Quenching for a reaction–diffusion equation with nonlinear memory

This paper is devoted to study the quenching phenomenon for a reaction–diffusion equation with nonlinear memory subject to positive Dirichlet boundary condition, u t = Δ u - α u - p ∫ 0 t u - q ( x , s ) ds . where p ⩾ 0, q, α > 0. The local existence and uniqueness of the solution are proved, moreover, there exists a critical length α∗ such that the solution quenches in finite time for α ⩾ α∗, and the blow-up of time-derivatives at the quenching point is verified. Under appropriate hypotheses, the quenching rate estimates are given. Finally, some numerical experiments are performed which illustrate our results.