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,ut=Δu-αu-p∫0tu-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.

► The local existence and uniqueness of the solution are proved. ► The solution quenches in finite time and the blow-up of time-derivatives at the quenching point are verifed. ► The quenching rate estimates are given.