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\(L^{\infty }\) decay estimates of solutions of nonlinear parabolic equation
Boundary Value Problems volume 2021, Article number: 2 (2021)
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.
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
and the degenerate evolution m-Laplacian equation
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
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
satisfies the \(L^{\infty }\) decay estimate
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
Estimate (1.5), or more general estimates
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
where \(a(x,t,s,\xi )\) is a Carathéodory function satisfying the following structure condition:
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
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.
- \((H_{1})\):
-
\(a(x,t,u, \xi )\) 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
Instead of (1.1), we consider the initial-boundary-value problem
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
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
and the \(L^{\infty }\) decay estimate
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
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 \(u(t)\in X=L^{\infty }(\mathbb{R}^{+}, L^{q}(\Omega ))\), \(\vert u \vert ^{(k-1)/m}u \in L^{m}_{\mathrm{loc}}((0, +\infty ); W_{0}^{1,m}( \Omega ))\), \(\vert u \vert ^{(k-1)/(m-1)}u \in L^{m-1}_{\mathrm{loc}}((0, \infty ); W_{0}^{1,m-1}( \Omega ))\),
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:
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
with \(A, \theta >0\), \(\mu \geq 0\). Then we have
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
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.
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
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
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
where
By \((H_{1})\) we have
Hence from (3.3) and (3.4) it follows that
Letting \(k\to \infty \) in (3.5) gives
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
where
Noting that \(\beta =(m-1)/(r+m-1)\), from (3.4) we get that
Hence from (3.7)–(3.9) it follows that
where \(M=m-1-\beta >0\). Then (3.10) implies that
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
Inserting this into (3.11) \((p=p_{n})\) yields
We claim that there exist bounded sequences \(\{\xi _{n}\}\) and \(\{\lambda _{n}\}\) such that
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
An application of Lemma 1 to (3.15) yields
Since
we see that there exists a constant \(\lambda _{0}>0\), independent of n, such that
Hence we define \(\xi _{n}\) inductively by
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
and
It is easy to show that
On the other hand, the definition of \(\xi _{n}\) gives
Hence
with some \(C_{0}>0\) independent of n. Then
and
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
if \(0< k\le 1\).
We now consider the following approximate problem for (1.2):
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
Note that
Then
This implies that
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
Note that
Hence from (4.7) and (4.8) it follows that
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
Inserting this into (4.9) \((p=p_{n})\) yields
As in the proof of Theorem 1, we can show that there exist bounded sequences \(\{\xi _{n}\}\) and \(\{\lambda _{n}\}\) such that
in which \(\lambda _{n}\to \lambda \) and \(\xi _{n}\le C_{0} \Vert u_{0} \Vert _{q}^{\mu }\) with
Letting \(n\to \infty \) in (4.12), we have
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.
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This work is supported by the Science Foundation of Xinjiang Uygur Autonomous Region (2016D01C383).
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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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DOI: https://doi.org/10.1186/s13661-020-01480-8