Blow-up behavior for a degenerate parabolic systems subject to Neumann boundary conditions

Abstract

The degenerate parabolic systems dealt with in this paper are as follows: $\{\begin{array}{ll}{\displaystyle \frac{\mathrm{\partial }y\left(u\right)}{\mathrm{\partial }t}=\mathrm{\nabla }\cdot \left(|\mathrm{\nabla }u{|}^r\mathrm{\nabla }u\right)+f(u,v,|\mathrm{\nabla }u{|}^2,t),}& \\ {\displaystyle \frac{\mathrm{\partial }z\left(v\right)}{\mathrm{\partial }t}=\mathrm{\nabla }\cdot \left(|\mathrm{\nabla }v{|}^s\mathrm{\nabla }v\right)+g(u,v,|\mathrm{\nabla }v{|}^2,t),}& x\in \mathrm{\Omega },t\in (0,t^{\ast }),\\ {\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=0,\frac{\mathrm{\partial }v}{\mathrm{\partial }\nu }=0,}& x\in \mathrm{\partial }\mathrm{\Omega },t\in (0,t^{\ast }),\\ {\displaystyle u(x,0)=u_0\left(x\right),v(x,0)=v_0\left(x\right),}& x\in \overline{\mathrm{\Omega }},\end{array}$ where r and s are two positive constants, spatial region $\mathrm{\Omega }\subset {\mathbb{R}}^N(N\ge 2)$ is bounded, and $\mathrm{\partial }\mathrm{\Omega }$ is smooth. The conditions that make the solution $(u,v)$ blow up at a certain finite time $t^{\ast }$ are obtained. An upper bound of $t^{\ast }$ and an upper estimate of the blow-up rate of $(u,v)$ are also derived.

Keywords:

• Blow-up behavior
• degenerate parabolic systems
• Neumann boundary conditions

AMS CLASSIFICATIONS:

• 35K51
• 35B44

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 61473180].