Global Existence, Large Time Behavior and Life Span of Solutions of a Semilinear Parabolic Cauchy Problem

Abstract

We investigate the behavior of the solution u(x, t) of |f=Aw + " inR" x(0, 7*), u(x, 0) = tp(x) inR", where A = /L] ^2/9^ is the Laplace operator, p > 1 is a constant, T > 0 , and tp is a nonnegative bounded continuous function in R" .The main results are for the case when the initial value tp has polynomial decay near x = oo .Assuming tp ~ A(l + \x\)~" with I, a > 0, various questions of global (in time) existence and nonexistence, arge time behavior or life span of the solution u(x, i) are answered in terms of simple conditions on X, a, p and the space dimension n .+ / / i^tt l v,W2e-]x-yl2/4{'-s)up(y,s;<p)dyds.