Existence of solutions to an initial Dirichlet problem of evolutional p(x)p(x)-Laplace equations

The existence and uniqueness of weak solutions are studied to the initial Dirichlet problem of the equationut=div(|∇u|p(x)−2∇u)+f(x,t,u),ut=div(|∇u|p(x)−2∇u)+f(x,t,u), with infp(x)>2. The problems describe the motion of generalized Newtonian fluids which were studied by some other authors in which the exponent p   was required to satisfy a logarithmic Hölder continuity condition. The authors in this paper use a difference scheme to transform the parabolic problem to a sequence of elliptic problems and then obtain the existence of solutions with less constraint to p(x)p(x). The uniqueness is also proved.