Global Existence And Explosiveness Of Solutions Of Two Parabolic Equations

In this paper, we investigate the existence of global solution, the solution blow-up at finite time, bounds for the blow-up time and blow-up rate estimates.In chapter 1, using some suitable auxiliary functions and the first-order differen-tial inequality technique, we study the bounds for the blow-up time and blow-up rate estimates for nonlinear parabolic equations where Ω(?) RN(N≥ 3) is a smooth bounded domain,1< p≤2. b is a C2(R+) function satisfying 1≤b’m≤ b’(s)≤b’M, b"(s)≤0 for all s> 0.In chapter 2, we consider some properties of the solution for the following p(x)-Laplace equation where Ω is a smooth bounded domain of RN, u0 ∈L∞(Ω) n W0l,P(x)(Ω), u0(?)0, p(·), q(·) ∈ C(Ω) and satisfy 1< p-≤p(x)≤p+<q-≤q(x)≤q+<+∞. We obtain the existence of global solution to the problem by employing the method of potential wells. On the other hand, we show that the solution will blow up in finite time with u0(?)0 and nonpositive initial energy functional J(u0). By defining a positive function F(t) and using the method of concavity we find an upper bound for the blow-up time.