Wave propagation in reaction diffusion equations via parabolic and hyperbolic singular limits

Abstract

In this thesis we consider reaction diffusion equations with diffusivity depends on the solution and several limit problems related to scaling. In Chapter 2 we consider an Allen-Cahn equation with nonlinear diffusivity. Given density dependent diffusion function $\varphi$ and a bistable reaction term $f$ satisfying certain conditions, the solution of the Allen-Cahn equation generates a steep transition layer within a short time. And after the generation, the transition layer propagates with speed proportional to the mean curvature of the layer times a constant which depends on $\varphi$ and $f$. In Chapter 3 we consider similar equation in Chapter 2, but adding a stochastic perturbation in time. This time, generated interface propagates with speed which depends not only on the mean curvature but also on the stochastic perturbation. In Chapter 4 and 5 we consider the biological invasion in heterogeneous environment. To this purpose, we consider a hyperbolic singular limit problem of reaction diffusion equation with porous medium diffusivity and logistic reaction term. Unlike homogeneous environment, where the invasion speed is constant, the invasion speed depends on the heterogeneous environment and the power of the porous medium diffusion.