Life span of solutions of the Cauchy problem for a semilinear heat equation
Asymptotic behaviour for the life span of the solutions of the Cauchy problem \[ u_ t = \Delta u + u^ p, \quad x \in \mathbb{R}^ n,\;t>0, \qquad u(x,0) = \lambda \varphi (x), \quad x \in \mathbb{R}^ n \tag{CP} \] is studied. Here \(p>1\) and the initial function is assumed to be nonnegative, bounded, continuous function, \(\varphi \not \equiv 0\), and \(\lambda\) is a positive parameter. It is a well known fact that there exists a time \(T_ \lambda > 0\) such that (CP) admits a unique classical solution \(u(x,t; \lambda) \in C^{2,1} (\mathbb{R}^ n \times (0,T_ \lambda)) \cap C(\mathbb{R}^ n \times [0,T_ \lambda))\) that is bounded in the interval \([0,T']\) for \(T'<T_ \lambda\), and \(| u (\cdot, t; \lambda) |_{L^ \infty (\mathbb{R}^ n)} \to 0\) as \(t \to T_ \lambda\) if \(T_ \lambda\) is finite. As usual \(T_ \lambda\) is referred as the life span of the solution \(u(x,t; \lambda)\). The authors improve a theorem due to \textit{T.-Y. Lee} and \textit{W.-M. Ni} [Trans. Am. Math. Soc. 333, No. 1, 365-378 (1992; Zbl 0785.35011)] that concerns the asymptotic behaviour of \(T_ \lambda\) as \(\lambda \to 0\) or \(\lambda \to \infty\). More precisely, the following results are proved: (i) \(\lim_{\lambda \to \infty} T_ \lambda \cdot \lambda^{p-1} = (1/(p-1)) | \varphi |^{1-p}_{L^ \infty (\mathbb{R}^ n)}\); (ii) If \(\lim_{| x | \to \infty} \varphi (x) = \varphi_ \infty>0\), then \(\lim_{\lambda \to 0} T_ \lambda \cdot \lambda^{p- 1} = 1/ (p-1) \varphi_ \infty^{1-p}\). Further, a result similar to that in case (ii) above is derived for special initial data \(\varphi (x)\) decaying to zero at infinity.
scientific article; zbMATH DE number 714691