数理解析研究所講究録
第 1896 巻 2014 年 1-11

1Time-dependent singularities
in the heat equation
Jin Takahashi
Department of Mathematics,
Tokyo Institute of Technology

1Introduction

In this article, we review recent works with Yanagida [16] and Kan [8] on removable and non-removable time-dependent singularities in parabolic equations. Throughout this article, we only consider the case $N\geq 3$ . The case
$N=2$ is considered in [16] and [8].
This article is organized as follows. In Section 1.1, we give a necessary
and sufficient condition for the removability of a time-dependent singularity
in the linear heat equation. Non-removable singularities are considered in
Section 1.2. We devote Section 1.3 to state results on effects of a motion of
the singular point. In Section 2, we prove Theorem 1.2.

1.1

Removable singularities

For a solution of the Laplace equation, the removability of a singularity isdefined as follows. Let be a solution of$u$$\triangle u=0$in$\Omega\backslash \{\xi_{0}\},$

where is a domain inand. We say that the singularity of$x=\xi_{0}$at the pointis removable if there exists a classical solution of the
Laplace equation in such thatin
. It is well known thatthe singularity of at is removable if and only if the singularity is weaker
than that of the fundamental solution of the Laplace equation, that is, thecondition for the removability is$\Omega$$\mathbb{R}^{N}$$\xi_{0}\in\Omega$$u$$\tilde{u}$$\tilde{u}\equiv u$$\Omega$$u$$\Omega\backslash \{\xi_{0}\}$$\xi_{0}$$|u(x)|=o(|x-\xi_{0}|^{2-N})$as$xarrow\xi_{0}.$2For nonlinear elliptic equations, the removability of a singularity has beenstudied in many papers and various results have been obtained (see, e.g.,
Brezis and V\’eron [1], V\’eron [17], .-L. Lions [10], Gidas and Spruck [3], themonograph V\’eron [18] and references cited therein).
Similarly, for the heat equation$P$$u_{t}=\triangle u$in$\Omega\backslash \{\xi_{0}\}\cross(0, T)$with $T>0$ , Hsu [6] proved that the singularity ofat $x=\xi_{0}$ is removable$u$

if and only if$|u(x, t)|=o(|x-\xi_{0}|^{2-N})$as$xarrow\xi_{0}$for every $t\in(0, T)$ . The proof is based on precise estimates of the heat
kernel. Later, Hui [7] gave a simpler proof for this result by utilizing Schauderestimates and the maximum principle. For the semilinear parabolic equation$u_{t}=\triangle u+u^{p}, x\in \mathbb{R}^{N}, t>0$

(1.1)

,

Hirata [5] extended Hsu and Hui’s result by an iteration technique.
For the case where a singular point may move in time, the problem on the
be a continuous
removability is formulated as follows. Let :such thatfor any $t\in[0, T]$ andcurve. We take a domain
define$D_{\Omega}:=\{(x, t)\in \mathbb{R}^{N+1}:x\in\Omega\backslash \{\xi(t)\}, t\in(O, T)\}$.
(1.2)
$\xi$

$\Omega\subset \mathbb{R}^{N}$$[0, T]arrow \mathbb{R}^{N}$$\xi(t)\in\Omega$

For a solution of$u_{t}-\Delta u=0$in

$D_{\Omega}$

,

(1.3)

the time-dependent singularity at $x=\xi(t)$ is said to be removable if there
in the classical sense
exists which satisfies the heat equation inandon . Our first theorem gives a necessary and sufficient condition
for the removability of the time-dependent singularity. Roughly speaking, if
has some H\"older continuity, then the removability is analogous to Hsu andHui’s result. More precisely, the results is the following:$\Omega\cross(0, T)$$\tilde{u}$$\tilde{u}\equiv u$

$D$

$\xi$

Theorem 1.1 ([16]). Suppose that is 1/2-H\"older continuous on $[0, T]$
and that satisfies (1.3) in the classical sense. Then the singularity of at$t_{2}\in(0, T)$ with $t_{1}<t_{2}$ and$x=\xi(t)$ is removable if and only if for any
and such that
there exists $r\in(0,1)$ depending on
$\xi$

$u$

$u$

$t_{1},$$\epsilon\in(0,1)$$t_{1},$$t_{2}$$\epsilon$$|u(x, t)|\leq\epsilon|x-\xi(t)|^{2-N}$

for any$x\in \mathbb{R}^{N}$

with $0<|x-\xi(t)|<r$ and for any$t\in[t_{1}, t_{2}].$

3
The proof is based on a construction of a suitable cut-off function. In order to construct the desired function, we suppose that has 1/2-H\"older continuity. However, 1/2 is the critical H\"older exponent in some sense. Indeed,in Section 1.3, we see that the shape of time-dependent singular solutions
are distorted when the motion of is quicker than or equal to 1/2-H\"older
continuous.
$\xi$

$\xi$

1.2

Non-removable singularities

In what follows, we consider singular solutions whose singularity moves intime and is not removable. For the semilinear heat equation (1.1), Sat andYanagida [11] constructed the solution with a time-dependent singularity to
the Cauchy problem for $N/(N-2)<p<(N+2\sqrt{N-1})/(N-4+2\sqrt{N-1})$ .
The solution is singular on given any smooth curve. Moreover, they also
proved that the leading term of the expansion at $x=\xi(t)$ has the same form
as that of the singular steady state of this equation, that is, the solution$0$$\xi(t)$satisfies$u(x, t)=L|x-\xi(t)|^{-m}+o(|x-\xi(t)|^{-m})$ ,as $x=\xi(t)$ , where $m:=2/(p-1)$ and $L:=\{m(N-m-2)^{1/(p-1)}\}$ . Afterward, they studied various properties of time-dependent singular solutions,for instance, the time-global existence [12], convergence to singular steadystates $[15]$ and appearance of anomalous singularities [13, 14].
In this article, we turn to the linear heat equation. To begin with, werecall that the linear heat equation has the singular steady state$\Psi(x):=A_{N}|x|^{2-N} (A_{N}:=4^{-1}\pi^{-\frac{N}{2}}\Gamma(\frac{N}{2}-1))$(1.4)

We remark that is the fundamental solution of the Laplace equation. Indeed, $A_{N}=1/N(N-2)\omega_{N}$ , whereis the volume of the unit ball in
Analogous to the semilinear heat equation, it is expected that there exists a
singular solution whose singular point moves in time, and the leading term of
the expansion is $\Psi(x-\xi(t))$ . Indeed, in [16], such solutions were constructedby utilizing the following equation:$\Psi$$\mathbb{R}^{N}.$$\omega_{N}$$u_{t}-\triangle u=\delta_{\xi(t)}$in$\mathbb{R}^{N}\cross(0, T)$

,

(1.5)

where
is the Dirac distribution concentrated at the point
for each $t\in(0, T)$ . We may represent a solution of (1.5) by using the$\delta_{\xi(t)}$$\xi(t)\in \mathbb{R}^{N}$4
representation formula for the inhomogeneous heat equation. More precisely,
we denote the heat kernel by $\Phi(x, t)=(4\pi t)^{-N/2}\exp(-|x|^{2}/4t)$ and define $F$inby$\mathbb{R}^{N}\cross(0, T)$$F(x, t) := \int_{0}^{t}\Phi(x-\xi(s), t-\mathcal{S})ds.$

Then, $F$ satisfies the following:

continuous. Then
Theorem 1.2 ([16]). Suppose that :inin the distributional sense and (1.3) insatisfies (1.5) inis given by (1.2).
the classical sense, where$F$$[0, T]arrow \mathbb{R}^{N}i\mathcal{S}$$\xi$$\mathbb{R}^{N}\cross(0, T)$$D_{\mathbb{R}^{N}}$$D_{\mathbb{R}^{N}}$

Remark 1.1. Theorem 1.2 also holds if $N=1$ and $N=2.$
It was also shown that the leading term of the expansion of $F(x, t)$ at$x=\xi(t)$ is $\Psi(x-\xi(t))$ if
has some H\"older continuity.
$\xi$

Theorem 1.3 ([16]). Suppose that issome $\alpha>1/2$ . Then for each $t\in(0, T)$ ,$\xi$$\alpha$-H\"older continuous on $[0, T]$ with$F(x, t)=\Psi(x-\xi(t))+o(|x-\xi(t)|^{2-N})$

as $xarrow\xi(t)$ , where$A_{N}$

is given by (1.4).

Remark 1.2. Another proof of Theorems 1.2 and 1.3 were given by Karchand Zheng [9, Section 4]. Their method is based on the Fourier transform.

1.3

Effects of a motion of the singular point

Let us consider the effect of the motion of the singular point. To measure, we makeinstantaneous quickness of the motion of the singular point
the following definition. In this article, we say that has an -velocity at$\xi(t)$

$\xi$

$t$

$\alpha$if$\lim_{s\uparrow t}\frac{\xi(t)-\xi(s)}{(t-s)^{\alpha}}$exists. When has an -velocity at , we call the above limit -velocity. Throughout this subsection, let us consider
vector and denote it by
the case where is continuous on $[0, T]$ , and
$\xi$

$t$

$\alpha$$\alpha$$v_{\alpha}(t)$$\xi$non-zero vector$v_{\alpha}(t_{0})$exists for some $\alpha\in(0,1] and t_{0}\in(0, T)$ .

5
We introduce notation before stating our results. Put
and
. For
, we write $r=|z|,$ $\omega=z/|z|$ anddenoteby the angle between and, that is,With this notation, we have the decomposition $\omega=-(\cos\theta)v_{0}+(\sin\theta)n$ forsomewith $|n|=1,$ $n\cdot\nu_{0}=0.$
In what follows, the case $\alpha=1,$ $\alpha\in(1/2,1),$ $\alpha=1/2$ and $\alpha\in(0,1/2)$
are considered, respectively. If $\alpha=1$ , then the expansion of at $x=\xi(t_{0})$is as follows.
$\rho_{0}$$v_{0};=v_{\alpha}(t_{0})/|v_{\alpha}(t_{0})|$$\theta\in[0, \pi]$$:=|v_{\alpha}(t_{0})|$$z\in \mathbb{R}^{N}\backslash \{0\}$$\omega$$-v_{0}$$\cos\theta=-\omega\cdot v_{0}.$$n\in \mathbb{R}^{N}$$F$

Theorem 1.4 ([8]). Suppose $\alpha=1$ . Then the following (i) and (ii) hold as$z:=x-\xi(t_{0})arrow 0.$(i) If $N=3$ , then$F(x, t_{0})= \Psi(z)+(4\pi)^{-\frac{3}{2}}[\Gamma(\frac{1}{2})\rho_{0}\cos\theta$$+ \int_{0}^{t_{0}}\tau^{-\frac{3}{2}}(e^{-\frac{1}{4}\tau^{-1}|\xi(t_{0})-\xi(t_{0}-\mathcal{T})|^{2}}-1)d\tau+\frac{2}{\sqrt{t_{0}}}].$(ii) If $N\geq 4$ , then$F(x, t_{0})= \Psi(z)+\frac{\rho_{0}\cos\theta}{8\pi^{\frac{N}{2}}}\Gamma(\frac{N}{2}-1)r^{3-N}+o(r^{3-N})$.

Remark 1.3. The integral in Theorem 1.4 (i) is finite.
If $\alpha\in(1/2,1)$ , then the effect of the motion also appears in the second
term of the expansion of $F.$

Theorem 1.5 ([8]). Suppose $\alpha\in(1/2,1)$ . Then
$F(x, t_{0})= \Psi(z)+\frac{\rho_{0}\cos\theta}{2^{2\alpha+1}\pi^{\frac{N}{2}}}\Gamma(\frac{N}{2}-\alpha)r^{2\alpha+1-N}+o(r^{2\alpha+1-N})$

as $z:=x-\xi(t_{0})arrow 0.$
When $\alpha=1/2$ , the effect appears in the leading term of the expansion.
The expansion in the next result implies that the shape of the solution is
distorted towards the back of the singular point.

6
Theorem 1.6 ([8]). If $\alpha=1/2$ , then$F(x, t_{0})=(4 \pi)^{-\frac{N}{2}}e^{-n_{4}^{2}}\rho(\int_{0}^{\infty}\sigma^{\frac{N}{2}-2}e^{-\frac{1}{4}(\sigma-2\sqrt{\sigma}\rho_{0}\cos\theta)}d\sigma)r^{2-N}+o(r^{2-N})$$a\mathcal{S}Z:=x-\xi(t_{0})arrow 0.$

In what follows, let us consider the case $\alpha<1/2$ . We remark that under
this assumption, the integral$\int_{0}^{t_{0}}(t_{0}-s)^{-\frac{N}{2}}\exp\{-\frac{|\xi(t_{0})-\xi(s)|^{2}}{4(t_{0}-s)}\}ds$. Therefore the value ofis finite, because the integrand is bounded in$F(x, t_{0})$ at $x=\xi(t_{0})$ can be defined as a finite value.
This fact suggests
iscontaining the pointsuch thatthat there is some region
andbounded in . The problems in this case are to find such a region$xarrow\xi(t_{0})$ .
also to specify the behavior of $F(x, t_{0})$ whenand $M>0,$
In order to state our result, we define for$(0, t_{0})$$\xi(t_{0})$$\mathcal{N}$$F(\cdot, t_{0})$$\mathcal{N}$$\mathcal{N}$$x\not\in \mathcal{N},$$\epsilon>0$$S_{\epsilon}:= \{z\in \mathbb{R}^{N}\backslash \{0\};1-\cos\theta\geq 2\rho_{0}^{\frac{1}{\alpha}}(\frac{N-3}{2\alpha}+1)(1+\epsilon)r^{\frac{1}{\alpha}-2}\log\frac{1}{r}\},$$T_{M}:=\{z\in \mathbb{R}^{N}\backslash \{0\};1-\cos\theta\leq Mr^{\frac{1}{\alpha}-2}\}.$

Our main result is the following.
Theorem 1.7 ([8]). Suppose that $\alpha\in(0,1/2)$ and that
(1.6)$\xi(t_{0})-\xi(s)=(t_{0}-s)^{\alpha}v_{\alpha}(t_{0})+(t_{0}-s)^{\frac{1}{2}}w_{0}+o((t_{0}-s)^{\frac{1}{2}})$for some$w_{0}\in \mathbb{R}^{N}$as$s\uparrow t_{0}$. Then, for any$\epsilon>0$

and $M>0,$$x- \xi(t_{0})\in S_{\epsilon}\lim_{xarrow\xi(t_{0})}F(x, t_{0})=F(\xi(t_{0}), t_{0})$,$x- \xi(t_{0})\in T_{M}\lim_{xarrow\xi(t_{0})}(r^{\frac{N-3}{2\alpha}+1}e^{\frac{1}{4}J(x-\xi(t_{0}))}F(x, t_{0}))=(4\pi)^{-\frac{N-1}{2}}\alpha^{-1}\rho^{\frac{N-3}{0^{2\alpha}}}e^{-\frac{1}{4}c0},$

where $J(z)$$:=2\rho^{\frac{1}{0^{\alpha}}}r^{-(\frac{1}{\alpha}-2)}(1-\cos\theta)+2\rho^{\frac{1}{0^{2\alpha}}}(n\cdot w_{0})r^{-(\frac{1}{2\alpha}-1)}\sin\theta$$|w_{0}|^{2}-(\nu_{0}\cdot w_{0})^{2}$

Furthermore,$\lim_{xarrow\xi}\inf_{(t_{0})}F(x, t_{0})=F(\xi(t_{0}), t_{0})$,$\lim_{xarrow\xi}\sup_{(t_{0})}(r^{\frac{N-3}{2\alpha}+1}F(x, t_{0}))=(4\pi)^{-\frac{N-1}{2}}\alpha^{-1}\rho^{\frac{N-3}{0^{2\alpha}}}$and$c_{0}$$:=$

7
Remark 1.4 ([8]). If $N=2$ and $\alpha\in(0,1/2)$ , then we obtain$\lim F(x, t_{0})=F(\xi(t_{0}), t_{0})$$xarrow\xi(t_{0})$without using (1.6).
Theorem 1.7 implies that the shape of the solution is more distorted thanthat of the case $\alpha\in[1/2,1]$ . In particular, the solution is continuous alongsome directions and is not continuous towards the back of the singular point.
To observe this phenomenon, we give a simpler version of Theorem 1.7. In
this version, we only consider the limit of when approaches
along
the direction
$F$

$x$

$\xi(t)$$\omega.$

Corollary 1.1. Let$\omega\in S^{N-1}$

Suppose that $\alpha\in(0,1/2)$ and that$\xi(t_{0})-\xi(s)=(t_{0}-s)^{\alpha}v_{\alpha}(t_{0})+o((t_{0}-s)^{\frac{1}{2}})$

as$s\uparrow t_{0}$(i) If. Then the following (i) and (ii) hold.

$\omega=-v_{\alpha}(t_{0})/|v_{\alpha}(t_{0})|$, then$F(x, t_{0})=(4\pi)^{-\frac{N-1}{2}}\alpha^{-1}\rho^{\frac{N-3}{0^{2\alpha}}}|x-\xi(t_{0})|^{-\frac{N-3}{2\alpha}-1}+o(|x-\xi(t_{0})|^{-\frac{N-3}{2\alpha}-1})$as(ii) If$xarrow\xi(t_{0})$along the direction$\omega\neq-v_{\alpha}(t_{0})/|v_{\alpha}(t_{0})|$-$v_{\alpha}(t_{0})/|v_{\alpha}(t_{0})|.$, then$F(x, t_{0})=F(\xi(t_{0}), t_{0})+o(1)$$a\mathcal{S}Xarrow\xi(t_{0})$2along the direction$\omega.$

Proof of Theorem 1.2

We give a proof of Theorem 1.2 for $N\geq 1$ . The proof is similar to [16,Section4]. In this article, we say that satisfies (1.5) in the distributional
sense if belongs toand satisfies
$u$

$u$

$L_{1oc}^{1}(\mathbb{R}^{N}\cross(0, T))$$\int_{0}^{T}\int_{\mathbb{R}^{N}}(-\phi_{t}-\triangle\phi)$

for any$\phi\in C_{0}^{\infty}(\mathbb{R}^{N}\cross(0, T))$.

udxdt $= \int_{0}^{T}\phi(\xi(t), t)dt$

(2.1)

8
proof of Theorem 1.2. Since$\int_{0}^{T}\int_{\mathbb{R}^{N}}F(x, t)dxdt=\frac{1}{2}T^{2}<\infty,$

the function$F$. In particular, $F$ belongs to
is integrable on. In the following, we show that $F$ satisfies (2.1) for all, we define. For each$\mathbb{R}^{N}\cross(0, T)$$L_{1oc}^{1}(\mathbb{R}^{N}\cross(0, T))$$\phi\in C_{0}^{\infty}(\mathbb{R}^{N}\cross(0, T))$$n\in \mathbb{N}$$F_{n}(x, t);= \int_{0}^{\frac{n}{n+1}t}\Phi(x-\xi(s), t-s)ds.$

Then, the integrating by parts yields$\int_{0}^{T}\int_{\mathbb{R}^{N}}(-\phi_{t}-\triangle\phi)F_{n}dxdt=\frac{n}{n+1}I_{n},$where$I_{n} := \int_{0}^{T}\int_{\mathbb{R}^{N}}\phi(x, t)\Phi(x-\xi(\frac{n}{n+1}t),\frac{1}{n+1}t)dxdt.$

First, we prove that$\lim_{narrow\infty}\frac{n}{n+1}I_{n}=\int_{0}^{T}\phi(\xi(t), t)dt$

.

(2.2)

To prove this, we rewrite$I_{n}= \int_{0}^{T}\phi(\xi(t), t)dt+I_{n}’,$where$I_{n}’:= \int_{0}^{T}J_{n}(t)dt$

and$J_{n}(t) := \int_{\mathbb{R}^{N}}\{\phi(x, t)-\phi(\xi(t), t)\}\Phi(x-\xi(\frac{n}{n+1}t), \frac{1}{n+1}t)dx.$

By a similar calculation to [2, Section 2.3.1], $\lim_{narrow\infty}J_{n}(t)=0$ for each$t\in(0, T)$ . Indeed, let $t\in(0, T)$ and, since is continuous on $(0,T)$ ,$\epsilon>0$

$\xi$

9
there exists $\delta>0$ such that $|\phi(x, t)-\phi(\xi(t), t)|<\epsilon$ for any$|x-\xi(t)|<\delta$$x\in \mathbb{R}^{N}$with. Then,$|J_{n}(t)| \leq\epsilon+C_{1}\int_{\{|x-\xi(t)|\geq\delta\}}\Phi(x-\xi(\frac{n}{n+1}t), \frac{1}{n+1}t)dx$

for some constant $C_{1}>0$ . Taking
such that, we
havewhen $|x-\xi(t)|\geq\delta$ . Thus, by the change ofvariables $r=|x-\xi(t)|$ and $s=\sqrt{n+1}r/4\sqrt{t}$ , we calculate that$n\in \mathbb{N}$$| \xi(t)-\xi(\frac{n}{n+1}t)|\leq\frac{1}{2}\delta$$\frac{1}{2}|x-\xi(t)|\leq|x-\xi(\frac{n}{n+1}t)|$$\int_{\{|x-\xi(t)|\geq\delta\}}\Phi(x-\xi(\frac{n}{n+1}t), \frac{1}{n+1}t)dx$$\leq C_{2}(\frac{t}{n+1})^{-\frac{N}{2}}\int_{\{|x-\xi(t)|\geq\delta\}}\exp\{-\frac{|x-\xi(t)|^{2}}{16(n+1)^{-1}t}\}dx$$\leq C_{3}\int_{4}^{\infty}\sqrt{\frac{n+1}{t}}^{s^{N-1}e^{-s^{2}}ds}$where$C_{2},$$C_{3}>0$’

are constants independent of . By $N\geq 1$ , we obtain$n$. Moreover, for any
and $t\in(0, T)$ , the integrandofis dominated by some constant $C_{4}>0$ . Hence, $\lim_{narrow\infty}I_{n}’=0$ by
Lebesgue’s dominated convergence thorem. Thus, (2.2) holds.
Next, direct calculation shows that$\lim_{narrow\infty}J_{n}(t)=0$$n\in \mathbb{N}$$I_{n}’$$| \int_{0}^{T}\int_{\mathbb{R}^{N}}(-\phi_{t}-\triangle\phi)(F_{n}-F)dxdt|\leq C_{5}\int_{0}^{T}\int_{\frac{n}{n+1}t}^{t}dsdt=\frac{C_{5}}{2(n+1)}T^{2}$for some constant $C_{5}>0$ . Taking$narrow\infty$, we obtain$\lim_{narrow\infty}\int_{0}^{T}\int_{\mathbb{R}^{N}}(-\phi_{t}-\triangle\phi)F_{n}dxdt=\int_{0}^{T}\int_{\mathbb{R}^{N}}(-\phi_{t}-\triangle\phi)$

Fdxdt.

(2.3)

Since (2.2) and (2.3) hold,satisfies (2.1). Therefore,satisfies (1.5) inin the distributional sense. Furthermore, (2.1) particularly showsthat for any
,
$F$

$F$

$\mathbb{R}^{N}\cross(0, T)$$\psi\in C_{0}^{\infty}(D_{\mathbb{R}^{N}})$$\int_{0}^{T}\int_{\mathbb{R}^{N}}(-\psi_{t}-\triangle\psi)$

Fdxdt $=0.$

By the Weyl lemma for the heat equation (see, e.g., [4, Section 6]), we conclude that $F$ satisfies (1.3) inin the classical sense.
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$solution\mathcal{S}$Department of Mathematics,
Tokyo Institute of Technology
Meguro-ku, Tokyo 152-8551, Japan-mail: takahashi. .ab@m.titech.ac.jp

$E$

$j$

sEff エ$\hat{}$$\ovalbox{\tt\small REJECT}$$k^{\succ^{\backslash }4}\mp Jk\mapsto^{\backslash }\mp\Re\Leftrightarrow\Phi I;^{\succ^{\backslash }4}\mp ffl_{J\iota}^{*}\ovalbox{\tt\small REJECT}^{\backslash }\backslash f\Re_{\mp\ovalbox{\tt\small REJECT}\iota p}^{\succ^{\backslash }4}rightarrow\cap^{r ゴ}\ovalbox{\tt\small REJECT} t_{-}^{-}$