On reaction-diffusion models with memory and mean curvature-type diffusion

Abstract

We propose and study a reaction-diffusion model with memory, where the linear diffusion operator is replaced by the mean curvature operator both in Euclidean and Lorentz–Minkowski spaces and the memory kernel is assumed to be of Jeffreys type. Regarding the reaction term, we consider a balanced bistable function f=−F′, with F a generic double well potential with wells of equal depth. In particular, we assume that the potential F has two global minima at ±1 and that F(u)∼|1±u|2+θ, for some θ>−1, when u≈±1, and we consider the corresponding equation in a bounded interval with homogeneous Neumann boundary conditions. We prove that if θ∈(−1,0), then there exist special steady states, named compactons, with a transition layer structure. In contrast, if θ≥0 the interface layers are not stationary and two different phenomena emerge: for θ=0 solutions exhibit a metastable behavior and maintain an unstable structure for an exponentially long time, while if θ>0 the exponentially slow motion is replaced by an algebraic one.