Nonlinear parabolic capacity and renormalized solutions for PDEs with diffuse measure data and variable exponent

Abstract

We extend the theory of capacity to generalized Sobolev spaces for the study of nonlinear parabolic equations. We introduce the definition and some properties of renormalized solutions and we show that diffuse measure can be decomposed in space and time. As consequence, we show the existence and uniqueness of renormalized solutions. The main technical tools used include estimates and compactness convergence.