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

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.

Keywords

• Degenerate parabolic equation
• L-infinity blowup
• gradient blowup
• global solution
• comparison principle