Existence and uniqueness of renormalized solutions to nonlinear multivalued parabolic problem with homogeneous Dirichlet boundary conditions involving variable exponent

In this paper, we prove the existence and the uniqueness of renormalized solution to a nonlinear multivalued parabolic problem β(u)t - div a(x,∇ u) ∋ f , with homogeneous Dirichlet boundary conditions and L1-data. The functional setting involves Lebesgue and Sobolev spaces with variable exponent. Some a-priori estimates are used to obtain our results.