Electron. J. Diff. Equ., Vol. 2013 (2013), No. 264, pp. 1-17.

Blowup and existence of global solutions to nonlinear parabolic equations with degenerate diffusion

Zhengce Zhang, Yan Li

Abstract:
In this article, we consider the degenerate parabolic equation
$$ u_t-\hbox{div}(|\nabla u|^{p-2}\nabla u) =\lambda u^m+\mu|\nabla u|^q $$
on a smoothly bounded domain $\Omega\subseteq\mathbb{R}^N\; (N\geq2)$, with homogeneous Dirichlet boundary conditions. The values of $p>2$, $q,m,\lambda$ and $\mu$ will vary in different circumstances, and the solutions will have different behaviors. Our main goal is to present the sufficient conditions for $L^\infty$ blowup, for gradient blowup, and for the existence of global solutions. A general comparison principle is also established.

Submitted January 18, 2013. Published November 29, 2013.
Math Subject Classifications: 35A01, 35B44, 35K55, 35K92.
Key Words: Degenerate parabolic equation; L-infinity blowup; gradient blowup; global solution; comparison principle.

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Zhengce Zhang
School of Mathematics and Statistics
Xi'an Jiaotong University
Xi'an 710049, China
email: zhangzc@mail.xjtu.edu.cn
Yan Li
School of Mathematics and Statistics
Xi'an Jiaotong University
Xi'an 710049, China
email: liyan1989@stu.xjtu.edu.cn

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