The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (234 KB) | Metadata | Table of Contents | ZAA summary
Volume 33, Issue 3, 2014, pp. 247–258
DOI: 10.4171/ZAA/1509

Published online: 2014-07-02

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

Lingling Zhang[1], Na Zhang[2] and Lixiang Li[3]

(1) Taiyuan University of Technology, China
(2) Taiyuan University of Technology, Taiyuan, Shanxi, China
(3) Taiyuan University of Technology, Taiyuan, Shanxi, China

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: \begin{equation*} \left\{ \begin{alignedat}{2} (g(u))_{t}&=\nabla\cdot(a(u)\nabla u)+f(x,u,|\nabla u|^{2},t)&\quad &\text{in} \ D\times(0,T)\\ %[0.5em] %\displaystyle \tfrac{\partial u}{\partial n}&=r(u)& &\rm{on} \ \partial D\times(0,T)\\ %%[0.5em] u(x,0)&=u_{0}(x)>0& &\rm{in} \ \overline{D}, \end{alignedat} \right. \end{equation*} 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.

Keywords: Reaction-diffusion equation, blow-up solution, global solution

Zhang Lingling, Zhang Na, Li Lixiang: Blow-Up Solutions and Global Existence for a Kind of Quasilinear Reaction-Diffusion Equations. Z. Anal. Anwend. 33 (2014), 247-258. doi: 10.4171/ZAA/1509