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

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• 作者：Sining Zheng ; Wei Wang
• 关键词：Quenching ; Non-simultaneous quenching ; Quenching set ; Quenching rate ; Nonlinear parabolic system ; Dirichlet boundary condition
• 刊名：Nonlinear Analysis
• 出版年：2008
• 出版时间：1 October 2008
• 年：2008
• 卷：69
• 期：7
• 页码：2274-2285
• 全文大小：318 K

文摘

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 cc25919ea34b4769db991e22a109"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">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.