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

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) similar to |1 +/- u|2+theta, for some 0 > -1, when u approximate to +/- 1, and we consider the corresponding equation in a bounded interval with homogeneous Neumann boundary conditions. We prove that if 0 is an element of (-1,0), then there exist special steady states, named compactons, with a transition layer structure. In contrast, if 0 >= 0 the interface layers are not stationary and two different phenomena emerge: for 0 = 0 solutions exhibit a metastable behavior and maintain an unstable structure for an exponentially long time, while if 0 > 0 the exponentially slow motion is replaced by an algebraic one.(c) 2023 Elsevier Inc. All rights reserved.