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 ) u(x,t) of \[ { ∂ u ∂ t = Δ u + u p a m p ; in R n × ( 0 , T ) , u ( x , 0 ) = φ ( x ) a m p ; in R n , \left \{ {\begin {array}{*{20}{c}} {\frac {{\partial u}} {{\partial t}} = \Delta u + {u^p}} \hfill & {{\text {in}}\;{\mathbb {R}^n} \times (0,T),} \hfill \\ {u(x,0) = \varphi (x)} \hfill & {{\text {in}}\;{\mathbb {R}^n},} \hfill \\ \end {array} } \right . \] where Δ = ∑ i = 1 n ∂ 2 / ∂ x i 2 \Delta = \sum \nolimits _{i = 1}^n {{\partial ^2}/\partial _{{x_i}}^2} is the Laplace operator, p > 1 p > 1 is a constant, T > 0 T > 0 , and φ \varphi is a nonnegative bounded continuous function in R n {\mathbb {R}^n} . The main results are for the case when the initial value φ \varphi has polynomial decay near x = ∞ x = \infty . Assuming φ ∼ λ ( 1 + | x | ) − a \varphi \sim \lambda {(1 + |x|)^{ - a}} with λ \lambda , a > 0 a > 0 , various questions of global (in time) existence and nonexistence, large time behavior or life span of the solution u ( x , t ) u(x,t) are answered in terms of simple conditions on λ \lambda , a a , p p and the space dimension n n .