Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

This paper deals with a homogeneous Neumann problem of a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. We prove the existence of weak solutions using Galerkin’s approximation and we derive suitable energy estimates. To this end, we establish the needed Poincaré type inequality for variable exponents related to the Neumann boundary problem. Furthermore, we show that the investigated problem possesses a unique weak solution and satisfies a stability estimate, provided some additional assumptions are fulfilled. In addition, we show under which conditions the solution is nonnegative.