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.
Citation: W. Y. Chan. Simultaneous and non-simultaneous quenching for a coupled semilinear parabolic system in a $ n $-dimensional ball with singular localized sources[J]. AIMS Mathematics, 2021, 6(7): 7704-7718. doi: 10.3934/math.2021447
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.
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W. Y. Chan. Simultaneous and non-simultaneous quenching for a coupled semilinear parabolic system in a $ n $-dimensional ball with singular localized sources[J]. AIMS Mathematics, 2021, 6(7): 7704-7718. doi: 10.3934/math.2021447
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