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 (...) where Δ = Σ 1=1 n ∂ 2 /∂ xi 2 is the Laplace operator, p > 1 is a constant, T > 0, and φ is a nonnegative bounded continuous function in R n . The main results are for the case when the initial value φ has polynomial decay near x = ∞. Assuming φ ∼ λ(1+|x|) #75a with λ, 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 λ, a, p and the space dimension n