Convection-diffusion problems involving measure data and p(.)-anisotropic operator in variable exponent space

Abstract

In this paper, we investigate a nonlinear diffusion-convection problem with measure data, involving a general anisotropic operator with variable exponent and a maximal monotone graph. Utilizing Yosida’s regularization, we apply an approximation technique to formulate a regularized problem. We then employ the theory of maximal monotone operators in Banach spaces to demonstrate the existence of at least one solution for the approximate problem. Subsequently, we show that the sequence of solutions to the approximate problem converges, in the limit, to a renormalized and/or entropic solution of the original problem. Finally, we establish the uniqueness of the solution under certain additional assumptions regarding the convection term.