Journal
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
Volume 23, Issue -, Pages 47-60Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.nonrwa.2014.11.002
Keywords
Lotka-Volterra cooperation system; Degenerate quasilinear equations; Positive minimal and maximal solutions; Asymptotic behavior and stability; Upper and lower solutions
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This paper deals with a class of Lotka-Volterra cooperation system where the densities of the cooperating species are governed by a finite number of degenerate reaction-diffusion equations. Three basic types of Dirichlet, Neumann, and Robin boundary conditions and two types of reaction functions, with and without saturation, are considered. The aim of the paper is to show the existence of positive minimal and maximal steady-state solutions, including the uniqueness of the positive solution, the existence and uniqueness of a global time-dependent solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Some very simple conditions on the physical parameters for the above objectives are obtained. Also discussed is the finite-time blow up property of the time-dependent solution and the non-existence of positive steady-state solution for the system with Neumann boundary condition. (C) 2014 Elsevier Ltd. All rights reserved.
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