Infiltration Equation with Degeneracy on the Boundary

Abstract

This paper is mainly about the infiltration equation

$$ {u_{t}}= \operatorname{div} \bigl(a(x)|u|^{\alpha }{ \vert { \nabla u} \vert ^{p-2}}\nabla u\bigr),\quad (x,t) \in \Omega \times (0,T), $$

where $p>1$, $\alpha >0$, $a(x)\in C^{1}(\overline{\Omega })$, $a(x)\geq 0$ with $a(x)|_{x\in \partial \Omega }=0$. If there is a constant $\beta$ such that $\int_{\Omega }a^{-\beta }(x)dx\leq c$, $p>1+\frac{1}{\beta }$, then the weak solution is smooth enough to define the trace on the boundary, the stability of the weak solutions can be proved as usual. Meanwhile, if for any $\beta >\frac{1}{p-1}$, $\int_{\Omega }a^{-\beta }(x)dxdt=\infty$, then the weak solution lacks the regularity to define the trace on the boundary. The main innovation of this paper is to introduce a new kind of the weak solutions. By these new definitions of the weak solutions, one can study the stability of the weak solutions without any boundary value condition.