqueness of a
measure related to the underlying dynamics of the problem. However, no
comparison principle is proved for (EP), and in fact no information is
provided on the behavior of sub- and supersolutions. These are crucial
elements towards obtaining large-time behavior for (1) in our approach and in
many of the cited works.

On this matter, we have found that a generalized simplicity holds for sub- and
supersolutions of (EP). We use “simplicity” here in the spirit of similar
results for e.g. the principal eigenfunction of second-order elliptic
operators (see e.g. [11]); and “generalized” since the property holds for also
for sub- and supersolutions, and not merely for solutions. These results are
proved in Propositions 1.2 and 2.11 and, as expected, are an essential part of
the proof of our main result, Theorem 1.3.

Returning to the question of large-time behavior, the authors studied this
problem in the superquadratic case ($m>2$) in [8], in collaboration with G.
Barles. In [8] it is shown that ergodic large-time behavior occurs assuming
minimal hypotheses on $f$ and $u_{0}$, although in this case we are able to
prove a very general comparison principle for the parabolic problem, and a
similar result was already available for the ergodic problem. Regarding the
data, for $f$ it is assumed that there exists a nondecreasing function
$\varphi:[0,+\infty)\to[0,+\infty)$ and constants $\alpha,c>0$ such that for
all $r\geq 0$, $c^{-1}r^{\alpha}\leq\varphi(r)$, and for all
$x\in\mathbb{R}^{N}$ and $r=|x|$, $c^{-1}\varphi(r)-c\leq f(x)\leq
c(\varphi(r)+1)$. In other words, $f$ is controlled by some radial function
that has at least some polynomial growth, but is not bounded in any way by
above.

In the present setting, we are able to obtain the result on large-time
behavior without the assumption of a “minimum growth rate” for $f$. We discuss
this further after providing the precise statements of our results.

###  1.1. Assumptions and main results

Our main result establishes, for $1<m\leq 2$, that the large-time behavior of
solutions of (1)-(2) is determined by the solutions of (EP). To this end, we
require the following assumptions on $f$.

  1. (H1)

$f\in W^{1,\infty}_{\textit{loc}}(\mathbb{R}^{N})$ is coercive; i.e.,
$f(x)\to+\infty$ as $|x|\to+\infty$.

  2. (H2)

$\limsup\limits_{x\to+\infty}\frac{|Df(x)|^{\frac{1}{2m-1}}}{|f(x)|^{\frac{1}{m}}}<+\infty.$

######  Remark 1.1.

Assumption (H2) may be interpreted as preventing $f$ from having large
oscillation at infinity. The fact that $\frac{1}{2m-1}<\frac{1}{m}$ for all
$m>1$ allows $f$ to have arbitrary growth.

We also introduce the generalized ergodic constant,

(3) $\lambda^{*}=\sup\\{\lambda\in\mathbb{R}\ |\
\text{\eqref{erg_prob}${}_{\lambda}$ has a }C^{2}\textrm{-subsolution}\\}.$

The relevant properties of $\lambda^{*}$ are provided in Section 2.

The following is a version of the simplicity results for (EP); for more
results of this type, see Section 2 below.

######  Proposition 1.2.

If $\chi$ and $v\in USC(\mathbb{R}^{N})$ are respectively a solution and a
subsolution of (EP)${}_{\lambda^{*}}$, both bounded from below, then there
exists $c\in\mathbb{R}$ such that $v(x)=\chi(x)+c$ for all
$x\in\mathbb{R}^{N}$.

The proof of Proposition 1.2 follows from combining an approximation of (EP)
over bounded domains with an appropriately chosen perturbation of solutions;
this perturbation relies on the homogeneity of the gradient term. In the end,
these computations allow us to exploit the coercivity of $f$ and “order
solutions at infinity”. It is remarkable that this result can be obtained from
such elementary techniques.

######  Theorem 1.3.

Assume $u_{0}\in C(\mathbb{R}^{N})$ is bounded from below and assumptions (H1)
and (H2) hold. Then, there exists a constant $\hat{c}\in\mathbb{R}$ such that

$u(\cdot,t)-\lambda^{*}t\to\phi\text{ locally uniformly over }\mathbb{R}^{N},$

where $u=u(x,t)$ is any solution of (1)-(2), $\lambda^{*}$ is defined by (3)
and $\phi$ is the unique solution of (EP)${}_{\lambda^{*}}$ given by
Proposition 1.2.

######  Remark 1.4.

Recall that in the case $1<m\leq 2$ there is no comparison principle available
for neither the ergodic nor the parabolic problems. In particular, the
uniqueness of solutions for the parabolic problem—in the generality of the
current setting—remains an open problem. Nevertheless, our result on large
time behavior holds for any solution of (1).

The proof of our main result, Theorem 1.3, begins by showing that a half-
relaxed limit for $u-\lambda^{*}t$ converges to the solution $\phi$ of (EP)
over a subsequence $t_{k}\to+\infty$; it is here that Proposition 1.2 is
essential. From here we construct parabolic barriers for some new initial
condition related to the previous subsequence, again using some of the ideas
behind the proof of the simplicity results.

Notation and organization of the article. The notation used in the article is
entirely standard. Section 2 contains preliminary results concerning (1) and
(EP), including the proof of Proposition 1.2 as well as others which are
collected from previous works for convenience. Our main result is proved in
Section 3.

##  2\. Preliminaries

We start by establishing some existence results for the parabolic problem.

######  Theorem 2.1 (Existence of solutions).

Assume $f,u_{0}\in W^{1,\infty}_{\textit{loc}}(\mathbb{R}^{N})$ are bounded
from below. Then, there exists a continuous, bounded from below solution of
(1)-(2).

###### Proof.

The proof is essentially the same as that of Proposition 3 in [8] for the case
$m>2$. We repeat the main points of the argument for convenience. Also, we
note that the difference in the generality allowed for the initial $u_{0}$ in
the is due to the lack of a comparison result for (1) in the present case,
$m\leq 2$. See Remark 2.2 below.

Let $\\{T_{R}\\}$ be a sequence such that where $T_{R}>0$ for all $R>0$ and
$T_{R}\nearrow+\infty$ as $R\to+\infty$, and consider the parabolic state-
constraints problem on $B_{R}\times(0,T_{R})$,

(4) $\displaystyle u_{t}-\Delta u+|Du|^{m}=f(x)$ $\displaystyle{}\quad\textrm{
in }B_{R}\times(0,T_{R}),$ (5) $\displaystyle u_{t}-\Delta u+|Du|^{m}\geq
f(x)$ $\displaystyle{}\quad\textrm{ in }\partial B_{R}\times(0,T_{R}),$ (6)
$\displaystyle u(\cdot,0)=u_{0}$ $\displaystyle{}\quad\textrm{ in
}\overline{B}_{R}.$

By the results of [5], this problem has a unique continuous viscosity
solution, which we denote by $u^{R}$.

It can be shown that if $R^{\prime}\geq R$, then $u^{R^{\prime}}(x,t)\leq
u^{R}(x,t)$ for all $(x,t)\in B_{R}\times[0,T_{R}]$. This follows from the
fact that solutions of (4) satisfying (5) are maximal. Additionally, a
constructive proof of this is provided in Lemma 2 in [8], and it is equally
valid for $1<m\leq 2$. The solution of (1)-(2) can be then obtained as a
locally uniform limit of the solutions $u^{R}$ as $R\to+\infty$ as follows.

Consider a fixed $\bar{R}$. For $R>2\bar{R}+1$, we have, assuming $u_{0},f\geq
0$

$0\leq u^{R}\leq u^{2\bar{R}+1}\quad\hbox{in
}\overline{B_{2\bar{R}}}\times[0,T_{2\bar{R}}]\;$

hence $\\{u^{R}\\}_{R>2\bar{R}+1}$ is uniformly bounded over
$\overline{B_{2\bar{R}}}\times[0,T_{2\bar{R}}]$.

Furthermore, using Theorem A.1 and Corollary A.2 in the Appendix, the
$C^{0,1/2}$-norm of $u^{R}$ on $\overline{B_{\bar{R}}}\times[0,T_{2\bar{R}}]$
remains also uniformly bounded for $R>2\bar{R}+1$.

Thus, by using the Ascoli-Arzela Theorem, we have the uniform convergence of
$u^{R}$ on $\overline{B_{\bar{R}}}\times[0,T_{2\bar{R}}]$. Then, by a diagonal
argument, we may extract a subsequence of $\\{u^{R}\\}_{R>0}$ that converges
locally uniformly to some $u\in C(\mathbb{R}^{N}\times[0,+\infty)$. By
stability, it follows that $u$ is a viscosity solution of (1)-(2). ∎

######  Remark 2.2.

(i) In [8], existence of solutions for all bounded-from-below $u_{0}\in
C(\mathbb{R}^{N})$ was obtained by approximating $u_{0}$ with functions in
$W^{1,\infty}_{\textit{loc}}(\mathbb{R}^{N})$ and employing a result analogous
to Theorem 2.1. A comparison result that is not available is a crucial tool in
this procedure, since it is used to control the “error” between the
corresponding solutions in terms of the error in the approximation of the
initial data. (ii) Theorem 1.3 holds for any solution of the parabolic problem
that may also exist for $u_{0}\in C(\mathbb{R}^{N})$.

######  Theorem 2.3.

Assume that $f\in W^{1,\infty}_{\textit{loc}}(\mathbb{R}^{N})$ is bounded from
below. Then $\lambda^{*}$ as defined in (3) is finite and
(EP)${}_{\lambda^{*}}$ has a $C^{2}$-solution.

###### Proof.

See Theorem 2.4 in [6]. ∎

######  Theorem 2.4.

Assume that $f\in W^{1,\infty}_{\textit{loc}}(\mathbb{R}^{N})$ is coercive,
i.e.,

$f(x)\to+\infty\quad\text{as}\quad|x|\to+\infty.$

Then (EP) has a $C^{2}$-solution which is bounded from below.

###### Proof.

See Theorem 2.6 in [6]. For further reference, we note that the desired
solution is therein obtained as the limit of solutions to the ergodic problem
set in $B_{R}$. More specifically, if $(\lambda_{R},\phi_{R})$ satisfy

(7)
$\left\\{\begin{array}[]{cl}\lambda_{R}-\Delta\phi_{R}+|D\phi_{R}|^{m}=f(x)&\textrm{in
}B_{R}\\\ \lambda_{R}-\Delta\phi_{R}+|D\phi_{R}|^{m}\geq f(x)&\textrm{on
}\partial B_{R}.\end{array}\right.$

then for some $\bar{\lambda}\in\mathbb{R}$ and $\bar{\phi}\in C^{2}$,
$\lambda_{R}\searrow\bar{\lambda}$ and $\phi_{R}-\phi_{R}(0)\to\bar{\phi}$
locally uniformly in $C^{2}$. By stability, it follows that
$(\bar{\lambda},\bar{\phi})$ is a $C^{2}$-solution of (EP). Furthermore, we
stress that the sequence $(\lambda_{R})$ is nonincreasing with respect to $R$
and that for all $R>0$, $\lambda_{R}\geq\min_{\mathbb{R}^{N}}f$ (thus
$\bar{\lambda}\geq\min_{\mathbb{R}^{N}}f$ as well). Both facts follow from the
extremal characterization of $\lambda_{R}$, analogous to (3); namely

(8) $\lambda_{R}=\sup\\{\lambda\ |\ \exists u:\ \lambda u-\Delta
u+|Du|^{m}\leq f(x)\text{ in }B_{R}\\}.$

∎

######  Proposition 2.5.

Denote by $(\psi_{R},\nu_{R})$ the solution pair of

$\lambda+\Delta\psi+|D\psi|^{m}=f_{R}\quad\textrm{in
}\mathbb{R}^{N}/2S_{R}\mathbb{Z}^{N},$

where $S_{R}$ is chosen so that $f(x)\geq R$ for $|x|\geq S_{R}$ and $f_{R}$
is the periodic extension of $\min\\{f,R\\}$ to
$\mathbb{R}^{N}/2S_{R}\mathbb{Z}^{N}$.

Then $\nu_{R}\to\bar{\nu}$ and $\psi_{R}-\psi_{R}(0)\to\bar{\psi}$ locally
uniformly in $C^{2}$, where $(\bar{\psi},\bar{\nu})$ are a solution of (EP),
and $\bar{\psi}$ is bounded from below.

###### Proof.

See [6], Proposition 4.2. ∎

######  Lemma 2.6.

Assume _(H1)_.

  1. (i)

If $\chi$ is a bounded-from-below supersolution of (EP), then

$\lim\limits_{|x|\to+\infty}\frac{\chi(x)}{|x|}=+\infty.$

  2. (ii)

Let $t_{0}>0$. If $v$ is a bounded-from-below supersolution of (1), then

$\lim\limits_{|x|\to+\infty}\frac{v(x,t_{0})}{|x|}=+\infty.$

###### Proof.

We will prove only (ii), since part (i) is analogous. Define, for all $r\geq
0$, $\varphi(r)=\inf_{\mathbb{R}^{N}\backslash B_{r}}f$. Observe that
$\varphi$ is non-decreasing and, for all $x\in\mathbb{R}^{N}$,
$f(x)\geq\varphi(|x|)$. Furthermore, assumption (H1) implies $\varphi$ is
coercive. We will argue by contradiction.

Assume that there exists some $t_{0}>0$ and a sequence
$(y_{n})_{n\in\mathbb{N}}$ with $|y_{n}|\to+\infty$ as $n\to+\infty$ such that
$\nicefrac{{v(y_{n},t_{0})}}{{|y_{n}|}}$ remains bounded as $n\to+\infty$.
Since $\varphi$ is coercive, this implies

(9)
$\liminf\limits_{n\to+\infty}\frac{v(y_{n},t_{0})}{|y_{n}|\varphi(y_{n})^{\frac{1}{m}}}=0.$

Define for $y\in B_{1}$,

$v_{n}(y,s)=\frac{v(y_{n}+\frac{|y_{n}|}{2}y,t_{0}+\frac{|y_{n}|^{2}}{4}s)}{|y_{n}|\varphi(\frac{1}{2}|y_{n}|)^{\frac{1}{m}}}.$

Formally, $v_{n}$ satisfies

$\displaystyle{}2^{2-m}|y_{n}|^{-1}\varphi(\frac{1}{2}|y_{n}|)^{\frac{1}{m}-1}\left(\partial_{t}v_{n}-\Delta
v_{n}\right)\,+\,|Dv_{n}|^{m}$
$\displaystyle{}\quad=2^{-m}\varphi(\frac{1}{2}|y_{n}|)^{-1}f\left(y_{n}+\frac{|y_{n}|}{2}y\right)\geq
2^{-m}$

where the last inequality follows from the definition of $\varphi$ and the
fact that
$|y_{n}+\frac{|y_{n}|}{2}y|\geq|y_{n}|-\frac{|y_{n}|}{2}|y|=\frac{1}{2}|y_{n}|$;
it is not hard to pass the computation onto a suitable test function. Writing
$\epsilon_{n}=2^{2-m}|y_{n}|^{-1}\varphi(\frac{1}{2}|y_{n}|)^{\frac{1}{m}-1}$,
we have $\epsilon_{n}\to 0$ since $\varphi$ is coercive and $1-\frac{1}{m}<0$,
hence

(10) $\epsilon_{n}(\partial_{t}v_{n}-\Delta v_{n})\,+\,|Dv_{n}|^{m}\geq
2^{-m}-o_{n}(1)\quad\hbox{in }B_{1}\times(t_{0},+\infty),$

where $o_{n}(1)\to 0$ as $n\to\infty$.

In order to pass to the limit, we lack some $L^{\infty}$-bound on $v_{n}$. To
overcome this difficulty, we set

$\tilde{v}_{n}(y,s)=\min\left(v_{n}(y,s),K(1-|y|)\right)\quad\hbox{in
}B_{1}\times(0,+\infty),$

for some suitably large $K$. For $n$ large enough, the concave function
$y\mapsto K(1-|y|)$ is also a supersolution of (10) and therefore so is
$\tilde{v}_{n}$, as the minimum of two supersolutions.

Hence, the half-relaxed limit
$\tilde{v}=\liminf\limits_{n\to\infty}\,\\!\\!^{*}\,\tilde{v}_{n}$ is well
defined and, by stability, we have in the limit $n\to+\infty$ that

$|D\tilde{v}|^{m}\geq 2^{-m}\quad\textrm{in }B_{1}\times(0,+\infty),$

in the viscosity sense. For all $n$, $\tilde{v}_{n}\geq 0$, hence also
$\tilde{v}\geq 0$ (recall that $v\geq 0$ may be assumed at the outset). Thus
$\tilde{v}$ is a supersolution of the eikonal equation, $|Du|^{m}=2^{-m}$ in
$B_{1}$ with homogeneous boundary condition. The latter has the unique
solution $\frac{1}{2}d(y,\partial B_{1})$. Therefore, by comparison,
$\tilde{v}(y,s)\geq\frac{1}{2}d(y,\partial B_{1})$ for all $y\in B_{1},\ s\geq
0$. We remark that comparison holds up to $s=0$ by the (more general) results
of [13].

Finally, using (9) we have

$0\leq\tilde{v}(0,0)\leq\liminf\limits_{n\to+\infty}\frac{v(y_{n},t_{0})}{|y_{n}|\varphi(\frac{1}{2}|y_{n}|)^{\frac{1}{m}}}=0.$

This implies $0=\tilde{v}(0,0)\geq\frac{1}{2}d(0,\partial B_{1})=\frac{1}{2}$,
a contradiction. ∎

######  Lemma 2.7 (“Simplicity”).

Let $(\bar{\lambda},\bar{\phi})$ denote the solution pair of (EP) obtained in
Theorem 2.4 and $v\in USC(\mathbb{R}^{N})$ be a bounded-from-below subsolution
of (EP)${}_{\bar{\lambda}}$. Then, there exists $c\in\mathbb{R}$ such that
$v(x)=\phi(x)+c$ for all $x\in\mathbb{R}^{N}$.

###### Proof.

We will first establish the following

(11) $\textrm{there exists }c\in\mathbb{R}\textrm{ such that
}{v}(x)\leq\phi(x)+c\quad\textrm{for all }x\in\mathbb{R}^{N}.$

To this end, we first prove that $\min_{\bar{B}_{R}}\phi_{R}-{v}$ is bounded
from below, uniformly in $R>0$. Let $\mu>1$—to be chosen later—and define
$\phi_{R}^{\mu}=\mu\phi_{R}$, where $(\lambda_{R},\phi_{R})$ are defined as in
the proof of Theorem 2.4. So defined, $\phi_{R}^{\mu}$ is a solution of

$-\Delta\phi_{R}^{\mu}+\mu^{1-m}|D\phi_{R}^{\mu}|^{m}=\mu(f(x)-\lambda_{R})\quad\textrm{in
}B_{R}.$

Since $\mu>1$ and $1-m<0$, we have $\mu^{1-m}\leq 1$, thus

(12)
$-\Delta\phi_{R}^{\mu}+|D\phi_{R}^{\mu}|^{m}\geq\mu(f(x)-\lambda_{R})\quad\textrm{in
}B_{R}.$

Set $\mu=\mu_{R}:=1+\lambda_{R}-\bar{\lambda}$. By the results of [6] (see
also the proof of Theorem 2.4 above), we indeed have that $\mu_{R}>1$, as well
as $\mu_{R}\searrow 1$ as $R\to+\infty$.

Consider $x^{R}\in\mathrm{argmax}_{\overline{B}_{R}}(v-\phi_{R}^{\mu_{R}})$.
This maximum is achieved in $B_{R}$ because $v\in USC$ and
$\phi_{R}^{\mu_{R}}\to+\infty$ as $x\to\partial B_{R}$. Since $v$ is a
subsolution of (EP)${}_{\bar{\lambda}}$ and $\phi_{R}^{\mu_{R}}\in C^{2}$, we
have

$\bar{\lambda}-\Delta\phi_{R}^{\mu_{R}}(x_{R})+|D\phi_{R}^{\mu_{R}}(x_{R})|^{m}\leq
f(x_{R}).$

Evaluating (18) at $x_{R}$ and subtracting from the previous equation we
obtain $0\geq(\mu_{R}-1)f(x^{R})-\mu_{R}\lambda_{R}+\bar{\lambda}$. From the
definition of $\mu_{R}$, we have

$\displaystyle(\mu_{R}-1)f(x^{R})\leq\mu_{R}\lambda_{R}-\bar{\lambda}=(1+\lambda_{R}-\bar{\lambda})\lambda_{R}-\bar{\lambda}$
$\displaystyle\quad=(\lambda_{R}-\bar{\lambda})(1+\lambda_{R}),$

hence

(13) $\displaystyle f(x_{R})\leq 1+\lambda_{R}\leq 1+\lambda_{1}.$

Here we have used the monotonicity of $\lambda_{R}$ (to clarify, $\lambda_{1}$
denotes the ergodic constant for the domain $B_{1}$). Since the right-hand
side is bounded uniformly with respect to $R$ and $f$ is coercive, $x_{R}$
must remain in a bounded set as $R\to+\infty$, say $B_{\bar{R}}$ for some
$\bar{R}>0$. We stress that $\bar{R}$ is independent of $R$ (thus also of
$\mu_{R}$).

Therefore, for $R>\bar{R}$,

$\displaystyle\max_{B_{R}}(v-\phi_{R}^{\mu_{R}})=(v-\phi_{R}^{\mu_{R}})(x_{R})=\max_{B_{\bar{R}}}(v-\phi_{R}^{\mu_{R}})\leq\left(\max_{B_{\bar{R}}}(v-\bar{\phi})\right)-1=:c,$

where we have used that $\phi_{R}^{\mu_{R}}\to\bar{\phi}$ uniformly over
$B_{\bar{R}}$ as $R\to+\infty$ (recall $\mu_{R}\to 1$ in said limit). This
implies that

$v(x)\leq\phi_{R}^{\mu_{R}}(x)+c\quad\textrm{for all }x\in\mathbb{R}^{N}.$

For any fixed $x\in\mathbb{R}^{N}$, sending $R\to+\infty$ leads to (11),
noting $c$ is independent of $R$.

We now define $\bar{\phi}^{\mu}:=\mu\bar{\phi}$, where again $\mu>1$; thus,

(14)
$-\Delta\bar{\phi}^{\mu}+|D\bar{\phi}^{\mu}|\geq\mu(f(x)-\bar{\lambda})\quad\textrm{in
}\mathbb{R}^{N}.$

Using (11), we have that

$v(x)-\bar{\phi}^{\mu}(x)=v(x)-\bar{\phi}(x)+(1-\mu)\bar{\phi}(x)\leq
c+(1-\mu)\bar{\phi}(x)\quad\textrm{for all }x\in\mathbb{R}^{N}.$

Since $\bar{\phi}$ is coercive (by Lemma 2.6) and $1-\mu<0$, it follows that
$v(x)-\bar{\phi}^{\mu}(x)\to-\infty$ as $|x|\to+\infty$. Therefore,
$v-\bar{\phi}^{\mu}$ attains a global maximum at some $x^{\mu}$, with
$x^{\mu}$ possibly depending on $\mu$. However, proceeding as before, we
obtain that

$\bar{\lambda}-\Delta\bar{\phi}^{\mu}(x^{\mu})+|D\bar{\phi}^{\mu}(x^{\mu})|^{m}\leq
f(x^{\mu}).$

Subtracting from (14) and using the coercivity of $f$, we again have that
$x^{\mu}$ remains in some ball $B_{\tilde{R}}$, with $\tilde{R}$ depending
only on $f$. Hence, for all $\mu>1$, $x^{\mu}\in\overline{B}_{R}$ and (up to a
subsequence) $x^{\mu}\to\hat{x}$ for some
$\hat{x}\in\overline{B}_{\tilde{R}}$.

Recalling now that each $x^{\mu}$ is a global maximum, we have, for all
$\mu>1$,

$\bar{v}(x)-\bar{\phi}^{\mu}(x)\leq
v(x^{\mu})-\bar{\phi}^{\mu}(x^{\mu})\quad\textrm{for all }x\in\mathbb{R}^{N}.$

Fixing $x\in\mathbb{R}^{N}$ and taking the limit $\mu\to 1^{+}$ in the
previous inequality, we obtain

$\bar{v}(x)-\bar{\phi}(x)\leq\bar{v}(\hat{x})-\phi(\hat{x})\quad\textrm{for
all }x\in\mathbb{R}^{N},$

recalling that $v\in USC$. We thus have that $\bar{v}$ and
$\bar{\phi}+\bar{v}(\hat{x})-\phi(\hat{x})$ are respectively sub- and
supersolution of (EP)${}_{\bar{\lambda}}$, satisfy
$\bar{v}\leq\bar{\phi}+\bar{v}(\hat{x})-\phi(\hat{x})$ in $\mathbb{R}^{N}$,
and are equal at $\hat{x}$. Therefore, they are equal by the strong comparison
principle (see e.g. [15], Theorem 3.1 and Remark 3.6), and with this we
conclude. ∎

######  Corollary 2.8.

Using the notation of Theorem 2.3 and the proof of Theorem 2.4,
$\bar{\lambda}=\lambda^{*}$.

###### Proof.

The fact that $(\bar{\lambda},\bar{\phi})$ is a solution pair of (EP) and the
definition of $\lambda^{*}$ given by (3) imply at once that
$\bar{\lambda}\leq\lambda^{*}$.

Assume $\bar{\lambda}<\lambda^{*}$. Again by (3), there exist
$\tilde{\lambda}\in(\bar{\lambda},\lambda^{*})$ and $\tilde{v}\in
C^{2}(\mathbb{R}^{N})$ such that

$\tilde{\lambda}-\Delta\tilde{v}+|D\tilde{v}|^{m}\leq f(x)\quad\text{in
}\mathbb{R}^{N}.$

But $\bar{\lambda}<\tilde{\lambda}$ implies that

$\bar{\lambda}-\Delta\tilde{v}+|D\tilde{v}|^{m}<f(x)\quad\text{in
}\mathbb{R}^{N},$

therefore, by Lemma 2.7, $\tilde{v}\equiv\bar{\phi}+c$ for some
$c\in\mathbb{R}$. We thus obtain

$\bar{\lambda}-\Delta\tilde{v}+|D\tilde{v}|^{m}=f(x)\quad\text{in
}\mathbb{R}^{N},$

a contradiction. ∎

###### Proof of Proposition 1.2.

The result is an immediate consequence of Lemma 2.7 and Corollary 2.8. It
shows, however, the full reach of the previous arguments. ∎

######  Theorem 2.9 (Uniqueness of the ergodic constant).

Assuming (H1) and (H2), the generalized ergodic constant of (EP) is unique in
the following sense: if $(\lambda,\psi)$ is a solution of (EP) and $\psi$ is
bounded from below, then $\lambda=\lambda^{*}$.

###### Proof.

For $m=2$, this result is contained in [6], Theorem 3.1; for $m\in(1,2)$, it
is covered in part (a) of Theorem 2.1 in [1]. ∎

######  Corollary 2.10 (Uniqueness of solutions for (EP)).

Assuming (H1) and (H2), solutions for (EP) are unique up to an additive
constant. More specifically, if $(\lambda,\phi)$ and $(\psi,\nu)$ are two
bounded-from-below solutions of (EP), then $\lambda=\nu$ and
$\psi(x)=\phi(x)+c$ for all $x\in\mathbb{R}^{N}$ for some $c\in\mathbb{R}$.

###### Proof.

This an easy consequence of Theorem 2.9 and the arguments of Lemma 2.7. ∎

######  Proposition 2.11.

Assume (H1) and (H2). Let $(\lambda^{*},\phi)$ denote the unique solution pair
of (EP) given by Theorem 2.9 and $w\in LSC(\mathbb{R}^{N})$ be a supersolution
of (EP)${}_{\lambda^{*}}$. Then, there exists $c\in\mathbb{R}$ such that
$w(x)=\phi(x)+c$ for all $x\in\mathbb{R}^{N}$.

######  Remark 2.12.

The statement of Proposition 2.11 is of course symmetrical to that of Lemma
2.7. We have chosen to state them separately since we are not able to obtain
the analogue of Corollary 2.8 for supersolutions. See Remark 2.13 below.

###### Proof.

The proof is similar to that of Lemma 2.7, so we merely point out a couple of
differences in the construction.

For $\gamma<1$, let $\psi_{R}^{\gamma}:=\gamma\psi_{R}$, where $\psi_{R}$ is
defined as in Proposition 2.5. Thus $\psi_{R}^{\gamma}$ satisfies

$-\Delta\psi_{R}^{\gamma}+|D\psi_{R}^{\gamma}|^{m}\leq\gamma(f_{R}(x)-\nu_{R})\leq\gamma(f(x)-\nu_{R}),$

where the second inequality follows from the definition of $f_{R}$ as an
extension of $\min\\{f,R\\}$.

Consider $\min_{\mathbb{R}^{N}}(w-\psi_{R}^{\gamma})$. This minimum is
attained since $\psi_{R}$ is periodic and therefore bounded in
$\mathbb{R}^{N}$ (though the bound may depend on $R$), while $w$ is coercive
by Lemma 2.6. Thus we may use $\psi_{R}^{\gamma}\in C^{2}$ as a test function
for (EP).

Setting $\gamma_{R}=1+\nu_{R}-\lambda^{*}$, and noting the definition of
$\lambda^{*}$ implies that $\gamma_{R}<1$, we can argue as in Lemma 2.7 to
show that $\min_{\mathbb{R}^{N}}(w-\psi_{R}^{\gamma_{R}})$ is attained in a
fixed compact set. Letting $R\to+\infty$ gives $w(x)\geq\phi(x)+c$ for some
$c\in\mathbb{R}$.

To show equality, we argue exactly as in the second part of Lemma 2.7,
considering the difference $w-\phi^{\gamma}$ for $\gamma<1$, where $\phi$ is
the unique solution of (EP). ∎

######  Remark 2.13.

Both $(\lambda_{R},\phi_{R})$ and $(\nu_{R},\psi_{R})$ are approximations of
the unique bounded-from-below solution $(\lambda^{*},\phi)$ given by Theorem
2.9 and Proposition 1.2. However, there are a couple of differences which
allow us to obtain Lemma 2.7 and Corollary 2.8 for the family
$(\lambda_{R},\phi_{R})$ without appealing to the results of [1], while the
analogous results for $(\nu_{R},\psi_{R})$ cannot be obtained in this way, at
least with similar techniques: first, since $\lambda_{R}$ is related to the
ergodic problem with state-constraints boundary condition, we have the
extremal characterization (8), which implies the monotonicity of
$(\lambda_{R})_{R}$, and in particular that $\lambda_{R}\geq\bar{\lambda}$;
second, $\lambda^{*}$ is defined as a critical value for the existence of
subsolutions of (EP), and the argument of Corollary 2.8 relies heavily on this
fact.

##  3\. Large-time behavior

######  Proposition 3.1.

For any compact $K\subset\mathbb{R}^{N}$, $u(x,t)-\lambda^{*}t$ is bounded
over $K\times(0,+\infty)$. Equivalently,

$\frac{u(\cdot,t)}{t}\to\lambda^{*}\quad\text{locally uniformly in
}\mathbb{R}^{N}.$

###### Proof.

Let $K\subset\mathbb{R}^{N}$. For $R>0$, We define
$V_{R}(x,t)=\phi_{R}(x)+\lambda_{R}t+\sup_{B_{R}}(u_{0}-\phi_{R})$, where
$(\phi_{R},\lambda_{R})$ are defined as in 2.4. We note that $V_{R}$ and $u$
are respectively a supersolution and a subsolution of the _parabolic_ state-
constraint problem in $B_{R}$,

$\left\\{\begin{array}[]{cl}v_{t}-\Delta v+|Dv|^{m}=f(x)&\textrm{in }B_{R}\\\
v_{t}-\Delta v+|Dv|^{m}\geq f(x)&\textrm{on }\partial B_{R}.\\\
v(\cdot,0)=u_{0}&\textrm{in }\overline{B}_{R}.\\\ \end{array}\right.$

Hence, by the comparison principle over a bounded domain (see [5]),
$V_{R}(x,t)\geq u(x,t)$ in $B_{R}\times(0,+\infty)$. It follows that

$\displaystyle\frac{u(x,t)}{t}\leq\lambda_{R}+\frac{\phi_{R}(x)+\sup_{B_{R}}(u_{0}-\phi_{R})}{t},$
$\displaystyle\limsup_{t\to+\infty}\frac{u(x,t)}{t}\leq\lambda_{R}.$

By Theorem 2.4 and Corollary 2.8, we have $\lambda_{R}\to\lambda^{*}$ as
$R\to+\infty$. Thus, taking $R\to+\infty$ in the last inequality we have

$\limsup_{t\to+\infty}\frac{u(x,t)}{t}\leq\lambda^{*}.$

The lower limit for $\frac{u(x,t)}{t}$ is obtained similarly, by considering
the solution pair $(\psi^{R},\nu_{R})$ of Proposition 2.5. In this case we
compare the solution $u(x,t)$ to

$U_{R}(x,t):=\psi_{R}(x)+\nu_{R}t+\inf_{B_{R}}(u_{0}-\psi_{R})$

over $\mathbb{R}^{N}\times(t_{0},+\infty)$ for some $t_{0}>0$. Despite lacking
a general comparison principle for equation (1), this comparison is possible
because we know $\psi_{R}$ is bounded (since it is periodic) and
$u(x,t_{0})\to+\infty$ as $|x|\to+\infty$ for any fixed $t_{0}>0$, by Lemma
2.6. The limit on the right hand side gives the desired lower bound by Theorem
2.9, which in particular implies that $\nu_{R}\to\lambda^{*}$. ∎

###  3.1. Proof of Theorem 1.3

###### Proof of Theorem 1.3.

Step 1: Let $v(x,t)=u(x,t)-\lambda^{*}t$ and define
$\bar{v}(x,t)=\limsup\limits_{t\to+\infty}\ \\!\\!^{*}v(x,t)$. By stability,
$\bar{v}$ is a subsolution of (EP)${}_{\lambda^{*}}$, and is bounded from
below by Proposition 3.1. Hence, by Lemma 1.2, there exists a constant
$\hat{c}$ such that $\bar{v}(x)=\phi(x)+\hat{c}$ for all $x\in\mathbb{R}^{N}$.

Step 2: We claim that there exists a sequence $(t_{k})_{k\in\mathbb{N}}$ such
that

(15) $v(\cdot,t_{k})\to\phi+\hat{c}\quad\textrm{uniformly over }K\textrm{ as
}t_{k}\to+\infty,$

for any compact $K\subset\mathbb{R}^{N}$. This follows from the same
compactness argument given in the proof of Theorem 2 in [8] for the case
$m>2$, which we include here for convenience.

Fix $\hat{x}\in\mathbb{R}^{N}$. By the definition of half-relaxed limits,
there exists a sequence $(x_{n},t_{n})\in\mathbb{R}^{N}\times(0,+\infty)$ such
that $x_{n}\to\hat{x},\ t_{n}\to\infty$, and
$v(x_{n},t_{n})\to\bar{v}(\hat{x})$. Consider
$v_{n}(\cdot):=v(\cdot,t_{n}-1)$. By Proposition 3.1, the sequence $(v_{n})$
is uniformly bounded over compact sets. Furthermore, by the local gradient
bound of Theorem A.1 (in the Appendix), it is also uniformly equicontinuous
over compact sets. Thus, there exists $w_{0}\in C(\mathbb{R}^{N})$ such that,
given a compact $K\subset\mathbb{R}^{N}$, there exists a subsequence
$(v_{n^{\prime}})$ such that $v_{n^{\prime}}\to w_{0}$ uniformly over $K$ as
$n^{\prime}\to\infty$.

Consider now

$w_{n^{\prime}}(x,t)=v(x,t+t_{n^{\prime}}-1)\quad\textrm{for
}(x,t)\in\mathbb{R}^{N}\times(0,+\infty).$

The sequence $(w_{n^{\prime}})$ is again uniformly equicontinuous (in both
space and time variables) due to Corollary A.2. Thus, given also $T>0$, we
have that

(16) $w_{n^{\prime}}\to w\textrm{ uniformly over }K\times[0,T]\quad\textrm{
for some }w\in C(\mathbb{R}^{N}\times(0,+\infty))$

(again passing to a subsequence if necessary—this is omitted for ease of
notation). By the definition of the half-relaxed limit $\bar{v}$,
$w(x,t)\leq\bar{v}(x)=\phi(x)+\hat{c}$ for all
$(x,t)\in\mathbb{R}^{N}\times(0,+\infty)$, and by construction,

(17)
$w(\hat{x},1)=\lim_{n^{\prime}}v(\hat{x},t_{n^{\prime}})=\phi(\hat{x})+\hat{c}.$

Hence, by the Strong Maximum Principle (see Lemma A.3 in the Appendix),
$w-(\phi+\hat{c})$ is constant in $\mathbb{R}^{N}\times[0,1]$. In particular,
$w_{0}(x)=\phi(x)+\hat{c}$ for all $x\in\mathbb{R}^{N}$. Thus, given a compact
$K\subset\mathbb{R}^{N}$, there exists a subsequence $(v_{n^{\prime}})$ such
that $v_{n^{\prime}}\to\phi(x)+\hat{c}$ uniformly over $K$ as
$n^{\prime}\to\infty$, and the claim follows from a simple diagonal argument.

Step 3: Let $(\lambda^{*},\phi)$ denote the solution pair of (EP) such that
$\min_{\mathbb{R}^{N}}\phi=0$; and let $(\lambda_{R},\phi_{R})$ denote the
solution pair of (7), normalized so that $\phi_{R}\to\phi$ locally uniformly
in $\mathbb{R}^{N}$.

As in Lemma 2.7, set $\mu_{R}=1+\lambda^{*}-\lambda_{R}>1$, define
$\phi_{R}^{\mu_{R}}=\mu_{R}\phi_{R}$, and denote by $x_{R}$ any point where
$m_{R}:=\min_{B_{R}}(\phi_{R}^{\mu_{R}}-\phi)$ is achieved. We will show that
$m_{R}=o(1)$ as $R\to+\infty$.

Indeed, arguing as in Lemma 2.7, we have that $x_{R}$ remains in a fixed
compact set as $R\to+\infty$, say $\bar{B}_{\bar{R}}$. Thus, using the uniform
convergence of $\phi_{R}^{\mu_{R}}\to\phi$ over $\bar{B}_{\bar{R}}$, we have

$\displaystyle
m_{R}=\min_{B_{R}}(\phi_{R}^{\mu_{R}}-\phi)=\min_{B_{\bar{R}}}(\phi_{R}^{\mu_{R}}-\phi)\to
0,\quad\text{as }R\to+\infty.$

Let $\hat{K}\subset\mathbb{R}^{N}$, $\epsilon>0$, and $R>0$ such that
$\hat{K}\subset B_{R}$. Using the conslusion of Step 2 above, let
$n_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$,

$v(x,t_{n})\leq\phi(x)+\hat{c}+\epsilon,\quad\textrm{for all
}x\in\bar{B}_{R}.$

We will construct a supersolution of the _parabolic_ state-constraints problem
in $B_{R}$ to bound $v(x,t+t_{n})$ by above for $x\in\hat{K}$ and $t>0$. Write
$M_{R}=(-m_{R})^{+}$, and define

$V_{R}(x,t)=\phi_{R}^{\mu_{R}}+\hat{c}+\epsilon+(\mu_{R}\lambda_{R}-\lambda^{*})t+M_{R},\quad
x\in B_{R},\ t>0.$

Recalling $\phi_{R}^{\mu_{R}}$ satisfies

(18)
$-\Delta\phi_{R}^{\mu_{R}}+{\mu_{R}}^{1-m}|D\phi_{R}^{\mu_{R}}|^{m}={\mu_{R}}(f(x)-\lambda_{R})\quad\textrm{in
}B_{R},$

we have

$\displaystyle\partial_{t}V_{R}-\Delta V_{R}+|DV_{R}|^{m}={}$
$\displaystyle{\mu_{R}}(f(x)-\lambda_{R})+{\mu_{R}}\lambda_{R}-\lambda^{*}$
$\displaystyle\geq{}$ $\displaystyle f(x)-\lambda^{*}\qquad\textrm{in
}B_{R}\times(0,+\infty),$

where we have used that $\mu_{R}>1$ and the assumption that $f\geq 0$.
Furthermore, for any $t\geq 0$,

$V_{R}(x,t)\to+\infty\quad\textrm{as }x\to\partial B_{R},$

since $V_{R}(x,t)\geq\phi_{R}^{\mu_{R}}(x)$ and $\phi_{R}^{\mu_{R}}$ has this
property. Finally,

$\displaystyle V_{R}(x,0)$
$\displaystyle{}=\phi_{R}^{\mu_{R}}(x)+M_{R}+\hat{c}+\epsilon$
$\displaystyle{}\geq\phi_{R}^{\mu_{R}}(x)+\max_{B_{R}}(\phi-\phi_{R}^{\mu_{R}})+\hat{c}+\epsilon$
$\displaystyle{}\geq\phi(x)+\hat{c}+\epsilon,\quad\textrm{for all }x\in
B_{R}.$

Thus, by comparison (over a bounded domain) we have

$V_{R}(x,t)\geq v(x,t+t_{n})\quad\text{in }B_{R}\times(0,+\infty).$

In particular, by the initial choice of $R>0$, this holds in
$\hat{K}\times(0,+\infty)$.

Fix $\hat{t}>0$. The previous inequality establishes that

$v(x,\hat{t}+t_{n})\leq\phi_{R}^{\mu_{R}}(x)+({\mu_{R}}\lambda_{R}-\lambda^{*})\hat{t}+M_{R}+\hat{c}+\epsilon,\quad\text{for
all }x\in\hat{K}.$

Sending $R\to+\infty$, by the uniform convergence of
$\phi_{R}^{\mu_{R}}\to\phi$ over $\hat{K}$, using also that $M_{R}\to 0$ and
$\mu_{R}\to 1$, we have

(19) $v(x,\hat{t}+t_{n})\leq\phi(x)+\hat{c}+\epsilon,\quad\text{for all
}x\in\hat{K}.$

Since $\hat{K}$ and the right-hand side of the previous inequality is
independent of $\hat{t}$, this provides the desired upper bound.

Step 4. To obtain the lower bound corresponding to (19), we define

$\tilde{U}_{R}(x,t)=\psi_{R}^{\gamma_{R}}(x)+\hat{c}-\epsilon+(\gamma_{R}\nu_{R}-\lambda^{*})t+\tilde{m}_{R},$

where $\tilde{m}_{R}=\min_{\mathbb{R}^{N}}(\phi-\psi_{R}^{\gamma_{R}})$.
Arguing as before, we can show that this minimum is achieved in a fixed
compact set and therefore $\tilde{m}_{R}\to 0$ as $R\to+\infty$.

Furthermore,

$\tilde{U}_{R}(x,0)=\psi_{R}^{\gamma_{R}}(x)+\hat{c}-\epsilon+\tilde{m}_{R}\leq\phi(x)+\hat{c}-\epsilon,$

and

$\displaystyle\partial_{t}\tilde{U}_{R}-\Delta\tilde{U}_{R}+|D\tilde{U}_{R}|^{m}\leq\gamma_{R}(f_{R}(x)-\nu_{R})+\gamma_{R}\nu_{R}-\lambda^{*}$
$\displaystyle\qquad=f_{R}(x)-\lambda^{*}\leq f(x)-\lambda^{*}.$

Therefore, $\tilde{U}_{R}(x,t)\leq v(x,t+t_{n})$ for all $x\in\mathbb{R}^{N},\
t>0$. In particular, this holds over $\hat{K}\times(0,+\infty)$. Hence, for
fixed $\hat{t}>0$, taking $R\to+\infty$ in $\tilde{U}_{R}(x,t)\leq
v(x,\hat{t}+t_{n})$ we arrive at

$\phi(x)+\hat{c}-\epsilon\leq v(x,\hat{t}+t_{n})\quad\text{for all
}x\in\hat{K},$

and with this we conclude. ∎

## Appendix A

######  Theorem A.1 (Local Gradient Bounds).

Let $R,\tau>0$.

  1. (a)

There exists $K_{2}>0$ depending only on $m$ and $N$, such that for any $R\geq
R^{\prime}+1>0$ the solution of (EP) satisfies

$\sup_{B_{R^{\prime}}}|D\phi|\leq
K_{2}(1+\sup_{B_{R}}|f|^{\frac{1}{m}}+\sup_{B_{R}}|Df|^{\frac{1}{2m-1}}).$

  2. (b)

If $u$ is a solution of

(20) $\displaystyle u_{t}-\Delta u+|Du|^{m}=f(x)$
$\displaystyle{}\quad\textrm{in }\Omega\times(0,T],$ (21) $\displaystyle
u(x,0)=u_{0}(x)$ $\displaystyle{}\quad\textrm{in }\overline{\Omega},$

where $\Omega$ is a domain of $\mathbb{R}^{N}$ such that
$B_{R+1}\subset\Omega$ and $f\in W^{1,\infty}_{\textit{loc}}(\mathbb{R}^{N})$,
then $u$ is Lipschitz continuous in $x$ in $B_{R}\times[\tau,+\infty)$ and
$|Du(x,t)|\leq L$ for a.e. $x\in B_{R}$, for all $t\geq\tau$, where $L$
depends on $R$ and $\tau$. Moreover this result holds with $\tau=0$ if $u_{0}$
is locally Lipschitz continuous in $\Omega$.

Both results in Theorem A.1 are classical. The estimate in (a) appears as
stated in [17], but can also be inferred from the results of [19] (see also
[23], [22]).

The conclusion of (b) can also be adapted from the results of [21] (that
recovers some of the results from [23]). For a proof closer to our
setting—namely, within the context of viscosity solutions, via the _weak
Bernstein method_ —we refer the reader to Theorem 4.1 in [3]. The viscous
Hamilton-Jacobi Equation (20) is easily shown to satisfy the structure
conditions required therein.

######  Corollary A.2.

Let $R,\tau>0$. The solution $u$ of (20)-(21) is Hölder-continuous of order
$1/2$ in $B_{R}\times[\tau,+\infty)$ and
$\|u\|_{C^{0,1/2}(B_{R}\times[\tau,+\infty))}\leq M$ for some $M>0$ depending
on $R,\tau$, the data $u_{0},f$ and universal constants. Moreover, this result
holds with $\tau=0$ if $u_{0}$ is locally Lipschitz continuous in $\Omega$.

###### Proof.

The proof of Corollary A.2 can be done in two ways: either by using classical
interior parabolic estimates (see [25], Theorem 4.19, and also Theorem 4.36 in
[18]), in which case the Hölder-regularity of the solution $u$ is of some
order $\alpha\in(0,1)$ depending on universal constants, or by the argument of
[4], which implies that a solution which is Lipschitz in $x$ is $1/2$-Hölder-
continuous in $t$.

In both proofs, the result relies on part (b) of Theorem A.1, which implies
that the solution $u$ of (1) satisfies $|Du|\leq L$ in
$B_{R}\times(\tau,+\infty)$, and therefore that $u_{t}-\Delta u$ is bounded in
$B_{R}\times(\tau,+\infty)$. This allows the use of either of the two
arguments just mentioned. ∎

######  Lemma A.3 (Strong Maximum Principle).

Let $R,C>0$. Any upper semicontinuous subsolution of

(22) $u_{t}-\Delta u-C|Du|=0\quad\text{in }B_{R}\times(0,+\infty)$

that attains its maximum at some $(x_{0},t_{0})\in B_{R}\times(0,+\infty)$ is
constant in $B_{R}\times[0,t_{0}]$.

###### Proof.

See, e.g., [13], Corollary 2.4 and [2], Corollary 1. (The latter result
concerns time-independent equations, but the method of proof equally applies
in this context.) ∎

Acknowledgments. A.Q. was partially supported by FONDECYT Grant # 1190282 and
Programa Basal, CMM. U. de Chile. A.R. was partially supported by Fondecyt,
Postdoctorado Nacional 2019, Proyecto Nº 3190858.

## References

  * [1] Ari Arapostathis, Anup Biswas, and Luis Caffarelli.  On uniqueness of solutions to viscous hjb equations with a subquadratic nonlinearity in the gradient.  Communications in Partial Differential Equations, 44(12):1466–1480, Jul 2019.
  * [2] Martino Bardi and Francesca Da Lio.  On the strong maximum principle for fully nonlinear degenerate elliptic equations.  Archiv der Mathematik, 73(4):276–285, 1999.
  * [3] Guy Barles.  Local gradient estimates for second-order nonlinear elliptic and parabolic equations by the weak Bernstein’s method.  Partial Differential Equations and Applications, 2(6):1–15, 2021\.
  * [4] Guy Barles, Samuel Biton, Mariane Bourgoing, and Olivier Ley.  Uniqueness results for quasilinear parabolic equations through viscosity solutions’ methods.  Calculus of Variations and Partial Differential Equations, 18(2):159–179, 2003.
  * [5] Guy Barles and Francesca Da Lio.  On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations.  Journal de Mathématiques Pures et Appliquées, 83(1):53–75, 2004.
  * [6] Guy Barles and Joao Meireles.  On unbounded solutions of ergodic problems in $\mathbb{R}^{m}$ for viscous Hamilton–Jacobi equations.  Communications in Partial Differential Equations, 41(12):1985–2003, 2016.
  * [7] Guy Barles, Alessio Porretta, and Thierry Tabet Tchamba.  On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton–Jacobi equations.  Journal de mathématiques pures et appliquées, 94(5):497–519, 2010.
  * [8] Guy Barles, Alexander Quaas, and Andrei Rodríguez-Paredes.  Large-time behavior of unbounded solutions of viscous hamilton-jacobi equations in rn.  Communications in Partial Differential Equations, 46(3):547–572, 2020.
  * [9] Guy Barles and Panagiotis E Souganidis.  Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations.  SIAM Journal on Mathematical Analysis, 32(6):1311–1323, 2001.
  * [10] Said Benachour, Grzegorz Karch, and Philippe Laurençot.  Asymptotic profiles of solutions to viscous hamilton–jacobi equations.  Journal de mathématiques pures et appliquées, 83(10):1275–1308, 2004.
  * [11] Henri Berestycki, Louis Nirenberg, and SR Srinivasa Varadhan.  The principal eigenvalue and maximum principle for second-order elliptic operators in general domains.  Communications on Pure and Applied Mathematics, 47(1):47–92, 1994\.
  * [12] Piotr Biler, Grzegorz Karch, and Mohammed Guedda.  Asymptotic properties of solutions of the viscous hamilton-jacobi equation.  Journal of Evolution Equations, 4(1):75–97, 2004.
  * [13] Francesca Da Lio.  Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations.  Communications on Pure and Applied Analysis, 3:395–416, 2004.
  * [14] Thierry Gallay and Philippe Laurençot.  Asymptotic behavior for a viscous hamilton-jacobi equation with critical exponent.  Indiana University mathematics journal, pages 459–479, 2007.
  * [15] Yoshikazu Giga and Masaki Ohnuma.  On strong comparison principle for semicontinuous viscosity solutions of some nonlinear elliptic equations.  International Journal of Pure and Applied Mathematics, 22(2):163–182, 2005.
  * [16] Razvan Gabriel Iagar, Philippe Laurençot, and Juan Luis Vázquez.  Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent.  arXiv preprint arXiv:1002.2094, 2010.
  * [17] Naoyuki Ichihara.  Large time asymptotic problems for optimal stochastic control with superlinear cost.  Stochastic Processes and their Applications, 122(4):1248–1275, 2012\.
  * [18] Cyril Imbert and Luis Silvestre.  An introduction to fully nonlinear parabolic equations.  In An introduction to the Kähler-Ricci flow, pages 7–88. Springer, 2013.
  * [19] Jean-Marie Lasry and Pierre-Louis Lions.  Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints.  Mathematische Annalen, 283(4):583–630, 1989.
  * [20] Philippe Laurençot.  Non-diffusive large time behavior for a degenerate viscous hamilton–jacobi equation.  Communications in Partial Differential Equations, 34(3):281–304, 2009.
  * [21] Pierre-Louis Lions.  Generalized solutions of Hamilton-Jacobi equations, volume 69.  London Pitman, 1982.
  * [22] Pierre-Louis Lions.  Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre.  Journal d’Analyse Mathématique, 45(1):234–254, 1985.
  * [23] PL Lions.  Résolution de problemes elliptiques quasilinéaires.  Archive for Rational Mechanics and Analysis, 74(4):335–353, 1980\.
  * [24] Thierry Tabet Tchamba.  Large time behavior of solutions of viscous Hamilton–Jacobi equations with superquadratic Hamiltonian.  Asymptotic Analysis, 66(3-4):161–186, 2010.
  * [25] Lihe Wang.  On the regularity theory of fully nonlinear parabolic equations: I.  Communications on pure and applied mathematics, 45(1):27–76, 1992\.

Alexander Quaas
Email<EMAIL_ADDRESS>
Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla
V-110, Avda. España 1680, Valparaíso, Chile.

Andrei Rodríguez-Paredes
Email<EMAIL_ADDRESS>
Departamento de Matemática y Ciencia de la Computación, Universidad de
Santiago de Chile, Avda. Libertador General Bernardo O’Higgins 3383, Santiago,
Chile.