Non-simultaneous versus simultaneous quenching in a coupled nonlinear parabolic system

This paper deals with finite-time quenching for the nonlinear parabolic system with coupled singular absorptions: u_t=Δu−v^−p, v_t=Δv−u^−q in Ω×(0,T) subject to positive Dirichlet boundary conditions, where p,q>0, Ω is a bounded domain in with smooth boundary. We obtain the sufficient conditions for global existence and finite-time quenching of solutions, and then determine the blow-up of time-derivatives and the quenching set for the quenching solutions. As the main results of the paper, a very clear picture is obtained for radial solutions with Ω=B_R: the quenching is simultaneous if p,q≥1, and non-simultaneous if p<1≤q or q<1≤p; if p,q<1 with , then both simultaneous and non-simultaneous quenching may happen, depending on the initial data. In determining the non-simultaneous quenching criteria of the paper, some new ideas have been introduced to deal with the coupled singular inner absorptions and inhomogeneous Dirichlet boundary value conditions, in addition to techniques frequently used in the literature.