Parabolic problems in non-standard Sobolev spaces of infinite order

Abstract

This paper is devoted to the study of the existence of solutions for the strongly nonlinear $p(x)$-parabolic equation$$\frac{\partial u}{\partial t} + Au + g(x,t,u) = f(x,t),$$where $A$ is a Leray-Lions operator acted from $V^{\infty,p(x)}(a_\alpha,Q_{T})$ into its dual. The nonlinear term $\>g(x,t,s)\>$ satisfies growth and sign conditions and the datum $\>f\>$ is assumed to be in the dual space $V^{-\infty,p'(x)}(a_\alpha,Q_{T})\>.