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

Abstract

In this paper, we prove the existence and uniqueness of entropy solution to nonlinear parabolic problem with homogeneous Dirichlet boundary conditions with $$L^1$$ -data. The functional setting involves Lebesgue and Sobolev spaces with variable exponent. The general assumptions and the nonlinear semigroup theory are considered to prove the existence and uniqueness of mild solution satisfying the $$L^1$$ -comparison principle. Moreover, under the same general assumptions and some a-priori estimations of the sequence of mild solutions, we obtain the existence and uniqueness of weak solution. Finally, we prove the existence and uniquess of the renormalized solution which is equivalent to the existence and uniqueness of entropy solution.