DCDS
Blow-up set for a superlinear heat equation and pointedness of the initial data
Yohei Fujishima
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 $f$.
keywords: Superlinear heat equation blow-up set blow-up problem comparison principle. small di usion
CPAA
On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation
Yohei Fujishima
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.
keywords: Semilinear heat equation blow-up problem blow-up set large initial data comparison principle

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