$L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

Abstract

In this paper, we are interested in $L^{\infty }$ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ decay estimates of weak solutiona.

Introduction

In this paper, we are interested in the $L^{\infty }$ decay estimate of the solution for the initial-boundary-value problem of the nonlinear parabolic equation in the divergence form

$$\begin{aligned} \textstyle\begin{cases} u_{t}=\operatorname{div}(a(x,t,u,\nabla u)) & \text{in } \Omega \times (0, +\infty ), \\ u(x,t)=0 & \text{on }\partial \Omega \times (0, +\infty ), \\ u(x,0)=u_{0}(x) & \text{in } \Omega , \end{cases}\displaystyle \end{aligned}$$

(1.1)

and the degenerate evolution m-Laplacian equation

$$\begin{aligned} \textstyle\begin{cases} u_{t}= \vert u \vert ^{k} \operatorname{div}( \vert \nabla u \vert ^{m-2}\nabla u)+b(u)\cdot \nabla u & \text{in } \Omega \times (0, +\infty ), \\ u(x,t)=0 & \text{on }\partial \Omega \times (0, +\infty ), \\ u(x,0)=u_{0}(x) & \text{in } \Omega , \end{cases}\displaystyle \end{aligned}$$

(1.2)

where $k>0$, Ω is a open set of $\mathbb{R}^{N}$ (not necessary bounded) with smooth boundary ∂Ω, and $a(x,t,u,\xi )$ is a Carathéodory function in $\Omega \times \mathbb{R}^{+}\times \mathbb{R}^{1}\times \mathbb{R}^{N}$, where $\mathbb{R}^{+}=[0, +\infty )$.

The model problem for (1.1) is the so-called doubly nonlinear equation

$$\begin{aligned} \textstyle\begin{cases} u_{t}= \operatorname{div}( \vert u \vert ^{r} \vert \nabla u \vert ^{m-2}\nabla u) & \text{in } \Omega \times (0, +\infty ), \\ u(x,t)=0 & \text{on }\partial \Omega \times (0, +\infty ), \\ u(x,0)=u_{0}(x) & \text{in } \Omega , \end{cases}\displaystyle \end{aligned}$$

(1.3)

with $r>0$ and $1< m< N$.

The interest in parabolic equations (1.1) and (1.2) comes from their mathematical structure. Many results concerning the global existence, blowup, and asymptotic behavior of solutions have been established; see [1–3, 8, 9, 13, 19, 20, 22, 23].

It is well-known that the solution $u(t)$ of the initial value problem

$$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u & \text{in } \mathbb{R}^{N} \times (0, +\infty ), \\ u(x,0)=u_{0}(x) & \text{in }\mathbb{R}^{N} \end{cases}\displaystyle \end{aligned}$$

(1.4)

satisfies the $L^{\infty }$ decay estimate

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{L^{\infty }(R^{N})}\le C \Vert u_{0} \Vert _{L^{q}(R^{N})} t^{-N/2q}, \quad t>0, \end{aligned}$$

(1.5)

with $u_{0}\in L^{q}(\mathbb{R}^{N})$, $q\ge 1$. Estimate (1.5) remains true for the solution of heat equation in a general open set Ω of $\mathbb{R}^{N}$ with zero Dirichlet boundary condition

$$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u & \text{in } \Omega \times (0, + \infty ), \\ u(x,t)=0 & \text{on } \partial \Omega \times (0, +\infty ), \\ u(x,0)=u_{0}(x) & \text{in } \Omega . \end{cases}\displaystyle \end{aligned}$$

(1.6)

Estimate (1.5), or more general estimates

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{L^{\infty }(\Omega )}\le C \Vert u_{0} \Vert ^{\alpha }_{L^{q}(\Omega )} t^{-\lambda }, \quad t>0, \end{aligned}$$

(1.7)

where α and λ are suitable positive constants, is known in the literature as $L^{\infty }$ decay estimates or ultracontractive estimates; see [6, 7, 11, 13, 17, 19].

These estimates have been proved not only for the heat equation but also for various differential problems, linear or nonlinear, degenerate or singular, for example, the evolution m-Laplacian equation, the porous media equation, the fast equation, and the doubly nonlinear equation; see [1–3, 8, 9, 11, 15, 17–19] and the references therein. The importance of estimate (1.7) describes the behavior of solution as $t\to 0$ and $t\to +\infty$.

The proofs of these estimates vary from problem to problem. In many cases, suitable families of logarithmic Sobolev inequalities are derived. These inequalities are similar to the well-known Gross logarithmic Sobolev inequalities [11].

Porzio [17] investigated the solution of the Leray–Lions-type problem

$$\begin{aligned} \textstyle\begin{cases} u_{t}=\operatorname{div} (a(x, t, u, \nabla u)) & \text{in } \Omega \times (0, +\infty ), \\ u(x,t)=0 & \text{on } \partial \Omega \times (0, +\infty ), \\ u(x,0)=u_{0}(x) & \text{in } \Omega , \end{cases}\displaystyle \end{aligned}$$

(1.8)

where $a(x,t,s,\xi )$ is a Carathéodory function satisfying the following structure condition:

$$\begin{aligned} a(x,t,s,\xi )\xi \ge \theta \vert \xi \vert ^{m}, \quad \forall (x,t,s,\xi ) \in \Omega \times \mathbb{R}^{+}\times \mathbb{R}^{1}\times \mathbb{R}^{N}, \end{aligned}$$

(1.9)

with $\theta >0$. By the integral inequalities method Porzio derived the $L^{\infty }$ decay estimate of the form (1.7) with $C=C(N,q,m,\theta )$, $\alpha =\frac{mq}{N(m-2)+mq}$, and $\lambda =\frac{N}{N(m-2)+mq}$. We see that the equation in problem (1.8) is in the divergence form.

Recently, Ghoul et al. [10] studied the Cauchy problem of the parabolic equation

$$\begin{aligned} \textstyle\begin{cases} u_{t}=-(-\Delta )^{m}u+u \vert u \vert ^{p-1} ,& (x,t)\in \mathbb{R}^{N} \times (0, +\infty ), \\ u(x,0)=u_{0}(x), & x\in \mathbb{R}^{N}, \end{cases}\displaystyle \end{aligned}$$

(1.10)

and derived an estimate for $\Vert u(t) \Vert _{L^{\infty }(\mathbb{R}^{N})}$ with $u_{0}\in L^{\infty }(\mathbb{R}^{N})$ by a formal approach based on spectral analysis. Similar consideration can been found in [12, 21].

In this paper, we derive the $L^{\infty }$ decay estimate like (1.7) for the solutions of problems (1.1) and (1.2). Our method is different from that in [17], and we will use a modified Moser technique as in [4, 5, 15] to get an $L^{\infty }$ decay estimate. Since the equation in (1.2) is not in the divergence form, it seems difficult to derive estimate (1.7) by the integral inequalities method in [17].

This paper is organized as follows. In Sect. 2, we state the main results and present some needed lemmas. In Sect. 3, we use these lemmas to derive $L^{\infty }$ decay estimates for the solutions of (1.1). The $L^{\infty }$ decay estimates for the solutions of (1.2) are established in Sect. 4.

Preliminaries and main results

We first make the following assumptions.

$(H_{1})$:

$a(x,t,u, \xi )$ is a Carathéodory function and satisfies the structure condition

$$\begin{aligned} a(x,t,u, \xi )\xi \ge \alpha _{0} \vert u \vert ^{r} \vert \xi \vert ^{m}, \quad \forall (x,t,u, \xi )\in \Omega \times \mathbb{R}^{+}\times \mathbb{R}^{1} \times \mathbb{R}^{N}, \end{aligned}$$

(2.1)

for some $\alpha _{0}>0$ and $r\ge 0$, where $1+\beta < m< N$ and $0<\beta =(m-1)(r+m-1)^{-1}\le 1$.

$(H_{2})$:

the initial data $u_{0}\in L^{q}(\Omega )$, $q\ge 1$.

As in [20], we introduce a new independent variable $u= \vert v \vert ^{\beta -1}v$. Then from (2.1) it follows that the principal part of the equation in (1.1) satisfies

$$\begin{aligned} a(x,t,u, \nabla u)\nabla v\ge \alpha _{0} \beta ^{m-1} \vert \nabla v \vert ^{m}. \end{aligned}$$

(2.2)

Instead of (1.1), we consider the initial-boundary-value problem

$$\begin{aligned} \textstyle\begin{cases} ( \vert v \vert ^{\beta -1}v)_{t}=\operatorname{div}(a(x,t, \vert v \vert ^{\beta -1}v,\nabla ( \vert v \vert ^{ \beta -1}v))) \quad \text{in } \Omega \times (0, +\infty ), \\ v(x,t)=0, \quad \text{on } \partial \Omega \times (0, +\infty ), \quad \quad v(x,0)=v_{0}(x), \quad \text{in } \Omega , \end{cases}\displaystyle \end{aligned}$$

(2.3)

with $v_{0}(x)= \vert u_{0}(x) \vert ^{-1+1/\beta }u_{0}(x)$.

Let $\Vert \cdot \Vert _{p}$ and $\Vert \cdot \Vert _{1,p}$ denote the norms in the Banach spaces $L^{p}(\Omega )$ and $W^{1,p}(\Omega )$, respectively, $1\le p\le \infty$. We often drop the letter Ω in these notations. In the following, we will consider (2.3) instead of (1.1), with v replaced by u in (2.3) for convenience.

Definition 1

A measurable function $u(x,t)$ on $\Omega \times (0, \infty )$ is said to be a global weak solution of problem (2.3) if $u(x,t)\in L_{\mathrm{loc}}^{\beta }(\mathbb{R}^{+}\times \Omega )$, $a(x,t, \vert u \vert ^{\beta -1}u,\nabla ( \vert u \vert ^{\beta -1}u)) \in L_{\mathrm{loc}}^{1}( \mathbb{R}^{+}; L^{1}(\Omega ))$, and the equality

$$\begin{aligned} \begin{aligned} \int _{0}^{t}& \int _{\Omega } \bigl\{ - \vert u \vert ^{\beta -1}u \varphi _{t}-a \bigl(x,\tau , \vert u \vert ^{\beta -1}u,\nabla \bigl( \vert u \vert ^{\beta -1}u \bigr) \bigr) \nabla \varphi \bigr\} \,dx \,d \tau \\ &= \int _{\Omega }{ \bigl\vert u_{0}(x) \bigr\vert ^{\beta -1}u_{0}(x)\varphi (x,0)- \bigl\vert u(x,t) \bigr\vert ^{ \beta -1}u(x,t)\varphi (x,t)}\,dx \end{aligned} \end{aligned}$$

(2.4)

is valid for any $\varphi \in C^{1}(\mathbb{R}^{+},C_{0}^{1}(\Omega ))$ and $t>0$.

Our first main result reads as follows.

Theorem 1

Assume $(H_{1})$–$(H_{2})$. If $u(t)$ is a global weak solution of (2.3), then it satisfies

$$\begin{aligned} u(t) \in L^{\infty } \bigl(\mathbb{R}^{+}; L^{q}(\Omega ) \bigr) \cap L_{\mathrm{loc}}^{m-1} \bigl((0, \infty ); W_{0}^{m-1}(\Omega ) \bigr) \end{aligned}$$

(2.5)

and the $L^{\infty }$ decay estimate

$$\begin{aligned}& \bigl\Vert u(t) \bigr\Vert _{q}\le \Vert u_{0} \Vert _{q}, \quad t>0, \end{aligned}$$

(2.6)

$$\begin{aligned}& \bigl\Vert u(t) \bigr\Vert _{\infty }\le C_{0} \Vert u_{0} \Vert ^{\mu }_{q} t^{-\lambda }, \quad t>0, \end{aligned}$$

(2.7)

with $\mu =\frac{mq}{MN+mq}$, $\lambda =\frac{N}{MN+mq}$, $M=m-1-\beta >0$, and $C_{0}=C_{0}(N,m,q)$.

Remark 1

The existence of a global weak solution for (2.3) can be established similarly as in [4, 15, 20].

For the degenerate evolution m-Laplacian problem (1.2), Passo and Luckhaus [16] considered the global existence and blowup of solution for $m=2$, $k=1$ by the lower and upper solution method. For $m=2$, $k>1$, blowup and asymptotic behavior of solution have been established by Wiegner [22] and Winkler [23]. Here we derive an $L^{ \infty }$ decay estimate for the solution of (1.2) with $k>0$, $1< m< N$.

For problem (1.2), we assume:

$(H_{3})$:

Let $B(u)=(B_{1}(u),B_{2}(u),\ldots, B_{N}(u))$, $B'(u)=(B'_{1}(u),B'_{2}(u),\ldots, B'_{N}(u))$, where $B'(u)=b(u)=(b_{1}(u),b_{2}(u),\ldots, b_{N}(u))$, $b_{i}(u)\in C^{1}(\mathbb{R}^{1})$, $i=1,2,\ldots,N$. There exist $k_{1}, \gamma \ge 0$, such that

$$\begin{aligned} \bigl\vert B(u) \bigr\vert \le k_{1} \vert u \vert ^{1+\gamma }, \quad\quad \bigl\vert B'(u) \bigr\vert \le k_{1} \vert u \vert ^{\gamma }, \quad \forall u\in \mathbb{R}^{1}; \end{aligned}$$

(2.8)

$(H_{4})$:

$u_{0}\in L^{q}(\Omega )$, $q\ge 1$.

Definition 2

([16, 22, 23])

A measurable function $u(t)=u(x,t)$ on $\Omega \times (0, +\infty )$ is said to be a global weak solution of problem (1.2) if $u(t)\in X=L^{\infty }(\mathbb{R}^{+}, L^{q}(\Omega ))$, $\vert u \vert ^{(k-1)/m}u \in L^{m}_{\mathrm{loc}}((0, +\infty ); W_{0}^{1,m}( \Omega ))$, $\vert u \vert ^{(k-1)/(m-1)}u \in L^{m-1}_{\mathrm{loc}}((0, \infty ); W_{0}^{1,m-1}( \Omega ))$,

$$\begin{aligned} \begin{aligned} & \int _{0}^{t} \int _{\Omega } \bigl\{ -u\phi _{t}+ \vert \nabla u \vert ^{m-2} \nabla u \cdot \nabla \bigl( \vert u \vert ^{k}\phi \bigr)+B(u)\cdot \nabla \phi \bigr\} \,dx\,d \tau \\ &\quad = \int _{\Omega }u(x,t)\phi (x,t)\,dx- \int _{\Omega } u_{0}(x)\phi (x,0)\,dx \end{aligned} \end{aligned}$$

(2.9)

for all $\varphi \in C^{1}(R^{+},C_{0}^{1}(\Omega ))$ and $t>0$.

Our second main result is the following:

Theorem 2

Suppose that $(H_{3})$–$(H_{4})$ hold and $k\ge 0$. If $u(t)$ is a global weak solution of (1.2), then $u(t)$ satisfies the following $L^{\infty }$ estimates:

$$\begin{aligned}& \bigl\Vert u(t) \bigr\Vert _{q}\le \Vert u_{0} \Vert _{q}, \quad t>0, \end{aligned}$$

(2.10)

$$\begin{aligned}& \bigl\Vert u(t) \bigr\Vert _{\infty }\le C_{0} \Vert u_{0} \Vert ^{\alpha }_{q} t^{-\lambda }, \quad t>0, \end{aligned}$$

(2.11)

with $\alpha =\frac{qm}{MN+mq}$, $\lambda =\frac{N}{MN+mq}$, $M=k+m-2>0$, and $C_{0}=C_{0}(N,m,q)$.

To derive above results, we will use the following lemmas.

Lemma 1

Let $y(t)$ be a nonnegative differentiable function on $(0, \infty )$ satisfying

$$\begin{aligned} y'(t)+At^{\mu }y^{1+\theta }(t)\le 0,\quad t\geq 0, \end{aligned}$$

with $A, \theta >0$, $\mu \geq 0$. Then we have

$$\begin{aligned} y(t)\le \bigl(A\theta / (1+\mu ) \bigr)^{-1/\theta }t^{-(1+\mu )/\theta },\quad t>0. \end{aligned}$$

Lemma 2

(Gagliardo–Nirenberg-type inequality)

Let Ω be a domain (not necessary bounded) in $\mathbb{R}^{N}$ with smooth boundary ∂Ω. Let $\beta \geq 0$, $N>m\geq 1$, $q\geq 1+\beta$, and $1\leq r \leq q \leq (1+\beta )Nm/(N-m)$. Then for $\vert u \vert ^{\beta }u\in W_{0}^{1,m}(\Omega )$, we have

$$\begin{aligned} \Vert u \Vert _{q}\leq C_{0}^{1/(\beta +1)} \Vert u \Vert _{r}^{1-\theta } \bigl\Vert \nabla \bigl( \vert u \vert ^{ \beta }u \bigr) \bigr\Vert _{m}^{\theta /(\beta +1)} \end{aligned}$$

with $\theta =(1+\beta )(r^{-1}-q^{-1})/(N^{-1}-m^{-1}+(1+\beta )r^{-1})$, where the constant $C_{0}$ depends only on m, N.

The proof of Lemma 2 can be obtained from the well-known Gagliardo–Nirenberg–Sobolev inequality and the interpolation inequality, and we omit it here.

Proof of Theorem 1

In this section, we assume that all assumptions in Theorem 1 are satisfied. As in [4, 5, 15], we derive a priori estimates of the smooth approximate solutions $u(t)$, and our argument will be justified through such an approximate procedure.

Proof of Theorem 1

First, we take $f_{n}(s)$ ($n=1,2,\ldots$) such that $f_{n}(s)\to f(s)= \vert s \vert ^{q-2}s$ uniformly in $\mathbb{R}^{1}$ as $n\to \infty$.

For $1< q<2$, we choose $f_{n}^{+}(s)=a_{n}s^{2}+b_{n}s$ if $0\le ns\le 1$ and $f_{n}^{+}(s)=s^{q-1}$ if $ns\ge 1$, where $a_{n}=(q-2)n^{3-q}$, $b_{n}=(3-q)n^{2-q}$. Further, let $f_{n}(s)$ be the odd extension of $f_{n}^{+}(s)$ in $\mathbb{R}^{1}$.

If $q\ge 2$, then we take $f_{n}(s)= \vert s \vert ^{q-2}s$. For $q=1$, we let

$$\begin{aligned} f_{n}(s)= \textstyle\begin{cases} 1, & s\ge 1/n, \\ ns(2-ns), & 0\le s \le 1/n, \\ -ns(2+ns), &-1/n\le s \le 0, \\ -1,& s< -1/n. \end{cases}\displaystyle \end{aligned}$$

(3.1)

Then we easily verify that $f_{n}(s)\in C^{1}(\mathbb{R}^{1})$, $f_{n}(s)\to f(s)= \vert s \vert ^{q-2}s$ uniformly in $\mathbb{R}^{1}$ as $n\to \infty$.

Let $\varphi _{n}^{+}(s)=s^{\beta -1}$ if $ns\ge 1$, $\varphi _{n}^{+}(s)=A_{n}s +B_{n}$ if $0\le ns\le 1$, where $A_{n}=(\beta -1)n^{2-\beta }$, $B_{n}=(2-\beta )n^{1-\beta }$. Further, let $\varphi _{n}(s)$ be the even extension of $\varphi _{n}^{+}(s)$ in $\mathbb{R}^{1}$. Obviously, $\varphi _{n}(s)\in C^{1}( \mathbb{R}^{1} )$, and $\varphi _{n}(s)\to \varphi (s)= \vert s \vert ^{\beta -1}$ uniformly in $\mathbb{R}^{1}$ as $n\to \infty$.

Let $u_{0,n}\in C_{0}^{2}(\Omega )$ and $u_{0,n}\to u_{0}$ in $L^{q}(\Omega )$ as $n\to \infty$. We take the approximate problem of (2.3) of the form

$$\begin{aligned} \textstyle\begin{cases} \varphi _{i}(u)u_{t}= \operatorname{div}(a(x,t, \vert u \vert ^{\beta -1}u,\nabla ( \vert u \vert ^{ \beta -1}u))) & \text{in } \Omega \times (0, \infty ), \\ u(x,t)=0 & \text{on }\partial \Omega \times (0, \infty ), \\ u(x,0)=u_{0,i}(x) & \text{in }\Omega , \end{cases}\displaystyle \end{aligned}$$

(3.2)

for $i=1,2,\ldots$ .

Then problem (3.2) has a unique smooth solution $u_{i}(x,t)$; see [14]. We further always write u instead of $u_{i}$ and $u^{p}$ for $\vert u \vert ^{p-1}u$ when $p>0$.

Multiplying the equation in (3.2) by $f_{k}(u)\varphi _{i}^{-1}(u)$, we obtain

$$\begin{aligned} \begin{aligned} & \int _{\Omega }f_{k}(u)u_{t}\,dx \\ &\quad =- \int _{\Omega }a \bigl(x,t, \vert u \vert ^{\beta -1}u, \nabla \bigl( \vert u \vert ^{\beta -1}u \bigr) \bigr) \nabla u \bigl(f'_{k}(u)\varphi _{i}(u)-\varphi ' _{i}(u)f_{k}(u) \bigr)\varphi _{i}^{-2}(u)\,dx, \end{aligned} \end{aligned}$$

(3.3)

where

$$ f'_{k}(u)\varphi _{i}(u)-\varphi '_{i}(u)f_{k}(u)\geq 0. $$

By $(H_{1})$ we have

$$\begin{aligned} \begin{aligned} &a \bigl(x,t, \vert u \vert ^{\beta -1}u,\nabla \bigl( \vert u \vert ^{\beta -1}u \bigr) \bigr) \nabla u \\ &\quad =\beta ^{-1}a \bigl(x,t, \vert u \vert ^{\beta -1}u,\nabla \bigl( \vert u \vert ^{\beta -1}u \bigr) \bigr)\nabla \bigl( \vert u \vert ^{ \beta -1}u \bigr) \vert u \vert ^{1-\beta } \\ &\quad \geq \alpha _{0}\beta ^{-1} \vert u \vert ^{\beta r} \bigl\vert \nabla \bigl( \vert u \vert ^{\beta -1}u \bigr) \bigr\vert ^{m} \vert u \vert ^{1- \beta }\geq 0. \end{aligned} \end{aligned}$$

(3.4)

Hence from (3.3) and (3.4) it follows that

$$\begin{aligned} \begin{aligned} \int _{\Omega }f_{k}(u)u_{t}\,dx\leq 0. \end{aligned} \end{aligned}$$

(3.5)

Letting $k\to \infty$ in (3.5) gives

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{q}\le \Vert u_{0} \Vert _{q},\quad t\ge 0. \end{aligned}$$

(3.6)

We now derive an $L^{\infty }$ decay estimate for the solution $u_{i}(t)$ of (3.2). Multiplying the equation in (3.2) by $\varphi _{i}^{-1}(u) \vert u \vert ^{p-2}u$, $p\geq 2$, we have

$$\begin{aligned} \begin{aligned} \frac{1}{p} \frac{d}{dt} \Vert u \Vert _{p}^{p}+ \int _{\Omega }a \bigl(x,t, \vert u \vert ^{ \beta -1}u, \nabla \bigl( \vert u \vert ^{\beta -1}u \bigr) \bigr)\nabla u E_{i}[u]\,dx=0, \end{aligned} \end{aligned}$$

(3.7)

where

$$\begin{aligned} \begin{aligned} E_{i}[u]= \bigl((p-1) \vert u \vert ^{p-2}\varphi _{i}(u)-\varphi _{i}^{-1}(u) \vert u \vert ^{p-2}u \bigr) \varphi _{i}^{-2}(u)\geq \frac{p-\beta }{4} \vert u \vert ^{p-\beta -1}. \end{aligned} \end{aligned}$$

(3.8)

Noting that $\beta =(m-1)/(r+m-1)$, from (3.4) we get that

$$\begin{aligned} \begin{aligned} a \bigl(x,t, \vert u \vert ^{\beta -1}u,\nabla \bigl( \vert u \vert ^{\beta -1}u \bigr) \bigr) \nabla u &\geq \beta ^{-1}\alpha _{0} \vert u \vert ^{\beta r} \bigl\vert \nabla \bigl( \vert u \vert ^{\beta -1}u \bigr) \bigr\vert ^{m} \vert u \vert ^{1- \beta } \\ &=\alpha _{0}\beta ^{m-1} \vert \nabla u \vert ^{m}. \end{aligned} \end{aligned}$$

(3.9)

Hence from (3.7)–(3.9) it follows that

$$\begin{aligned} \frac{1}{p}\frac{d}{dt} \Vert u \Vert _{p}^{p}+C_{1} p \biggl(\frac{m}{p+M} \biggr)^{m} \int _{\Omega } \bigl\vert \nabla u^{\frac{p+M}{m}} \bigr\vert ^{m}\,dx\leq 0, \end{aligned}$$

(3.10)

where $M=m-1-\beta >0$. Then (3.10) implies that

$$\begin{aligned} \frac{d}{dt} \bigl\Vert u(t) \bigr\Vert _{p}^{p}+C_{1} p^{2-m} \bigl\Vert \nabla u^{\frac{p+M}{m}} \bigr\Vert _{m}^{m}\le 0,\quad \forall t>0. \end{aligned}$$

(3.11)

Let C, $C_{j}$ be general constants independent of p, i, n changeable from line to line. We now employ Moser’s technique as in [4, 5, 15]. Set $R>1+M/q$, $p_{1}=q$, $p_{n}=Rp_{n-1}-M$, $\theta _{n}=RN(1-p_{n-1}p_{n}^{-1})(m+N(R-1))^{-1}$, $\beta _{n}=(p_{n}+M)\theta _{n}^{-1}$, $n=2,3,\ldots$ .

From Lemma 2 we see that

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{p_{n}}\leq C^{\frac{m}{p_{n}+M}} \Vert u \Vert _{p_{n-1}}^{1- \theta _{n}} \bigl\Vert \nabla u^{\frac{p_{n}+M}{m}} \bigr\Vert _{m}^{m\theta _{n}/(p_{n}+M)}. \end{aligned}$$

(3.12)

Inserting this into (3.11) $(p=p_{n})$ yields

$$\begin{aligned} \frac{d}{dt} \bigl\Vert u(t) \bigr\Vert _{p_{n}}+C_{1}C^{\frac{-m}{\theta _{n}}} p_{n}^{2-m} \Vert u \Vert _{p_{n-1}}^{M-\beta _{n}} \Vert u \Vert _{p_{n}}^{1+\beta _{n}}\leq 0, \quad \forall t>0. \end{aligned}$$

(3.13)

We claim that there exist bounded sequences $\{\xi _{n}\}$ and $\{\lambda _{n}\}$ such that

$$\begin{aligned} \begin{aligned} \bigl\Vert u(t) \bigr\Vert _{p_{n}} \leq \xi _{n}t^{-\lambda _{n}}, \quad \forall t>0, \end{aligned} \end{aligned}$$

(3.14)

where $\lambda _{n}=(1+\lambda _{n-1}(\beta _{n}-M))/\beta _{n}$. It is not difficult to show that $\lambda _{n}\to \lambda =\frac{N}{MN+mq}$ as $n\to \infty$.

In fact, let $\xi _{1}= \Vert u_{0} \Vert _{q}$ and $\lambda _{1}=0$. If (3.14) is true for $n-1$, the from (3.13) it follows that

$$\begin{aligned} \frac{d}{dt} \bigl\Vert u(t) \bigr\Vert _{p_{n}}+C_{1}C^{\frac{-m}{\theta _{n}}} p_{n}^{1-m} \xi _{n}^{M-\beta _{n}}t^{\lambda _{n-1}(\beta _{n}-M)} \Vert u \Vert _{p_{n}}^{1+ \beta _{n}}\leq 0, \quad \forall t>0. \end{aligned}$$

(3.15)

An application of Lemma 1 to (3.15) yields

$$\begin{aligned} \begin{aligned} \bigl\Vert u(t) \bigr\Vert _{p_{n}} &\leq \bigl(C_{1} C^{\frac{-m}{\theta _{n}}} p_{n}^{1-m} \xi _{n-1}^{M-\beta _{n}}\beta _{n}/ \bigl(1+\lambda _{n-1}(\beta _{n}-M) \bigr) \bigr)^{-1/ \beta _{n}}t^{-(1+\lambda _{n-1}(\beta _{n}-\mu )) / \beta _{n}} \\ &= \bigl(C_{1} C^{\frac{-m}{\theta _{n}}} \bigr)^{-1/ \beta _{n}}\lambda _{n}^{1/ \beta _{n}} p_{n}^{(m-1)/\beta _{n}} \xi _{n-1}^{(\beta _{n}-M)/ \beta _{n}} t^{-\lambda _{n}}. \end{aligned} \end{aligned}$$

(3.16)

Since

$$\begin{aligned} \lim_{n\to \infty }\frac{p_{n}}{\beta _{n}}=\frac{M+2}{N(M+1)}, \end{aligned}$$

we see that there exists a constant $\lambda _{0}>0$, independent of n, such that

$$\begin{aligned} \begin{aligned} \bigl\Vert u(t) \bigr\Vert _{p_{n}}\leq (\lambda _{0}p_{n})^{\lambda _{0}/p_{n}} \xi _{n-1}^{1-M/\beta _{n}} t^{-\lambda _{n}}, \quad t>0. \end{aligned} \end{aligned}$$

(3.17)

Hence we define $\xi _{n}$ inductively by

$$\begin{aligned} \xi _{n}=(\lambda _{0}p_{n})^{\lambda _{0}/p_{n}} \xi _{n-1}^{1-M/ \beta _{n}} \end{aligned}$$

(3.18)

for $n=2,3,\ldots$ with $\xi _{1}= \Vert u_{0} \Vert _{q}$. Here, setting $\omega _{n}=mp_{n}+MN$, $p_{1}=q$, and $p_{n}=Rp_{n-1}-M$, by direct calculation we get

$$\begin{aligned} \begin{aligned} \frac{\beta _{n}-M}{\beta _{n}}= \frac{\omega _{n}}{p_{n}}\cdot \frac{p_{n-1}}{\omega _{n-1}} \end{aligned} \end{aligned}$$

(3.19)

and

$$\begin{aligned} \begin{aligned} \prod _{k=2}^{n}\frac{\beta _{k}-M}{\beta _{k}}= \frac{\omega _{n}}{p_{n}} \cdot \frac{p_{1}}{\omega _{1}}= \frac{MN+p_{n}m}{p_{n}}\cdot \frac{q}{mq+MN}. \end{aligned} \end{aligned}$$

(3.20)

It is easy to show that

$$\begin{aligned} \begin{aligned} \lim_{n\to \infty } \prod_{k=2}^{n} \frac{\beta _{k}-M}{\beta _{k}}= \mu = \frac{mq}{mq+MN}. \end{aligned} \end{aligned}$$

(3.21)

On the other hand, the definition of $\xi _{n}$ gives

$$\begin{aligned} \begin{aligned} \log \xi _{n}&= \frac{\lambda _{0}}{p_{n}}(\log \lambda _{0}+ \log p_{n})+ \biggl(1- \frac{M}{\beta _{n}} \biggr)\log \xi _{n-1} \\ &=\frac{\lambda _{0}}{p_{n}}(\log \lambda _{0}+\log p_{n})+ \biggl(1- \frac{M}{\beta _{n}} \biggr) \biggl(\frac{\lambda _{0}}{p_{n}}(\log \lambda _{0}+ \log p_{n-1}) \\ &\quad {}+ \biggl(1-\frac{M}{\beta _{n-1}} \biggr)\log \xi _{n-2} \biggr) \\ &\leq \lambda _{0}\sum_{k=2}^{n} \frac{\log \lambda _{0}+\log p_{k}}{p_{k}}+\prod_{k=2}^{n} \biggl(1- \frac{M}{\beta _{k}} \biggr)\log \xi _{1}. \end{aligned} \end{aligned}$$

(3.22)

Hence

$$\begin{aligned} \begin{aligned} \log \xi _{n}\leq C_{0}+\frac{MN+p_{n}m}{p_{n}}\cdot \frac{q}{m q+MN}\log \xi _{1} \end{aligned} \end{aligned}$$

(3.23)

with some $C_{0}>0$ independent of n. Then

$$\begin{aligned} \begin{aligned} \log \xi _{n}\leq C_{0}+\mu \log \xi _{1} \end{aligned} \end{aligned}$$

(3.24)

and

$$\begin{aligned} \begin{aligned} \xi _{n}\leq e^{C_{0}}\xi _{1} ^{\mu }=C_{1} \Vert u_{0} \Vert _{q}^{ \mu } \quad t>0. \end{aligned} \end{aligned}$$

(3.25)

Then, letting $n\to \infty$ in (3.14), we obtain (2.7) and finish the proof of Theorem 1. □

Proof of Theorem 2

In this section, we derive $L^{\infty }$ decay estimates of solutions for the degenerate evolution m-Laplacian problem (1.2).

Similarly as in the proof of Theorem 1, we take $u_{0,n}\in C_{0}^{2}(\Omega )$ such that $u_{0,n}\to u_{0}$ in $L^{q}(\Omega )$. Further, we choose $\phi _{n}(s)\in C^{1}(\mathbb{R}^{1})$, $\phi _{n}(s)\to \phi (s)$ uniformly in $\mathbb{R}^{1}$.

In fact, for $n=1,2,\ldots$ , we define $\phi _{n}(s)= \vert s \vert ^{k}+n^{-k}$ if $k>1$ and

$$\begin{aligned} \phi _{n}(s)= \textstyle\begin{cases} \vert s \vert ^{k}+n^{-k} & \text{for } \vert s \vert \ge n^{-1}, \\ s^{2}n^{2-k}(3-k+(k-2)n \vert s \vert )+n^{-k} & \text{for } \vert s \vert \le n^{-1} \end{cases}\displaystyle \end{aligned}$$

(4.1)

if $0< k\le 1$.

We now consider the following approximate problem for (1.2):

$$\begin{aligned} \textstyle\begin{cases} u_{t}=\phi _{i}(u) \operatorname{div}(( \vert \nabla u \vert ^{2}+i^{-1})^{m/2}\nabla u)+b(u) \nabla u & \text{in } \Omega \times (0, \infty ), \\ u(x,t)= 0 & \text{on }\partial \Omega \times (0, \infty ), \\ u(x,0)=u_{0,i} & \text{in } \Omega , \end{cases}\displaystyle \end{aligned}$$

(4.2)

for $i=1,2,\ldots$ .

Problem (4.2) is a standard quasilinear parabolic equation and admits a unique smooth solution $u_{i}(x,t)$ for each i; see [4, 5, 14, 15]. For convenience, we denote $u_{i}$ by u and $\vert u \vert ^{p-1}u$ by $u^{p}$ if $p>0$.

Multiplying the equation in (4.2) by $\vert u \vert ^{q-2}u$ (if $q>1$), we obtain

$$\begin{aligned} \begin{aligned} \frac{1}{q} \frac{d}{dt} \int _{\Omega } \vert u \vert ^{q}\,dx+ \int _{ \Omega } \vert \nabla u \vert ^{m} \bigl(\phi '_{i}(u) \vert u \vert ^{q-2}u+(q-1)\phi _{i}(u) \vert u \vert ^{q-2} \bigr)\,dx \leq 0. \end{aligned} \end{aligned}$$

(4.3)

Note that

$$\begin{aligned} \begin{aligned} \phi '_{i}(u) \vert u \vert ^{q-2}u+(q-1)\phi _{i}(u) \vert u \vert ^{q-2} \,dx \ge 0. \end{aligned} \end{aligned}$$

(4.4)

Then

$$\begin{aligned} \begin{aligned} \frac{1}{q} \frac{d}{dt} \int _{\Omega } \vert u \vert ^{q}\,dx\le 0. \end{aligned} \end{aligned}$$

(4.5)

This implies that

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{q}\le \Vert u_{0} \Vert _{q}, \quad \forall t\ge 0. \end{aligned}$$

(4.6)

If $q=1$, then we multiply the equation in (4.2) by $f_{n}(u)$, where $f_{n}(u)$ is defined by (3.1). Similarly, we can get estimate (4.6).

To derive an $L^{\infty }$ decay estimate of solutions for (4.2), we multiply the equation in (4.2) by $\vert u \vert ^{p-2}u (p\geq q)$ and obtain

$$\begin{aligned} \begin{aligned} \frac{1}{p} \frac{d}{dt} \int _{\Omega } \vert u \vert ^{p}\,dx+ \int _{ \Omega } \vert \nabla u \vert ^{m} \bigl(\phi '_{i}(u) \vert u \vert ^{p-2}u+(p-1)\phi _{i}(u) \vert u \vert ^{p-2} \bigr)\,dx \leq 0. \end{aligned} \end{aligned}$$

(4.7)

Note that

$$\begin{aligned} \begin{aligned} \phi '_{i}(u) \vert u \vert ^{p-2}u+(p-1)\phi _{i}(u) \vert u \vert ^{p-2} \,dx \geq \frac{k+p-1}{4} \vert u \vert ^{k+p-2}. \end{aligned} \end{aligned}$$

(4.8)

Hence from (4.7) and (4.8) it follows that

$$\begin{aligned} \begin{aligned} \frac{d}{dt} \bigl\Vert u(t) \bigr\Vert _{p}^{p}+C_{1} p^{2-m} \bigl\Vert \nabla u^{ \frac{p+M}{m}} \bigr\Vert _{m}^{m}\leq 0,\quad \forall t>0, \end{aligned} \end{aligned}$$

(4.9)

where $M=k+m-2>0$.

Set $R>1+M/q$, $p_{1}=q$, $p_{n}=Rp_{n-1}-M$, $\theta _{n}=RN(1-p_{n-1}p_{n}^{-1})(m+N(R-1))^{-1}$, $\beta _{n}=(p_{n}+M)\theta _{n}^{-1}$, $n=2,3,\ldots$ . From Lemma 2 we see that

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{p_{n}}\leq C^{m/(p_{n}+M)} \Vert u \Vert _{p_{n-1}}^{1-\theta _{n}} \bigl\Vert \nabla u^{\frac{p_{n}+M}{m}} \bigr\Vert _{m}^{m\theta _{n}/(p_{n}+M)}. \end{aligned}$$

(4.10)

Inserting this into (4.9) $(p=p_{n})$ yields

$$\begin{aligned} \frac{d}{dt} \bigl\Vert u(t) \bigr\Vert _{p_{n}}+C_{2}C^{\frac{-m}{\theta _{n}}} p_{n}^{2-m} \Vert u \Vert _{p_{n-1}}^{M-\beta _{n}} \Vert u \Vert _{p_{n}}^{1+\beta _{n}}\leq 0 \quad t>0. \end{aligned}$$

(4.11)

As in the proof of Theorem 1, we can show that there exist bounded sequences $\{\xi _{n}\}$ and $\{\lambda _{n}\}$ such that

$$\begin{aligned} \begin{aligned} \bigl\Vert u(t) \bigr\Vert _{p_{n}} \leq \xi _{n}t^{-\lambda _{n}} \quad t>0, \end{aligned} \end{aligned}$$

(4.12)

in which $\lambda _{n}\to \lambda$ and $\xi _{n}\le C_{0} \Vert u_{0} \Vert _{q}^{\mu }$ with

$$\begin{aligned} \lambda =\frac{N}{mq +MN},\quad \quad \mu = \frac{qm}{qm+MN},\quad\quad M=k+m-2>0. \end{aligned}$$

(4.13)

Letting $n\to \infty$ in (4.12), we have

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{\infty }\leq C_{0} \Vert u_{0} \Vert _{q}^{\mu } t^{-\lambda }, \quad \forall t\ge 0. \end{aligned}$$

(4.14)

This finishes the proof of Theorem 2.