Periodic solutions to fast diffusion equations with nonLipschitz convective terms

Abstract

This work focuses on the study of periodic solutions to fast diffusion equations with highly nonlinear convective terms. First, the existence of a periodic solution to an intermediate problem restraint to a period T is proved and then the result is extended by periodicity to the whole real time. The approach involves an appropriate approximating problem whose periodic solution is proved via a fixed point theorem. Next, a pass to the limit technique ensures the existence of a weak solution to the original problem. Under the considered hypotheses this is not unique. Uniqueness can be obtained in the case of a globally Lipschitz convective function, dominated in some sense by the diffusivity. Results upon the solution asymptotic behaviour at large time are provided. The paper also includes the study of the degenerate case under the hypothesis of the convection absence. The existence of at least a solution less regular than that obtained in the nondegenerate case is proved and its asymptotic behaviour is discussed. The theoretical results are illustrated at the end by numerical applications to a real problem of water infiltration in nonlinear soils.