Life span of solution to Cauchy problem for a semilinear heat equation (Q1312917)

It is known that for \(1<p<+\infty\), \(n=2\) or \(1<p<(n+2)/(n-2)\), \(n \geq 3\), the elliptic problem \[ -\Delta u = u^ p-u,\;x \in \mathbb{R}^ n,\quad u(x)>0,\;u(x) \to 0 \quad \text{ as } \quad | x | \to + \infty \] has a unique solution \(\overline u(x)\) which is radially symmetric and satisfies \(\overline u'(0)=0\), \(\overline u'(r)<0\), \(\overline u(x) \sim Me^{-\alpha | x |^ 2}\) as \(| x | \to + \infty\). Let \(u(x,t)\) be the positive solution of \[ u_ t - \Delta u = u^ p-u,\;x \in \mathbb{R}^ n,\;0<t<T,\quad u(x,0) = \lambda \overline u(x),\;x \in \mathbb{R}^ n,\;\lambda>0. \] If \(\lambda>1\), then \(u(x,t)\) blows up in finite time. We denote the life span of \(u(x,t)\) by \(T_ \lambda\). In this note, we discuss the estimate of \(T_ \lambda\).