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We investigate the behavior of the solution      u ( x , t ) u(x,t)       of  \[    { ∂  u ∂  t = Δ  u + u p a m p ; in R n ×  ( 0 , T ) , u ( x , 0 ) = φ  ( x ) a m p ; in R n , \left \{ {\begin {array}{*{20}{c}} {\frac {{\partial u}} {{\partial t}} = \Delta u + {u^p}} \hfill &amp; {{\text {in}}\;{\mathbb {R}^n} \times (0,T),} \hfill \\ {u(x,0) = \varphi (x)} \hfill &amp; {{\text {in}}\;{\mathbb {R}^n},} \hfill \\ \end {array} } \right .     \]  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 &gt; 1 p &gt; 1       is a constant,      T &gt; 0 T &gt; 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 &gt; 0 a &gt; 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      .

Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem