Biological Competition Model Existence Of Global Solution, Staggered Diffusion Equations

This thesis deals with the global existence of the solution of the following quasilinear parabolic system:where Ω is a bounded domain inIn chapter 1,we introduce the biological background of the system (1) in the population model.And we briefly review the existed global results of the system (1).In chapter 2,we consider the global solution of the simplified model:First we establish an important embedding inequality (see Lemma 2.2.1 and Lemma 2.3.1). Then using this embedding inequality, we prove the fact ||△v(x,t)||L ∞(QT) ≤C(T) by iterative methods. From the classical theorem of the syetems with divergence form in O.A.Ladyzenskaja[7] and Gary.M.Lieber-mann[6],we prove the following estimate: for any fixed T > 0,we haveSo by H.Amann's results about the sufficient condition of the global existence, when the initial values satisfy u0(x),u0(x) > 0(x ), and u0(x),v0(x),we obtain the following: Theorem 2.1 When n = 3 , if B1,B2 satisfy the inequality :Then the system (2) has a unique global solution u(x, t), v(x, t) Theorem2.2 For any n >1,if B1,B2 satisfy the inequality:Then the system (2) has a unique global solution u(x, t), v(x, t) In chapter 3, we establish the global existence of classic solutions when both equations have self-diffusion term:We let w2 = (d2 + a22v)v ,then the equation of v becomesUsing the analogous method in Chapter 2,when the initial values satisfy , we get the following result:Theorem 3.1 When n = 3,the system (3) has an unique solution (u(x, t),...