Simultaneous and non-simultaneous quenching for a coupled semilinear parabolic system in a $ n $-dimensional ball with singular localized sources

Abstract

<abstract> In this paper, we investigate a coupled semilinear parabolic system with singular localized sources at the point $ x_{0} $: $ u_{t}-\Delta u = af\left(v\left(x_{0}, t\right) \right) $, $ v_{t}-\Delta v = bg\left(u\left(x_{0}, t\right) \right) $ for $ x\in B_{1}\left(x_{0}\right) $ and $ t\in \left(0, T\right) $ with the Dirichlet boundary condition, where $ a $ and $ b $ are positive real numbers, $ B_{1}\left(x_{0}\right) $ is a $ n $-dimensional ball with the center and radius being $ x_{0} $ and $ 1 $, and the nonlinear sources $ f $ and $ g $ are positive functions such that they are unbounded when $ u $ and $ v $ tend to a positive constant $ c $, respectively. We prove that the solution $ \left(u, v\right) $ quenches simultaneously and non-simultaneously under some sufficient conditions. </abstract>