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

Abstract

The existence and uniqueness of weak solutions are studied to the initial Dirichlet problem of the equation u_{t} = \mathrm{div}\left(|\mathrm{∇}u|^{p(x)−2}\mathrm{∇}u\right) + f(x,t,u), with \mathrm{\inf }\:p(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) . The uniqueness is also proved.