Global and blow-up solutions for quasilinear parabolic equations with a gradient term and nonlinear boundary flux

Lu Sun,Changjun Li,Zhong Bo Fang

doi:10.1186/1029-242x-2014-234

Abstract

Abstract This work is concerned with positive classical solutions for a quasilinear parabolic equation with a gradient term and nonlinear boundary flux. We find sufficient conditions for the existence of global and blow-up solutions. Moreover, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’ and an upper estimate of the global solution are given. Finally, some application examples are presented. MSC:35R45, 35K65, 34A12.

Highlights

In this paper, we consider the quasilinear parabolic equation with a gradient term g(u) t = ∇ · a(u)b(x)c(t)∇u + f (x, u, q, t) in D × (, T), ( . )subject to the nonlinear boundary flux and initial conditions∂u = h(x, t)r(u) on ∂D × (, T),∂n u(x, ) = u (x) in D.Here D ⊂ RN (N ≥ ) is a bounded domain with a smooth boundary ∂D, D is the closure of D, q = |∇u|, n is the outer normal vector and T is the maximum existence time of u(x, t). a(u)b(x)c(t), f (x, u, q, t) and h(x, t)r(u) are nonlinear diffusion coefficient, reaction term and boundary flux, respectively
The classical parabolic equation theory [, Section ] ensures that there exists a unique classical solution u(x, t) to problem ( . )-( . ) for some T >, and the solution is positive over D × [, T)
The aim of this paper is to obtain sufficient conditions for the existence of blow-up and global solution, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’ and an upper estimate of the global solution and to give some examples

Summary

Introduction

1 Introduction In this paper, we consider the quasilinear parabolic equation with a gradient term g(u) t = ∇ · a(u)b(x)c(t)∇u + f (x, u, q, t) in D × ( , T), Subject to the nonlinear boundary flux and initial conditions The classical parabolic equation theory [ , Section ] ensures that there exists a unique classical solution u(x, t) to problem

Objectives

Results

Conclusion