The simplest model of the Cauchy problem considered in this paper is the following ( ∗ ) (\ast ) \[ u t = Δ u + u p , a m p ; x ∈ R n , t > 0 , u ≥ 0 , p > 1 , u | t = 0 = ϕ ∈ C B ( R n ) , a m p ; ϕ ≥ 0 , ϕ ≢ 0. \begin {array}{*{20}{c}} {{u_t} = \Delta u + {u^p},} \hfill & {x \in {R^n},t > 0,u \geq 0,p > 1,} \hfill \\ {u{|_{t = 0}} = \phi \in {C_B}({R^n}),} \hfill & {\phi \geq 0,\phi \,\not \equiv \,0.} \hfill \\ \end {array} \; \] It is well known that when 1 > p ≤ ( n + 2 ) / n 1 > p \leq (n + 2)/n , the local solution of ( ∗ ) (\ast ) blows up in finite time as long as the initial value ϕ \phi is nontrivial; and when p > ( n + 2 ) / n p > (n + 2)/n , if ϕ \phi is "small", ( ∗ ) (\ast ) has a global classical solution decaying to zero as t → + ∞ t \to + \infty , while if ϕ \phi is "large", the local solution blows up in finite time. The main aim of this paper is to obtain optimal conditions on ϕ \phi for global existence and to study the asymptotic behavior of those global solutions. In particular, we prove that if n ≥ 3 n \geq

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