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We study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation    $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$  u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that    $A'(s)=a(s)\geq 0$  A ′ ( s ) = a ( s ) ≥ 0 ,    $A(s)$  A ( s ) is a strictly increasing function,    $A(0)=0$  A ( 0 ) = 0 ,    $b(x,t)\geq 0$  b ( x , t ) ≥ 0 , and    $\alpha (x,t)\geq 0$  α ( x , t ) ≥ 0 . If    $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$  b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type