Properties Of The Solution To Semilinear Parabolic Equations

The global existence and finite time blow-up of solutions on semilinear parabolic systems with nonlocal source have been studied by many authors. There are three basic techniques for proofs so far-super and sub solution method, convex method and energy method. In this paper we mainly utilize super and sub solution techniques, use comparision priciple and construct appropriate supersolu-tion and subsolution by eigenfunctions to discuss two kinds of nonlocal semilinear parabolic systems with homogeneous Dirichlet boundary condition.Charpte 2 is dedicate to studying the strongly coupled parabolic system with nonlocal source. Since the local existence and uniqueness of classical solution have been discussed, we pay attention to conditions on global existence and blow-up of the solution in finite time. In Chapter 3 a nonlocal degenerate singular semilinear parabolic system is investigated. There are degeneration and singularity, so the local existence and uniqueness of classical solution are established firstly. Under appropriate hypotheses, the conditions on the global existence and finite time blow-up of the positive solution are obtained, it is shows that for pq > 1 the solution exists globally for sufficiently small initial data while blows up in finite time for initial ones large enough. Furthermore, the precise blow-up rate is also established.