The Global Structure Of The Solution Set For Two Reaction-diffusion Systems

Reaction-diffusion systems,i.e.,systems of parabolic partial differential equations,which include one or several parameters are considered in this dissertation.The purpose of this dissertation is to develop both analytical and numerical methods of finding a branch of nontrivial steady-state solutions which connects a bifurcation point with singularly perturbed solutions.Here we are interested not only in the existence of solutions but also in the spectral property of the linearized operator around a stationary solution to get much better understanding of the global structure.”Global structure” means the dependence of the number of stationary solutions and their stability/instability property on parameters.In many of the well-known reaction-diffusion systems,the stability or instability of these bifurcating solutions or singularly perturbed solutions are known.However,the results are local in the sense that the existence and stability results are valid only in a small range of parameters or in the small neighborhood of the steady-state solutions thus constructed.In practice,the realistic parameter values are often out of these ranges.Therefore,it is very important to know what happens for the parameter values between a bifurcation point and the singular perturbation realm.The dissertation is organized as follows.1.Marciniak-Czochra model with Turing instability and hysteresis which is a semilinear parabolic equation coupled to an ODE system is studied.Under the condition of different parameter values,the existence of constant steady states of Marciniak-Czochra model is studied,and through strict calculation constant steady states of the model are considered.In an abstract setting the properties of the spectrum of the linearized operator are studied.By applying the bifurcation theory,spatially heterogeneous steady states are discussed in the neighborhood of a constant steady-state.Furthermore,the behavior of the critical eigenvalue is considered.2.The solution of Marciniak-Czochra model is studied by taking advantage of the fact that finding its steady states is reduced to finding solutions of the boundary value problem for a single equation.It is emphasized that the construction of monotone increasing steady states and their dependence on the initial data and the diffusion coefficient are discussed.The global asymptotical behaviors of the bifurcating branches of the non-constant steady-state solutions are considered.3.A diffusion equation coupled to an ordinary differential equation with Fitz HughNagumo type nonlinearity is studied.The steady states of the Fitz Hugh-Nagumo model are constructed near and far from the constant steady states,respectively.The spectrum of the linearized operator and the stability of the continuous steady states is studied.4.By finding(weak)solutions of Fitz Hugh-Nagumo model for the reduced boundary value problem of a single equation,discontinuous steady states are discussed where the single equation is solved by using an approach.Starting with the construction of monotone increasing solutions of the boundary value problem,various types of steady states with jump discontinuities are constructed and their stability is studied.