Abstract
The degenerate reaction diffusion system has been applied to a variety of physical and engineering problems. This paper is extended the existence of solutions from the quasimonotone reaction functions (e.g., inhibitor‐inhibitor mechanism) to the mixed quasimonotone reaction functions (e.g., activator‐inhibitor mechanism). By Schauder fixed point theorem, it is shown that the system admits at least one positive solution if there exist a coupled of upper and lower solutions. This result is applied to a Lotka‐Volterra predator‐prey model.
Highlights
We consider a quasilinear reaction diffusion system in a bounded domain under coupled nonlinear boundary conditions
Our approach to the existence problem is by the method of coupled upper and lower solutions which are defined as follows
We show that u 1 and u 1 are coupled upper and lower solutions of 1.1
Summary
We consider a quasilinear reaction diffusion system in a bounded domain under coupled nonlinear boundary conditions. By use of upper and lower solutions and its associated monotone iterations, 4, 5 deal with the scalar equation and the system endowed with the nonlinear Neumann-Robin boundary conditions, respectively. The paper in 6 is concerned with the existence, uniqueness, and asymptotic behavior for the quasilinear parabolic systems with the Dirichlet boundary condition. This paper relaxed the condition to mixed quasimonotone reaction functions, which leads to the difficult point that the ordered upper and lower solutions do not exist. The purpose of this paper is to study the existence for the system 1.1 by the Schauder fixed point theorem.
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