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

Lingling Zhang; Na Zhang; Lixiang Li

doi:10.4171/zaa/1509

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

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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:

⎩ ⎨ ⎧ (g (u))_{t} \frac{\partial u}{\partial n} u (x, 0) = \nabla \cdot (a (u) \nabla u) + f (x, u, ∣\nabla u ∣^{2}, t) = r (u) = u_{0} (x) > 0 in D \times (0, T) on \partial D \times (0, T) in \overline{D},

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