American Institute of Mathematical Sciences

Journals

DCDS
Discrete & Continuous Dynamical Systems - A 2014, 34(11): 4617-4645 doi: 10.3934/dcds.2014.34.4617
We study the blow-up problem for a superlinear heat equation $$\label{eq:P} \tag{P} \left\{ \begin{array}{ll} \partial_t u = \epsilon \Delta u + f(u), x\in\Omega, \,\,\, t>0, \\ u(x,t)=0, x\in\partial\Omega, \,\,\, t>0, \\ u(x,0)=\varphi(x)\ge 0\, (\not\equiv 0), x\in\Omega, \end{array} \right.$$ 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:
CPAA
Communications on Pure & Applied Analysis 2018, 17(2): 449-475 doi: 10.3934/cpaa.2018025
This paper concerns the blow-up problem for a semilinear heat equation
 $$$\label{eq:P}\tag{P}≤\left\{\begin{array}{ll}\partial_t u=Δ u+u^p, &x∈ Ω, \, \, \, t>0, \\ u(x, t)=0, &x∈\partialΩ, \, \, \, t>0, \\ u(x, 0)=u_0(x)≥ 0, &x∈ Ω, \end{array}\right.$$$
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: