On a weak solution matching up with the double degenerate parabolic equation

Abstract

Abstract The well-posedness of weak solutions to a double degenerate evolutionary p(x) $p(x)$-Laplacian equation ut=div(b(x,t)|∇A(u)|p(x)−2∇A(u)), $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)\bigr), $$ is studied. It is assumed that b(x,t)|(x,t)∈Ω×[0,T]>0 $b(x,t)| _{(x,t)\in \varOmega \times [0,T]}>0$ but b(x,t)|(x,t)∈∂Ω×[0,T]=0 $b(x,t) | _{(x,t)\in \partial \varOmega \times [0,T]}=0$, A′(s)=a(s)≥0 $A'(s)=a(s)\geq 0$, and A(s) $A(s)$ is a strictly monotone increasing function with A(0)=0 $A(0)=0$. A weak solution matching up with the double degenerate parabolic equation is introduced. The existence of weak solution is proved by a parabolically regularized method. The stability theorem of weak solutions is established independent of the boundary value condition. In particular, the initial value condition is satisfied in a wider generality.

Keywords

• Double degenerate parabolic equation
• Well-posedness
• Existence
• Initial value