Some Research Into Structure Of Solutions For Parabolic And Elliptic Equations(Systems)

This paper investigates some results of solutions to parabolic systems and elliptic equations. This investigation contains the problems of. local existence and uniqueness of solutions, global existence and finite time blow up, the boundedness of global positive solutions, etc.In chapter 1, the existence of positive radial singular rupture solutions of the problem div(|â–½u|^p-2â–½u) = f(u) with f(0) =∞in a finite, ball is obtained via the Pohozaev identity and some comparison arguments.In chapter 2, we investigates the local existence of the nonnegative solution and the finite time blow-up of solution and boundary layer profiles of diffusion equations with nonlocal reaction sources, we prove that the solutions have global blow-up and that the rate of blow- up is uniform in all compact subsets of the domain, the blow-up rate of |u(t)|_∞is precisely determined.In chapter 3, we deal with p-Laplacian system with null Dirichlet boundary conditions in a bounded domainΩ= B_r(?)Rⁿ, where p, q≥2,α,β≥1. We first get the nonexistence result for a related elliptic systems of nonincreasing positive solutions. Secondly by using this non-existence result, blow up estimates for above p-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained. Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exist globally or blow up in finite time.