JournalszaaVol. 33, No. 3pp. 247–258

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

\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.

where DRND\subset R^{N} is a bounded domain with smooth boundary D\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,af, g, r,a, and initial value u0u_{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