On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation

Yohei Fujishima

doi:10.3934/cpaa.2018025

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2018, Volume 17, Issue 2: 449-475. Doi: 10.3934/cpaa.2018025

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On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation

Yohei Fujishima

Department of Mathematical and Systems Engineering, Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu 432-8561, Japan

Received: October 31, 2016

Revised: April 30, 2017

Published: June 2018

The first author is partially supported by by the Grant-in-Aid for Encouragement of Young Scientists (B)(No. 15K17573) from Japan Society for the Promotion of Science.

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Abstract

This paper concerns the blow-up problem for a semilinear heat equation

$\begin{equation}\label{eq:P}\tag{P}≤\left\{\begin{array}{ll}\partial_t u=Δ u+u^p, &x∈ Ω, \, \, \, t>0, \\ u(x, t)=0, &x∈\partialΩ, \, \, \, t>0, \\ u(x, 0)=u_0(x)≥ 0, &x∈ Ω, \end{array}\right.\end{equation}$

where $\partial_t=\partial/\partial t$, $p>1$, $N≥ 1$, $Ω\subset {\bf R}^N$, $u_0$ is a bounded continuous function in $\overline{Ω}$. For the case $u_0(x)=λ\varphi(x)$ for some function $\varphi$ and a sufficiently large $λ>0$, it is known that the solution blows up only near the maximum points of $\varphi$ under suitable assumptions. Furthermore, if $\varphi$ has several maximum points, then the blow-up set for (P) is characterized by $Δ\varphi$ at its maximum points. However, for initial data $u_0(x)=λ\varphi(x)$, it seems difficult to obtain further information on the blow-up set such that effect of higher order derivatives of initial data. In this paper, we consider another type large initial data $u_0(x)=λ+\varphi(x)$ and study the relationship between the blow-up set for (P) and higher order derivatives of initial data.

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

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