Studying Of Properties Of Solutions For Several Types Of Nonlinear Parabolic Systems

This paper deals with the properties of the solution for three types of nonlinear parabolic systems.This thesis is divided into five chapters.In the first two chapters,we present introduction and some basic knowledge.In the third chapter,we consider a nonlinear parabolic system ut=f(u)(△u+ uqi(x,t)ep1v(x0,t),vt=f(v)(△v+vq2(x,t)ep2u(x0,t),t>0,x∈Ω,with a localized reaction source and a nonlocal boundary condition.The method of upper and lower solutions is used to obtain the sufficient conditions for the global existence and blow up of positive solutions for the system.In the fourth chapter,we discuss the existence of generalized solutions for a coupling evolution (p1(x,t),p2(x,t)).Laplacian systemsut-div(a0|▽u|p1(z)-2▽u) =f1(z,u,v),vt--div(b0|▽v|p2(z)-2▽v)=f2(z,u,v),with Dirichlet boundary conditions in a bounded domain　Ω(?)RN,where z=(x,t),a0>0,b0>0,QT=Ω×(0,T],ΓT (?)aΩ×[0,T],and (?)Ω is Lipschitz-continuous.By making a sequence of priory estimates to the solutions,we prove the weak convergence of the approximation solution sequence and justify the existence of generalized solutions.In the fifth chapter,we discuss the properties of weak solutions to a degenerate viscous Hamilton-Jacobi system (?)tu-div(|▽u|p1-2▽u)=|▽v|q1,(?)tv-div(|▽v|p2-2▽v)= |▽u|q2,with Dirichlet boundary conditions in a bounded domain　Ω(?) RN,where pi>2, qi>max{(p1-1),(p2-1)},i=1,2.Firstly；we get a maximal solution(u,v)∈W1,∞× W1,∞ time.Furthermore,a regularizing effect for (ut,vt)is obtained.