Global existence and nonexistence of solutions for quasilinear parabolic equation

Xianghui Xu,Yong-Hoon Lee,Zhong Bo Fang

doi:10.1186/1687-2770-2014-33

Abstract

Abstract This work is concerned with the global existence and nonexistence of solutions for a quasilinear parabolic equation with null Dirichlet boundary condition. Based on the Galerkin approximation technique and the theory of a family of potential wells, we obtain the invariant sets and vacuum isolating of global solutions including critical case, and we also give global nonexistence. MSC:35A01, 35B06, 35B08.

Highlights

Our main interest lies in the following quasilinear p-Laplacian parabolic equation:|ut|m– ut = div |∇u|p– ∇u + bu +q, x ∈, t >, ( . )subject to homogeneous Dirichlet boundary and initial conditions u(x, t) =, x ∈ ∂, t >, u(x, ) = u (x), x ∈, where b >, ⊂ RN (N ≥ ) is a bounded domain with smooth boundary, p < + q < ∞ if N ≤ p; p < + q
Tsutsumi [ ] studied the homogeneous Dirichlet initial boundary value problem of the nonlinear parabolic equation ut = div |∇u|p– ∇u + u +q, x ∈, t >, ( . ). He obtained the sufficient conditions of the existence of global weak solutions and the solutions blow up in finite time for the case p < + q
6 Existence of global solution with critical initial conditions we prove the result of global existence with critical initial conditions

Summary

Introduction

Our main interest lies in the following quasilinear p-Laplacian parabolic equation:. subject to homogeneous Dirichlet boundary and initial conditions u(x, t) = , x ∈ ∂ , t > , u(x, ) = u (x), x ∈ , where b > , ⊂ RN (N ≥ ) is a bounded domain with smooth boundary, p < + q < ∞. Tsutsumi [ ] studied the homogeneous Dirichlet initial boundary value problem of the nonlinear parabolic equation ut = div |∇u|p– ∇u + u +q, x ∈ , t > , and he obtained the sufficient conditions of the existence of global weak solutions and the solutions blow up in finite time for the case p < + q. Liu and Zhao [ ] proved the global existence and nonexistence of solutions, but they obtained the vacuum isolating of solutions of the initial boundary value problem for semilinear hyperbolic equations and parabolic equations. As far as we know, there are fewer papers on the global existence and nonexistence of weak solutions for nonlinear parabolic equations by using the theory of a family of potential wells.

This implies that

Let λn n