Blow-up Properties Of Solutions For Two Classes Of Nonlinear Parabolic Equations

In this thesis, we first consider an initial-boundary value problem for a semi-linear heat equation with variable reaction of the form where Ω is a bounded domain in R3 assumed to be star-shaped and convex in two orthogonal directions. Whose solution may blow up in finite time under some appropriate assumptions on the exponent and the initial data. When blow-up does occur, by constructing auxiliary function and applying differential inequality technique, we give the lower bound on blow-up time of solutions.Then, we study the blow-up of solutions of the nonlinear parabolic equations of the form under nonnegative initial condition and homogeneous Dirichlet boundary con-dition. Where Ω is a bounded domain with sufficiently smooth boundary in RN(N ≥ 3). When the solution of the problem blows up, by modifying the first differential inequality technique by Payne and Schaefer, we get a lower bound for the blow-up time of the solution and blow-up rate. Furthermore, for the special case of ρ(x) = xp-2/2, f(u)=uq, that is, for the equation subject to nonnegative initial condition and homogeneous Dirichlet boundary con-dition. By constructing the auxiliary function and estimating differential inequal-ity, we prove that if p> q+1≥ 2, the solution u(x, t) cannot blow up in L2-norm; If 2< p< q+1, the solution u(x,t) blows up in L2-norm. Moreover, we can obtain a lower bound and a upper bound for blow-up time of the solution and blow-up rate.