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 $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)\bigr), $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) ) , is studied. It is assumed that $b(x,t)| _{(x,t)\in \varOmega \times [0,T]}>0$ b ( x , t ) | ( x , t ) ∈ Ω × [ 0 , T ] > 0 but $b(x,t) | _{(x,t)\in \partial \varOmega \times [0,T]}=0$ b ( x , t ) | ( x , t ) ∈ ∂ Ω × [ 0 , T ] = 0 , $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 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.