Exponentially slow motion for a one-dimensional Allen-Cahn equation with memory
Raffaele Folino

Abstract. A reaction-diffusion equation with memory kernel of Jeffreys type and with a balanced bistable reaction term is considered in a bounded interval of the real line. Taking advantage of the fact that in this case the integro-differential equation can be transformed into a local partial differential equation, it is proved that there exist solutions which evolve very slowly in time and maintain a transition layer structure for an exponentially long time T_ε≥ c₁ exp(c₂/ε) as ε→ 0⁺, where ε² is the diffusion coefficient. Hence, we extend to reaction-diffusion equations with memory kernel of Jeffreys type the well-known results valid for the classic Allen-Cahn equation

Rend. Mat. Appl. (7) 42 (2021) 253-270; pdf