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


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.

1 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

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

and the degenerate evolution m-Laplacian equation

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

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

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

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

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

satisfies the \(L^{\infty }\) decay estimate

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

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

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

Estimate (1.5), or more general estimates

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

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

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

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

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

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

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

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.

2 Preliminaries and main results

We first make the following assumptions.


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

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

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


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

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

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

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

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

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

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

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

and the \(L^{\infty }\) decay estimate

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

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:


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

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

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

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 \(u(t)\in X=L^{\infty }(\mathbb{R}^{+}, L^{q}(\Omega ))\), \(\vert u \vert ^{(k-1)/m}u \in L^{m}_{\mathrm{loc}}((0, +\infty ); W_{0}^{1,m}( \Omega ))\), \(\vert u \vert ^{(k-1)/(m-1)}u \in L^{m-1}_{\mathrm{loc}}((0, \infty ); W_{0}^{1,m-1}( \Omega ))\),

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

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:

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

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.

3 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

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

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

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

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

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


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

By \((H_{1})\) we have

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

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

An application of Lemma 1 to (3.15) yields

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


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

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

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

Hence we define \(\xi _{n}\) inductively by

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

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

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


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

It is easy to show that

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

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

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


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

with some \(C_{0}>0\) independent of n. Then

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


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

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

4 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

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

if \(0< k\le 1\).

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

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

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

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

Note that

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


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

This implies that

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

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

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

Note that

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

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

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

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

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

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

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

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

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

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

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

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

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

This finishes the proof of Theorem 2.

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


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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Wang, H., Chen, C. \(L^{\infty }\) decay estimates of solutions of nonlinear parabolic equation. Bound Value Probl 2021, 2 (2021).

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