Abstract.
This paper is concerned with a Cauchy problem
\( \mbox{(P)} \;\;\; \left \{ \begin{array}{ll} u _t = \Delta u + u^p & \quad \mbox{ in }{\bf R}^N \times (0, \infty), u (x,0) = \lambda \varphi(x) & \quad \mbox{ in } {\bf R}^N, \end{array} \right.\)
where \( p > p_{\ast} \equiv (N+2)/(N-2), \lambda > 0 \) and \( \varphi\) is a nonnegative radially symmetric function in \( C^1({\bf R}^N) \) with compact support. Denote the solution of (P) by \(u_{\lambda} \). Let \(p^{\ast} = \infty \) if \( 3 \leq N \leq 10 \) and $p^{\ast} = 1+6/(N-10) \( if \) N \geq 11 \(. We show that if \) p_{\ast} < p < p^{\ast} \(, then there is \)\lambda_{\varphi} > 0 $ such that:
(i) If $ \lambda < \lambda_{\varphi} \(, then \) u_{\lambda} $ exists globally in time in the classical sense and \( u_{\lambda}(t) \) converges to zero locally uniformly in \({\bf R}^N \) as \( t \to \infty \).
(ii) If \( \lambda = \lambda_{\varphi} \), then $ u_{\lambda} $ blows upincompletely in finite time.
(iii) If \( \lambda > \lambda_{\varphi} \), then \( u_{\lambda} \) blows upcompletely in finite time.
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Received: 20 December 1999; in final form: 26 May 2000 / Published online: 4 May 2001
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Mizoguchi, N. On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity. Math Z 239, 215–229 (2002). https://doi.org/10.1007/s002090100292
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DOI: https://doi.org/10.1007/s002090100292