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 (equation formula) is the Laplace operator, p > 1 is a constant, T > 0, and φ is a nonnegative bounded continuous function in ℝⁿ. The main results are for the case when the initial value φ has polynomial decay near x = ∞. Assuming φ φ λ(l + |x|) ⁿ with λ, 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.