# Blow-Up Solutions and Global Existence for a Kind of Quasilinear Reaction-Diffusion Equations

### Lingling Zhang

Taiyuan University of Technology, China### Na Zhang

Taiyuan University of Technology, Taiyuan, Shanxi, China### Lixiang Li

Taiyuan University of Technology, Taiyuan, Shanxi, China

## Abstract

In this paper, we study the blow-up solutions and global existence for a quasilinear reaction-diffusion equation including a gradient term and nonlinear boundary condition:

where $D\subset R^{N}$ is a bounded domain with smooth boundary $\partial D$. The sufficient conditions are obtained for the existence of a global solution and a blow-up 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 are specified under some appropriate assumptions for the nonlinear system functions $f, g, r,a$, and initial value $u_{0}$ by constructing suitable auxiliary functions and using maximum principles.

## Cite this article

Lingling Zhang, Na Zhang, Lixiang Li, Blow-Up Solutions and Global Existence for a Kind of Quasilinear Reaction-Diffusion Equations. Z. Anal. Anwend. 33 (2014), no. 3, pp. 247–258

DOI 10.4171/ZAA/1509