数理解析研究所講究録
第 1984 巻 2016 年 46-51

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Eﬀect of higher order derivatives of initial data on
the blow-up set for a semilinear heat equation
大阪大学大学院基礎工学研究科 藤嶋陽平

Yohei Fujishima
Graduate School of Engineering Science, Osaka University

1Introduction

This paper deals with the blow-up problem for a semilinear heat equation(1.1)$\{\begin{array}{ll}\partial_{t}u=\Delta u+u^{p}, x\in\Omega, t>0,u(x, t)=0, x\in\partial\Omega, t>0,u(x, O)=u_{0}(x)\geq 0, x\in\Omega,\end{array}$where $p>1,$ $N\geq 1,$ is a domain inandis a nonnegative initialfunction. We denote by $T(u_{0})$ the maximal existence time of the unique classical solutionfor problem (1.1). If $T(u_{0})<\infty$ , the solution satisfies$R^{N}$$\Omega$$u_{0}\in C(\overline{\Omega})\cap L^{\infty}(\Omega)$$\lim\sup\Vert u(t)\Vert_{L^{\infty}(\Omega)}=+\infty.$$t\nearrow T(u_{0})$Then we say that the solution blows up in finite time and call $T(u_{0})$ the blow-up time ofthe solution for problem (1.1). Furthermore, for the solution of (1.1) with $T(u_{0})<\infty$ , wedefine the blow-up set $B(u_{0})$ by$B(u_{0}):=\{x\in\overline{\Omega}$: there exists a sequencesuch that$\{(x_{n}, t_{n})\}\subset\overline{\Omega}\cross(0, T(u_{0}))$$\lim_{narrow\infty}(x_{n}, t_{n})=(x, T(u_{0}))$and$\lim_{narrow\infty}u(x_{n}, t_{n})=\infty\}.$The purpose of this paper is to characterize the location of the set $B(u_{0})$ for “large” initial
data . We explain the meaning of “large” initial data later.
We first focus on the case where $u_{0}(x)=\lambda\varphi(x)$ . Here $\lambda>0$ is a suﬃciently large
parameter andis a nonnegative function on St. For problem (1.1) with thistype initial data, it is well known that, for suﬃciently large, the blow-up set $B(u_{0})$is approximated by the one for corresponding ordinary diﬀerential equation$u_{0}$$\varphi\in C_{0}^{\infty}(\Omega)$$\lambda>0$$\partial_{t}u=u^{p}, x\in\Omega, t>0, u(x, O)=\lambda\varphi(x) , x\in\Omega$.

(1.2)

In fact, the author of this paper and Ishige in [4] proved the following result: Assume thatthe solution of (1.1) satisfies$\sup_{\lambda>\lambda_{0}}\sup_{0<t<T(\lambda\varphi)}(T(\lambda\varphi)-t)^{1/(p-1)}\Vert u(t)\Vert_{L\infty(\ddagger l)}<\infty$

(1.3)

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for some$\lambda_{0}>0$, then, for any $\delta>0$ , there exists a constant$\lambda_{\delta}>0$

such that$B(\lambda\varphi)\subset\{x\in\overline{\Omega}:\varphi(x)\geq\Vert\varphi\Vert_{L^{\infty}(\Omega)}-\delta\}$for any. Since $\delta>0$ is arbitrary, this results implies that the solution blows up only
near the maximum points of if is suﬃciently large. Furthermore, the set of maximum
points of corresponds the blow-up set for ODE (1.2), so the blow-up set for (1.1) isapproximated by the one for ODE if is suﬃciently large. Here we give one remark. Wedefine$v(x, t)=\lambda^{-1}u(x, \lambda^{-(p-1)}t)$ .
$\lambda>\lambda_{\delta}$$\lambda$$\varphi$$\varphi$$\lambda$

Then$v$satisfies$\{\begin{array}{ll}\partial_{t}v=\lambda^{-(p-1)}\Delta v+v^{p} in \Omega\cross(0, \lambda^{p-1}T(\lambda\varphi)) ,v(x, t)=0 on \partial\Omega\cross(0, \lambda^{p-1}T(\lambda\varphi)) ,v(x, 0)=\varphi(x) in \Omega.\end{array}$

Therefore, a semilinear heat equation with large initial data is equivalent to a semilinear
heat equation with small diﬀusion, and the above results hold true for$\partial_{t}u=\epsilon\triangle u+u^{p},$$x\in\Omega,$$t>0,$$u(x, t)=0,$$x\in\partial\Omega,$$t>0,$$u(x, O)=\varphi(x)$,$x\in\Omega,$with suﬃciently small
It is natural to ask what happens for the case where has several maximum points.
are maximum points
Concerning this question, it has been proved in [5] that, ifof and,$\epsilon>0.$$\varphi$$\alpha,$$\beta\in\Omega$$\varphi$$\Delta\varphi(\alpha)>\Delta\varphi(\beta)$

for suﬃciently large
then there exists a constant $\delta>0$ such that
under the condition (1.3). Therefore we can characterize the location of the blow-up setby using
at maximum points, and the solution does not blow-up near the maximum
point . In particular, the solution blows up near the maximum point if has only twomaximum points a and . One might ask the natural question$\lambda$$B(\lambda\varphi)\cap B(\beta, \delta)=\emptyset$$\Delta\varphi$$\alpha$$\beta$$\varphi$$\beta$

How can we characterize the location of the blow-up set$B(\lambda\varphi)$

if$\Delta\varphi(\alpha)=\Delta\varphi(\beta)$?

However, unfortunately, it seems diﬃcult to consider this case for problem (1.1) with theinitial data of the form $u0(x)=\lambda\varphi(x)$ . In this paper, we consider another type of “large”
initial data $u_{0}(x)=\lambda+\varphi(x)$ with suﬃciently large $\lambda>0$ , and characterize the locationof the blow-up set for problem (1.1). In particular, we show the relationship between theblow-up set and higher order derivatives of the initial data.

Before stating our main results, we introduce some notation. For any $x\in R^{N}$ andbe the$r>0$ , we denote the open ball of radius and center by $B(x, r)$ . Let$\phi\inBC_{+}(R^{N})$, we denoteset of nonnegative bounded continuos functions on . For anywithinequationofheattheuniquethebybounded solution$BC_{+}(\overline{\Omega})$

$x$

$r$

$\overline{\Omega}$$\partial_{t}u=\Delta u$$e^{t\Delta}\phi$$u(x, 0)=\phi(x)$ in$R^{N}$$R^{N}\cross(0, \infty)$, that is,$(e^{t\triangle} \phi)(x)=(4\pi t)^{-\frac{N}{2}}\int_{R^{N}}e^{-\frac{|x-y|^{2}}{4t}\phi(y)dy}, (x, t)\in R^{N}\cross(0, \infty)$.

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Furthermore, we denote by $M(\phi)$ the set of maximum points ofour main results.

$\phi$. We are ready to state

Theorem 1.1 Let $p>1,$ $N\geq 1,$
be a domain inandsatisfyonand. For any $\lambda>0$ , letbe the blow-up time and the blow-up set of the solution for problem (1.1) withand$u0(x)=\lambda+\varphi(x)$ . Assume thatconsists of only two points and and that
$R^{N}$$\Omega$$\partial\Omega$$\varphi(x)<1\varphi\Vert_{L^{\infty}(\Omega)}$$T_{\lambda}$$\varphi\in C^{4}(\Omega)\cap BC_{+}(\overline{\Omega})$$\lim\sup_{|x|arrow\infty}\varphi(x)<\Vert\varphi\Vert_{L^{\infty}(\Omega)}$$B_{\lambda}$$M(\varphi)$$\alpha$$\Delta\varphi(\alpha)=\Delta\varphi(\beta) , \Delta^{2}\varphi(\alpha)>\Delta^{2}\varphi(\beta)$

Furthermore, assume that there exist positive constant $C$ and$\lambda_{0}$$\beta$.
such that(1.4)$\sup_{0<t<T_{\lambda}}(T_{\lambda}-t)^{1/(p-1)}\Vert u(t)\Vert_{L^{\infty}(tl)}\leq C$

for all$\lambda>\lambda_{0}$. Then there exist positive constant$\delta$and$\lambda_{*}$

such that$B_{\lambda}\subset B(\alpha_{;}\delta)$

for all$\lambda>\lambda_{*}.$

Remark 1.1 (i) Assumption (1.4) can be proved under suitable assumptions on
and$\Omega=R^{N}$. In particular, ifandis suﬃ ciently large, then we can prove (1.4) with$p,$$\Omega$$\lambda>0$$\varphi$

the aid of the argument of [1].
(ii) Let $u_{0}(x)=\lambda+\varphi(x)$ and assume the same situation as in [5]. Let
only two points and , and assume that

$M(\varphi)$

consist of$\beta$$\alpha$$\Delta\varphi(\alpha)>\triangle\varphi(\beta)$.

Then the solution blows up only near the maximum points
if is suﬃciently large.
Therefore, the similar result as in [5] also holds for initial data of the type $u_{0}(x)=\lambda+\varphi(x)$ .
$\lambda$$\alpha$Unfortunately, we can not prove further results in general. However, under some
restriction on , we can show the eﬀect of higher order derivatives on the blow-up set for(1.1).
$p$

Theorem 1.2 For any $m\in N$ with $m\geq 3$ , let $1<p<1+1/2(m-2)$ . Let $N\geq 1,$
be a domain insatisfyandonand, let. For any
andbe the blow-up time and theblow-up set of the solution for problem (1.1) with $u_{0}(x)=\lambda+\varphi(x)$ . Assume thatconsists of only two points and and that
$R^{N}$$\Omega$$\varphi\in C^{4}(\Omega)\cap BC_{+}(\overline{\Omega})$$\lambda>0$$\lim\sup_{|x|arrow\infty}\varphi(x)<\Vert\varphi\Vert_{L^{\infty}(\ddagger 1)}$$\varphi(x)<\Vert\varphi\Vert_{L^{\infty}((l)}$$\partial\Omega$$B_{\lambda}$$T_{\lambda}$$M(\varphi)$$a$$\beta$$\Delta^{k}\varphi(\alpha)=\Delta^{k}\varphi(\beta) (k=1, \ldots, m-1) , \Delta^{m}\varphi(\alpha)>\Delta^{m}\varphi(\beta)$

Assume (1.4). Then there exist positive constant$\delta$$B_{\lambda}\subset B(\alpha, \delta)$

for all$\lambda>\lambda_{*}.$

and$\lambda_{*}$

such that.

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Remark 1.2 Under the same assumptions as in Theorem 1.2, we have$T_{\lambda}= \frac{\lambda_{\varphi}^{-(p-1)}}{p-1}(1+\lambda_{\varphi}^{-(p-1)-1}|\Delta\varphi(\alpha)|-\sum_{k=2}^{m}\frac{\lambda_{\varphi}^{-k(p-1)-1}}{k!(p-1)^{k-1}}\Delta^{k}\varphi(\alpha)+o(\lambda^{-m(p-1)-1}))$for all suﬃciently large

2$\lambda>0$, where$\lambda_{\varphi}$$:=\lambda+\Vert\varphi\Vert_{L^{\infty}(\Omega)}.$Outline of the proof of Theorem 1.1

This section is devoted to explain the outline of the proof of Theorem 1.1. In order toprove Theorem 1.1, we study the profile of the solution just before the blow-up time. Infact, we study the profile of the solution at$t=S_{\lambda}-\lambda_{\varphi}^{-3(p-1)-1},$

where$S_{\lambda}:= \frac{\lambda_{\varphi}^{-(p-1)}}{p-1}(1+\lambda_{\varphi}^{-(p-1)-1}|\Delta\varphi(\alpha)|-\frac{\lambda_{\varphi}^{2(p-1)-1}}{2(p-1)}\Delta^{2}\varphi(\alpha))$.

One of the most important point in the proof of Theorem 1.1 is to get the profile of thesolution at$t=S_{\lambda}-\lambda_{\varphi}^{-2(p-1)-1}.$

Once we get the profile of the solution at this time, we can easily obtain the profile of theby the argument as in [5].
solution at
In order to get the profile of the solution just before the blow-up time, we constructcomparison functions. For the construction of subsolutions, let be the solution of$t=S_{\lambda}-\lambda_{\varphi}^{-3(p-1)-1}$$z$$\partial_{t}z=\Delta z$

in$\Omega\cross(0, \infty)$,$z(x, t)=0$ in$\partial\Omega\cross(0, \infty)$,$z(x, 0)=\lambda+\varphi(x)$in$\Omega,$

and put$U_{0}(x, t) :=(z(x, t)^{-(p-1)}-(p-1)t)^{-1/(p-1)}$

Then we can easily check that the function
is a subsolution for problem (1.1), and wegetprofile
below.
theoffromcanthe solution
For the construction of supersolutions, we employ the cut-oﬀ technique. For a positive
parameter , which will be chosen later, we put$U_{0}$$\epsilon$$\varphi_{\lambda}(x):=\{\begin{array}{ll}m へ \{\varphi(x) , \Vert\varphi\Vert_{L^{\infty}(1l)}-\lambda_{\varphi}^{-(p-1)+\epsilon}\} if x\in\Omega,\Vert\varphi\Vert_{L\infty(\zeta))}-\lambda_{\varphi}^{-(p-1)+\epsilon} if x\not\in\Omega.\end{array}$

Then we have$u(x, 0)\leq\lambda+\varphi_{\lambda}(x)$in$\Omega$

and$\Vert\nabla\varphi_{\lambda}\Vert_{L^{\infty}(R^{N})\sim}<\lambda_{\varphi}^{-(p-1)/2+\epsilon/2}$

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for all suﬃciently large . Consider$\lambda$(2.1)$\{\begin{array}{ll}\partial_{t}U=\Delta U+U^{p}, x\in R^{N}, t>0,U(x, O)=\lambda+\varphi_{\lambda}(x) , x\in R^{N}.\end{array}$

For the construction of supersolutions for problem (1.1), it is enough to construct superby
solutions for problem (2.1). For any $\sigma>0$ , we define the function$U_{\sigma}$$U_{\sigma}(x, t):=([\lambda+(e^{t\Delta}\varphi_{\lambda})(x)]^{-(p-1)}-(p-1)(1+\sigma)t)^{-1/(p-1)}$

Then we see that, if$U_{\sigma}$satisfies$p( \inf_{x\in R^{N}}U(x, 0))^{-2p}U_{\sigma}(x, t)^{p-1}|\nabla(e^{t\triangle}\varphi_{\lambda})(x)|^{2}\leq\sigma,$then the function
is a supersolution for problem (2.1) as long as it exists, and we canget the profile of the solution from above. In order to get precise profile of the solution,
we have to take a parameter $\sigma>0$ as small as possible. For this purpose, we consider thefollowing partition of time interval. Put$U_{\sigma}$$\{\begin{array}{l}I_{0}:=[0, S_{\lambda}-\lambda_{\varphi}^{-(p-1)-1/2}],I_{k}:=[S_{\lambda}-\lambda_{\varphi}^{-(p-1)-k/2}, S_{\lambda}-\lambda_{\varphi}^{-(p-1)-(k+1)/2}] (k=1, \ldots, 2[p-1I:=[S_{\lambda}-\lambda_{\varphi}^{-(p-1)-[p-1]-1/2}, S_{\lambda}-\lambda_{\varphi}^{-2(p-1)-1}].\end{array}$

Then we have$[0, S_{\lambda}- \lambda_{\varphi}^{-2(p-1)-1}]=I_{0}\cup(\bigcup_{k=1}^{2[p-1]}I_{k})\cup I.$

We construct supersolutions in each intervaland obtain the profile of the solution at
the profile of the solution at
We conclude the proof of Theorem 1.1. Put$I_{0},$$I_{k}$

and$I$

by following the above manner,
As a result, we finally get$t=S_{\lambda}-\lambda_{\varphi}^{-2(p-1)-1}$$t=S_{\lambda}-\lambda_{\varphi}^{-3(p-1)-1}$$v(x, \tau):=\lambda_{\varphi}^{-3-\frac{1}{p-1}}u(x, S_{\lambda}-\lambda_{\varphi}^{-3(p-1)-1}+\lambda_{\varphi}^{-3(p-1)-1_{\mathcal{T})}}.$

Then

$v$

satisfies(2.2)$\{\begin{array}{ll}\partial_{\tau}v=\lambda_{\varphi}^{-3(p-1)-1}\Delta v+v^{p}, x\in\Omega, \tau>0,v(x, \tau)=0, x\in\partial\Omega, \tau>0,v(x, O)=\lambda_{\varphi}^{-3-\frac{1}{p-1}}u(x, S_{\lambda}-\lambda_{\varphi}^{-3(p-1)-1}) , x\in\Omega.\end{array}$Furthermore,such that$v$$0$) satisfies the following properties: there exist positive constants$\delta_{1}$and$\delta_{2}$$\sup v(x, O)\leq\Vert v(0)\Vert_{L^{\infty}(t1)}-\delta_{2}$$x\in B(\beta,\delta_{1})$

(2.3)

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for all suﬃciently large . Furthermore, by (1.4) we have$\lambda$$\lim\sup\Vert v(0)\Vert_{L^{\infty}(\Omega)}<\infty.$$\lambdaarrow\infty$) can not take its maximum near . On the other
These imply that the function
hand, the diﬀusion coeﬃcient of problem (2.2) is suﬃciently small, so the solution blows) by the results of [4] and we conclude that theup only near the maximum points ofsolution blows only near the maximum point$v$$0$$\beta$$v$$0$$\alpha.$$\square$

References[1] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equation, Indiana Univ. Math. J. 34 (1985), 425-447.

[2] Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation with small
diﬀusion, J. Diﬀerential $Equati_{on\mathcal{S}}249^{1}(2010)$ , 1056-1077.
[3] Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation and pointedness of the initial data, Indiarla Univ. Math. J. 61 (2012), 627-663.