Bifurcation Analysis Of Diffusion System With Nonlinear Competiing Boundary Condition

In many situations,the chemical reactions or the biological phenomena occur in a narrow layer near the boundary or on the boundary surface of cell membrane,and these reaction-diffusion models are usually consisted of a linear diffusion equation(systems)in the spatial domain and a nonlinear reaction on the boundary,and they may represent some pattern formation mechanisms different from the classical ones derived from interior reaction and boundary conditions(Dirichlet boundary value,Neumann boundary value,etc).Because of this,it has great research value for the solution of equations(systems)with nonlinear boundary conditions.In this paper,the bifurcation analysis of diffusion system with nonlinear competing boundary condition is studied.We discuss the bifurcation that occur at semi-trivial solutions,and apply the Crandall-Rabinowitz local bifurcation theorem.We prove the existence of a smooth curve bifurcating from the appropriate semi-trivial branch,so as to obtain the local exact structure of non-constant positive solutions,and discuss the stability of semi-trivial solution and the stability of bifurcation branch.