Electron. J. Differential Equations, Vol. 2017 (2017), No. 207, pp. 1-10.

Solutions to polytropic filtration equations with a convection term

Huashui Zhan

We introduce a new type of the weak solution of the polytropic filtration equations with a convection term,
 {u_t}= \hbox{div} (a(x)|u|^{\alpha}{| {\nabla u} |^{p-2}}\nabla u)
 +\frac{\partial b^{i}(u^m)}{\partial x_i}.
Here, $\Omega\subset\mathbb{R}^N$ is a domain with a $C^2$ smooth boundary $\partial \Omega$, $a(x)\in C^1(\overline{\Omega})$, $p>1$, $m=1+\frac{\alpha}{p-1}$, $\alpha >0$, $a(x)>0$ when $x\in \Omega$ and $a(x)=0$ when $x\in \partial \Omega$. Since the equation is degenerate on the boundary, its weak solutions may lack the needed regularity to have a trace on the boundary. The main aim of the paper is to establish the stability of the weak solution without any boundary value condition.

Submitted February 16, 2017. Published September 8, 2017.
Math Subject Classifications: 35L65, 35K85, 35R35.
Key Words: Polytropic filtration equation; convection term; stability; boundary value condition.

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Huashui Zhan
School of Applied Mathematics
Xiamen University of Technology
Xiamen, Fujian 361024, China
email: 2012111007@xmut.edu.cn

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