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.

Show me the PDF file (330 KB), TEX file, and other files for this article.

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

Return to the EJDE web page