Blow-up set for a superlinear heat equation  and pointedness of the initial data

Yohei Fujishima

doi:10.3934/dcds.2014.34.4617

Article Contents

2014, Volume 34, Issue 11: 4617-4645. Doi: 10.3934/dcds.2014.34.4617

This issue Previous Article Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations Next Article Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity

Blow-up set for a superlinear heat equation and pointedness of the initial data

Yohei Fujishima^1,

1.
Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka 560-8531

Received: October 2013

Revised: March 2014

Published: November 2014

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Abstract

We study the blow-up problem for a superlinear heat equation \begin{equation} \label{eq:P} \tag{P} \left\{ \begin{array}{ll} \partial_t u = \epsilon \Delta u + f(u), x\in\Omega, \,\,\, t>0, \\ u(x,t)=0, x\in\partial\Omega, \,\,\, t>0, \\ u(x,0)=\varphi(x)\ge 0\, (\not\equiv 0), x\in\Omega, \end{array} \right. \end{equation} where $\partial_t=\partial/\partial t$, $\epsilon>0$ is a sufficiently small constant, $N\ge 1$, $\Omega\subset {\bf R}^N$ is a domain, $\varphi\in C^2(\Omega)\cap C(\overline{\Omega})$ is a nonnegative bounded function, and $f$ is a positive convex function in $(0,\infty)$. In [10], the author of this paper and Ishige characterized the location of the blow-up set for problem (p) with $f(u)=u^p$ ($p>1$) with the aid of the invariance of the equation under some scale transformation for the solution, which played an important role in their argument. However, due to the lack of such scale invariance for problem (p), we can not apply their argument directly to problem (p). In this paper we introduce a new transformation for the solution of problem (p), which is a generalization of the scale transformation introduced in [10], and generalize the argument of [10]. In particular, we show the relationship between the blow-up set for problem (p) and pointedness of the initial function under suitable assumptions on $f$.

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Mathematics Subject Classification: Primary: 35K91; Secondary: 35B44.

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