Threshold Results For Semilinear Parabolic Equations

Let p> 1.Ωbe a bounded domain in Rn.In this thesis we study initial boundary value problems of semilinear parabolic equations.First let us consider the problem and its steady-state problem Using convex method, Lacey A.A. [43] studied the initial threshold phenomena of this kind of question, and obtained an initial threshold for existence and nonex-istence of global solution. In this thesis, using the characters of solutions of(2). a priori estimate for the global solution of (1) and combining with the long time behavior,we prove that any positive solution of (2) is an initial datum threshold for the existence and nonexistence of global solutions to (1). More precisely, we present asTheorem 1:Assume U(x) is an arbitrary solution of (2).Then the following conclusions hold(1)If 0≤u0(x)≤U(x), and uo(x)(?) U(x),then (1) has a global solution u(x,t;u0). Moreover.(2)If u0(x)≥U(x). and u0(x)(?) U(x),then the solution u(x, t;u0) of (1) blows up in finite time.That is, there exists 0< T<+∞, such thatThe method we established not only solved the homogeneous equation, but also expressed significant advantage in solving inhomogeneous equation. Next we study the problem and its steady-state problemThe difference between(1)and(3)is that any two distinct solution of(4)may not intersect.But the structure of solution set of(4)is very good.That is,there exists a positiveλf,such that ifλ>λf,then(4)has no solution:while if 0<λ≤λf,then (4)only has a minimal solution Uλ(x)[32].Morover,any two different solution which are distinct to Uλ(x)either equivalent identity or intersect somewhere.Using the same method,we prove the following theoremTheorem 2:For any initial data u0(x),the solution of(3)blows up in finite time whenλ>λf.Theorem 3:Suppose 0<λ≤λf,and Uλ(x)is the minimal solution of (4).If uλ(x)is an arbitrary solution of(4)which is distinct to Uλ(x),then we have(1)If 0