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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.

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