Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Uniqueness of nonzero positive solutions of Laplacian elliptic equations arising in combustion theory

Pages: 849 - 861, Volume 21, Issue 3, May 2016      doi:10.3934/dcdsb.2016.21.849

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Kunquan Lan - Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada (email)
Wei Lin - School of Mathematical Sciences and Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China (email)

Abstract: Uniqueness of nonzero positive solutions of a Laplacian elliptic equation arising in combustion theory is of great interest in combustion theory since it can be applied to determine where the extinction phenomenon occurs. We study the uniqueness whenever the orders of the reaction rates are in $(-\infty,1]$. Previous results on uniqueness treated the case when the orders belong to $[0,1)$. When the orders are negative or 1, it is physically meaningful and the bimolecular reaction rate corresponds to the order 1, but there is little study on uniqueness. Our results on the uniqueness are completely new when the orders are negative or 1, and also improve some known results when the orders belong to $(0,1)$. Our results provide exact intervals of the Frank-Kamenetskii parameters on which the extinction phenomenon never occurs. The novelty of our methodology is to combine and utilize the results from Laplacian elliptic inequalities and equations to derive new results on uniqueness of nonzero positive solutions for general Laplacian elliptic equations.

Keywords:  Laplacian elliptic equations, Laplacian elliptic inequalities, uniqueness, nonzero positive solutions, combustion theory.
Mathematics Subject Classification:  Primary: 35J61; Secondary: 35R45, 49J40, 80A25.

Received: September 2014;      Revised: November 2015;      Available Online: January 2016.