GLOBAL EXISTENCE, LARGE TIME BEHAVIOR AND LIFE-SPAN OF SOLUTIONS OF A SEMILINEAR PARABOLIC CAUCHY-PROBLEM

We investigate the behavior of the solution u(x, t) of(~)[GRAPHICS] where DELTA = SIGMA(i=1)n partial derivative 2/partial derivative(xi)2 is the Laplace operator, p > 1 is a constant, T > 0, and phi is a nonnegative bounded continuous function in R(n). The main results are for the case when the initial value-phi has polynomial decay near x = infinity. Assuming phi approximately lambda(1 + Absolute value of x)-a with lambda, a > 0, various questions of global (in time) existence and nonexistence, large time behavior or life span of the solution u(x, t) are answered in terms of simple conditions on lambda, a, p and the space dimension n.