Finite time blow-up and global solutions for a class of semilinear parabolic equations at high energy level

doi:10.1016/j.nonrwa.2011.07.025

Nonlinear Analysis: Real World Applications

Volume 13, Issue 1, February 2012, Pages 197-202

https://doi.org/10.1016/j.nonrwa.2011.07.025 Get rights and content

Abstract

In this paper we study the initial boundary value problem of a class of semilinear parabolic equation. Our main tools are the comparison principle and variational methods. In this paper, we will find both finite time blow-up and global solutions at high energy level.

Introduction

In 1972, Tsutsumi [1] studied the problem of nonlinear parabolic equation $\frac{\partial u}{\partial t} = \sum_{i = 1}^{N} \frac{\partial}{\partial x_{i}} ({| \frac{\partial u}{\partial x_{i}} |}^{p - 2} \frac{\partial u}{\partial x_{i}}) + u^{1 + α}, x \in Ω, t > 0,$ $u (x, 0) = u_{0} (x), x \in Ω,$ $u (x, t) = 0, x \in \partial Ω, t \geq 0,$ where $Ω \subset R^{N}$ is an open bounded domain. And $\sum_{i = 1}^{N} \frac{\partial}{\partial x_{i}} ({| \frac{\partial u}{\partial x_{i}} |}^{p - 2} \frac{\partial u}{\partial x_{i}})$ is called $p$ -Laplace operator. For the case $p < 2 + α$ , he obtained the existence of global weak solutions in the application of potential well. In 2004, Tao Zhong and Yao Zheng-an [2] changed the nonlinearity $u^{1 + α}$ into ${| u |}^{q - 2} u$ . And they proved that the global solutions with the initial data in some “stable set” converge strongly to zero in $W_{0}^{1, p} (Ω)$ . Then, using the potential well, Liu Yacheng and Zhao Junsheng [3] proved that if $0 \leq u_{0} (x) \in W_{0}^{1, p} (Ω), J (u_{0}) = d, I (u_{0}) > 0$ or $I (u_{0}) = 0, 0 < J (u_{0}) \leq d$ , then the corresponding problem admits a global solution $u (t) \in L^{\infty} (0, \infty; W_{0}^{1, p} (Ω))$ with $u_{t} (t) \in L^{2} (0, \infty; L^{2} (Ω))$ and $u (t) \in \bar{W}$ for $0 \leq t < \infty$ exists.

So far the situation $J (u_{0}) ⩽ d$ has been discussed. But the depth of potential well “ $d$ ” for the problem (1.1), (1.2), (1.3) is usually very small so that such initial data $u_{0}$ do not satisfy the high energy case $J (u_{0}) > d$ . In fact, many people have discussed the high energy level conditions for some kinds of partial differential equations. In 2009, Chen Shangjie and Tang Chunlei [4] studied the existence of infinitely many large energy solutions for the superlinear Schrödinger–Maxwell equations. In 2005, Gazzola and Weth [5] found finite time blow-up and global solutions for a kind of parabolic equations at high energy level. In this paper we will also use the way in [5] to consider the equation with high energy level $\frac{\partial u}{\partial t} = \sum_{i = 1}^{N} \frac{\partial}{\partial x_{i}} ({| \frac{\partial u}{\partial x_{i}} |}^{p - 2} \frac{\partial u}{\partial x_{i}}) + u^{1 + α},$ where $p, α$ satisfy $p < 2 + α if N \leq p; p < 2 + α \leq \frac{N p}{N - p} if N > p .$ And, we will prove that we can both find finite time blow-up and global solutions while $J (u_{0}) > d$ .

Section snippets

Setup and notations

We denote the $L^{q} (Ω)$ norm by ${‖ \cdot ‖}_{q}$ for $1 \leq q \leq \infty$ and by ${‖ \nabla \cdot ‖}_{p}$ the Dirichlet norm in $W_{0}^{1, p} (Ω)$ . And we define the cone of nonnegative functions $K = {u \in W_{0}^{1, p} (Ω) ∣ u \geq 0 a.e. in Ω} .$

For any $u \in W_{0}^{1, p} (Ω)$ , we denote its positive part by $u^{+} (x) ≔ max {u (x), 0},$ and its negative part by $u^{-} (x) ≔ min {u (x), 0} .$

We also define the energy functional and the Nehari functional $J (u) = \frac{1}{p} {‖ \nabla u ‖}_{p}^{p} - \frac{1}{2 + α} {‖ u ‖}_{2 + α}^{2 + α}, I (u) = {‖ \nabla u ‖}_{p}^{p} - {‖ u ‖}_{2 + α}^{2 + α} .$ By the Nehari manifold, we define $N = {u \in W_{0}^{1, p} (Ω) ∖ {0} | I (u) = 0},$ and the unbounded sets separated by $N$ $N_{+} = {u \in W_{0}^{1,}$

Comparison principle and a theorem about the stationary problem

In this section, we will also discuss the stationary problem $- \sum_{i = 1}^{N} \frac{\partial}{\partial x_{i}} ({| \frac{\partial u}{\partial x_{i}} |}^{p - 2} \frac{\partial u}{\partial x_{i}}) = u^{1 + α} in Ω,$ $u = 0 on \partial Ω .$

Lemma 3.1

Let $u_{0} \in W_{0}^{1, p} (Ω)$ be such that $T^{*} (u_{0}) = \infty$ . Then, we have the convergence of solution $S (t) u_{0}$ to the solution of problem (3.1)–(3.2).

The lemma has been proved by Chill, Fiorenza [7] and Guesmia [6].

In this paper, the methods in [5] are employed to prove the main results but the comparison principle is not ready for us to complete the main proof. So we first give the following comparison

The main results

Theorem 4.1

Let $u$ be a nontrivial solution of (3.1), and $u_{0} \in W_{0}^{1, p} (Ω), u_{0} ≢ u$ . Then

(i)
If $u^{+} ≢ 0$ , and $u_{0} \geq u$ , then $u_{0} \in B$ ;
(ii)
If $u^{-} ≢ 0$ , and $u_{0} \leq u$ , then $u_{0} \in B$ ;
(iii)
If $u > 0, u \geq u_{0} \geq 0$ , then $u_{0} \in G_{0}$ .

Proof

(i)
$u_{0}$ is not in $B$ , then when $t \to \infty$ , we have $S (t) u_{0} \to u^{'}$ which is a solution of (3.1). By comparison and $u_{0} ≢ \pm u$ , we also have $S (t) u_{0} > S (t) u$ . Taking the limits of both sides, we will have $u^{'} > u$ . By Theorem 3.5, we know it must be $u^{'} > 0 > u$ . But for $u^{+} ≢ 0$ , it is impossible. So $u_{0} \in B$ .
(ii)
Do it as it is done in (i).
(iii)
Since $u \geq u_{0} \geq 0$ , by comparison we have $u_{0} \in G$ and then $S (t$

Acknowledgments

This work was supported by National Natural Science Foundation of China (10871055, 10926149); Ph.D. Programs Foundation of Ministry of Education of China (20102304120022); Natural Science Foundation of Heilongjiang Province (A201014); Foundational Science Foundation of Harbin Engineering University, Fundamental Research Funds for the Central Universities (HEUCF20111101).

References (8)

Liu Yacheng et al.
Nonlinear parabolic equations with critical initial conditions $J (u_{0}) = d$ or $I (u_{0}) = 0$
Nonlinear Anal.
(2004)
Senoussi Guesmia
Some convergence results for quasilinear parabolic boundary value problems in cylindrical domains of large size
Nonlinear Anal.
(2009)
Ralph Chill et al.
Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations
J. Differential Equations
(2006)
Hongjun Yuan et al.
Positive solutions for a class of $p$ -Laplace problems involving quasi-linear and semi-linear terms
J. Math. Anal. Appl.
(2007)

There are more references available in the full text version of this article.

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