Properties Of Solutions To Two Reaction-Diffusion Systems

This dissertation is devoted to the global existence and uniqueness of solutions to the initial and boundary value problem for a class of degenerate reaction-diffusion system, and the uniform boundedness of solutions to systems in a three species model with cross-diffusions.The dissertation contains three parts:In Chapter 1, the basic application background, advanced studies and the main idea of this dissertation is introduced.In Chapter 2, we consider a class of degenerate reaction-diffusion system:the global existence and uniqueness of solutions to the initial and boundary value problem are discussed. Here, Ω is a Lipschitz bounded domain in Rⁿ with a smooth boundary Ω, v is the unit outward normal to Ω. n≥ 1, and vector function x = {x₁,x₂,...x_n).Byintroducing truncated function ,we apply compact theory to obtain the global existence and uniqueness of solutions to the initial and boundary value problem, where we assume that thenonnegative initial values u_i, (x,0) = u_i⁰ (x) L2 (Ω), i.e., in a "lower" functional space; the nonlinear diffusion term η,(u,) and the reaction term ft{x,t,u) satisfy some assumptions,too.In Chapter 3, we consider the uniform boundedness of solutions to systems in a three competition species model with cross-diffusions:Ω is a bounded domain in Rⁿ with a smooth boundary Ω , V is the unit outward normal to Ω,. Let the nonnegative initial values uo(x) W₂¹([0,1]), v0 (x) W₂¹ ([0,l]), ands0 (x) W₂²([0,l]). We apply in this chapter Gagliardo-Nirenberg type inequalities, embeddingtheorem in fractional power spaces and some technique in prior estimation to establish the uniform for the solutions.