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