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\(L^{\infty }\) 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 [13, 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 [13, 8, 9, 11, 15, 1719] 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 mN.

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.

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Acknowledgements

The authors would like to express their sincere gratitude to the anonymous reviewers for the valuable comments and suggestions.

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This work is supported by the Science Foundation of Xinjiang Uygur Autonomous Region (2016D01C383).

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HW and CC participated in the theoretical research and drafted the manuscript. Both authors read and approved the final manuscript.

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Correspondence to Hui Wang.

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Wang, H., Chen, C. \(L^{\infty }\) decay estimates of solutions of nonlinear parabolic equation. Bound Value Probl 2021, 2 (2021). https://doi.org/10.1186/s13661-020-01480-8

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MSC

  • 35K59
  • 35K65
  • 35K92

Keywords

  • Doubly nonlinear equation
  • Degenerate evolution m-Laplacian equation
  • \(L^{\infty }\) decay estimates of solution