Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 264, pp. 1-17.
Title: Blowup and existence of global solutions to nonlinear
parabolic equations with degenerate diffusion
Authors: Zhengce Zhang (Xi'an Jiaotong Univ., Xi'an, China)
Yan Li (Xi'an Jiaotong Univ., Xi'an, China)
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.