Type-II blowup for a semilinear heat equation

Abstract

The present paper is concerned with a Cauchy problem for a semilinear heat equation, \[ \left\{ \begin{array}{ll} u_t = \Delta u + u^p & \quad \mbox{ in } {{\bf R}}^N \times (0,\infty), \hspace{3.5cm}\\ u(x,0) = u_0(x) \geq 0 & \quad \mbox{ in } {{\bf R}}^N. \end{array} \right. \tag*{(P)} \] A solution $ u $ of (P) is said to exhibit type-II blowup at $ t = T < + \infty $ if \[ \limsup_{ t \nearrow T } (T-t)^{ 1/(p-1) } |u(t)|_\infty = + \infty \] with the supremum norm $ | \cdot |_\infty $ in $ {{\bf R}}^N $. We show the existence of type-II-blowup solutions. Though our proof is based on the argument due to [9, 10], it is simpler than theirs by taking account of the number of intersections with the singular steady state of (P).