Convergence to heterogeneous diffusion equation and wave propagation in reaction-diffusion equations

Abstract

In reality, when the environment is heterogeneous, the particles do not diffuse uniformly. Many researchers have continued to try to explain this heterogeneous diffusion. However, previous diffusion equations do not fully explain actual diffusion. In this study, we derived a new diffusion equation by taking the diffusive limit from the discrete velocity kinetic model. In this diffusion equation, the diffusion constant is expressed as a physical quantity and explains the existing diffusion phenomena well. In this paper, we presented a useful energy functional in the discrete velocity kinetic model and a method to induce strong convergence in heterogeneous situations using the Div-curl lemma. The traveling wave in reaction-diffusion equations has been extensively studied over the past few decades. In particular, when the reaction term is multistable, there is a peculiar wave phenomenon called propagating terrace. However, when the reaction term is Lipschitz continuous, the propagating terrace cannot be defined as a solution of the reaction-diffusion equation. In this study, the terrace solution was defined considering the discontinuous and multistable reaction term. We showed the existence, uniqueness, and stability of the terrace solution.