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 \[ \left \{ {\begin {array}{*{20}{c}} {\frac {{\partial u}} {{\partial t}} = \Delta u + {u^p}} \hfill & {{\text {in}}\;{\mathbb {R}^n} \times (0,T),} \hfill \\ {u(x,0) = \varphi (x)} \hfill & {{\text {in}}\;{\mathbb {R}^n},} \hfill \\ \end {array} } \right .\] where $\Delta = \sum \nolimits _{i = 1}^n {{\partial ^2}/\partial _{{x_i}}^2}$ is the Laplace operator, $p > 1$ is a constant, $T > 0$, and $\varphi$ is a nonnegative bounded continuous function in ${\mathbb {R}^n}$. The main results are for the case when the initial value $\varphi$ has polynomial decay near $x = \infty$. Assuming $\varphi \sim \lambda {(1 + |x|)^{ - a}}$ 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 $n$.