Abstract
In this paper, we mainly study a diffusive Lotka-Volterra competition-advection system with lethal boundary conditions in a general heterogeneous environment. By using the basic theory of partial differential equations and some nonlinear analysis techniques, we investigate the existence, uniqueness and global asymptotic behavior of steady-state solutions of the system equations. The existence, uniqueness and global asymptotic behavior of steady-state solutions are proved by upper and lower solutions, maximum principle and other methods. In theory, the methods and skills to deal with this kind of nonlinear problem are further developed, which provides a theoretical basis for understanding some practical problems.
Highlights
The existence, uniqueness and global asymptotic behavior of steady-state solutions are proved by upper and lower solutions, maximum principle and other methods
We mainly study a diffusive Lotka-Volterra competition-advection system with lethal boundary conditions in a general heterogeneous environment
By using the basic theory of partial differential equations and some nonlinear analysis techniques, we investigate the existence, uniqueness and global asymptotic behavior of steady-state solutions of the system equations
Summary
The existence, uniqueness and global asymptotic behavior of steady-state solutions are proved by upper and lower solutions, maximum principle and other methods. Let (U ( x,t ),V ( x,t )) be the solution of Equation (1.4) with U ( x, 0),V ( x, 0) ≥, ≡/ 0,U ( x, 0),V ( x, 0) ∈ C1 (Ω) , and vanishing on ∂Ω . V= ( x) e−η p , v ( x) c2θd2 ,d (x) , we have that (r1φ ( x),V ) is a lower solution of Equation (1.5) provided that 1− f ( x) c2 < 0, a ( x) > λ1d1 + c ( x) v ( x) , for r1 sufficiently small.
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