数理解析研究所講究録 1237 巻 2001 年 120-135

120
BLOW-UP TIME AND BLOW-UP SET OF THE SOLUTIONS FORSEMILINEAR HEAT EQUATIONS WITH LARGE DIFFUSION

名古屋大学・多元数理科学研究科

石毛和弘 (KAZUHIRO IsHIGE)

Graduate School of Mathematics
Nagoya University
1. Introduction. We consider the Cauchy-Neumann problem(1.1)$u_{t}=d\Delta u+u^{p}$

(1.2)
(1.3)

$D\cross(0, T)$$\frac{\partial}{\partial\nu}u(x, t)=0$

on$\partial D\cross(0, T)$$u(x, 0)=\varphi(x)\geq 0$

on

$D$

(1.4)

$\partial D$,,

where $d>0$ , $p>1,0<T<\infty$ , $D$ is acylindrical domain innormal vector to,in$R^{n}$

and

$\nu$

is the outer unit

. Throughout this paper we assume that

$D=D’\cross(0, L)$,$\varphi\in C(\overline{D})$,$\varphi\not\equiv 0$,$\varphi(x)\geq 0$

in

$D$

,

where $D’$ is asmooth bounded domain in $R^{n-1}$ and $L>0$ . In this paper we study theblow-up set of the solutions$u_{d}$

for the Cauchy-Neumann problem (1.1)-(1.3) with large

diffusion . Furthermore we give an estimate of the blow-up time of the solutions$d$

We denote by$T_{d}$

the supremum of all$\sigma$such that the solution$u_{d}$$u_{d}$.

of (1.1)-(1.3) exists

uniquely for a1H $t<\sigma$ . If $T_{d}<\infty$ , we have.

$\lim_{t\uparrow T_{d}}\mathrm{m}_{\frac{\mathrm{a}\mathrm{x}}{D}}u_{d}(x, t)=\infty x\in$Then we say that $ud$ blows up at the time$u_{d}$. We define the blow-up set$B_{d}(\varphi)=${$x\in\overline{D}|\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$$B_{d}(\varphi)$exist$T_{d}$, and callof the solution$x_{k}arrow x$and$t_{k}\uparrow T_{d}$$T_{d}$$u_{d}$

the blow-up time of the solution

bysuch that$\lim_{karrow\infty}u_{d}(x_{k},$$t_{k})=\infty$

Typeset by

}.
-qffl

$\mathrm{A}\mathcal{M}\mathrm{S}$121
F. B. Weissler [20] first proved that some solutions blow up only at asingle point

for the case $n=1$ . A. Friedman and B. McLeod [8] proved similar results for moregeneral domains under the Dirichlet boundary condition or the Robin boundary condition.

Subsequently, the blow-up sets of the blow-up solutions have been studied by various
peoples. Among others, for the case $n=1$ , X. Y. Chen and H. Matano [5] proved thatthe blow-up solution blows up at most at finite points in $D$ under the Dirichlet boundary
condition or the Neumann boundary condition. Furthermore, for the case $n=1$ , F. Merle[11] proved that, for any given finite points
blow-up set is exactly$\{x_{1}, \ldots, x_{k}\}$$x_{1}$,$\ldots$,$x_{k}\subset D$, there exists asolution whose. For the case $n\geq 2$ , J. J. L. Velazquez [19] provedthat the $(n-1)$ -dimensional HausdorfF measure of the blow-up set of nontrivial blow-up
solution for the case $D=R^{n}$ is bounded in compacts sets of$R^{n}$. (For further results on

the blow-up set, see [2-4], [6], [7], [9], [12-17], and references given there.) However, forthe case $n\geq 2$ , it seems to be difficult to study the arrangement of the blow-up set without

somewhat strong conditions on the initial data, even for the case that $D$ is acylindrical

domain.

Our main interest is to investigate the blow-up set$B_{d}(\varphi)$of the solutions of the

Cauchy-Neumann problem (1.1)-(1.3) with large diffusion . We prove that, for almost all$d$initial data$\varphi$, the blow-up set

for sufficiently large$d$$B_{d}(\varphi)$

consists of the points of the set$\overline{D’}\cross\{0, L\}\subset\partial D$. Furthermore, as aby-product, we give an estimate of the blow-up

time for sufficiently large .
$d$

Now we give our main result of this paper.
Theorem A. Consider the Cauchy-Neumann problem (1.1)-(1.3) under the condition(1.4). Assume that
(1.5)

.

$I( \varphi)\equiv\int_{D}\varphi\cos(\frac{\pi}{L}x_{n})dx\neq 0$

Then there exists a positive constantof the solution(1.6)$u_{d}$$d_{0}$

such that, for any$d\geq d_{0}$of (1.1)-(1.3) satisfies that
$B_{d}(\varphi)\subset\overline{D’}\cross\{0\}$if $I(\varphi)>0$$B_{d}(\varphi)\subset\overline{D’}\cross\{L\}$if $I(\varphi)<0$ .

and that(1.7), the blow-up set$B_{d}(\varphi)$

122
Here

$d_{0}$

depends only on , $D,$ ,
$n$

$p$

$I(\varphi)$, and$||\varphi||_{L}\infty(D)$.

We remark that the condition (1.5) holds for almost all initial data$\varphi$

physically. We mayfind the similar condition to (1.5) in the Rauch observation, which means that the hot
spots of the solutions of the heat equation under the zero Neumann boundary condition

move to the boundary, as$tarrow\infty$(see [1], [10], and [18]).

As aby-product of arguments in the proof of Theorem
blow-up time

$T_{d}$

for sufficiently large$d$$\mathrm{A}$, we have an estimate of the.

Theorem B. Consider the Cauchy-Neumann problem (1.1)-(1.3) under the condition(1.4). Then $T_{d}<\infty$ . Fuhhemooe there exist constants $C$ and(1.8)$|T_{d}-(p-1)(_{\overline{\varphi}}^{\underline{1}})^{p-1}| \leq C\frac{1\mathrm{o}\mathrm{g}d}{d}$for all$d\geq d_{0}$. Here $h$ depends only on ,
$n$

$D$

,

$\overline{\varphi}=\frac{1}{|D|}\int_{D}\varphi dx$, , and$p$such that$d_{0}$$||\varphi||_{L}\infty(D)$,.

The remainder of this paper is organized as follows. In Section 2, by the comparison
principle, we obtain aupper and alower estimates of the solution$u_{d}$. Furthermore weconstruct approximate solutions of (1.1)-(1.3), and give a $C^{2}(D)$ -norm estimate of thesolution and the approximate solutions. In Section 3we give an estimate of minimum
value of the solution$u_{d}$

at the blow-up time. In Section 4we prove Theorem$\mathrm{B}$by using

the results of Sections 2and 3. In Section 5we prove the monotonicity of the solution$ud$in the direction$x_{n}$

at some time. Furthermore, we apply the arguments in [5] and [8]together with the estimates in Sections 2and 3to our problem, and complete the proof of

Theorem A.
2. Preliminary Results. In this section, by the comparison principle, we obtain aupper
and alower estimates of the solution$u_{d}$. Furthermore we construct approximate solutionsof (1.1)-(1.3) by the Galerkin method, and give a $C^{2}(D)$ -norm estimate of the solution
and the approximate solutions.

Let
(2.1)

$\zeta(t:\alpha)$be asolution of$\zeta’=\zeta^{p}$,$\zeta(0)=\alpha\geq 0$.

Put$S_{\alpha}=(p-1)( \frac{1}{\alpha})^{p-1}$,$S=S_{\max_{x\in\overline{D}}\varphi}$.

$u_{d}$

123
Then$\zeta(\cdot :exists on the interval\alpha)$Proposition 2.1. Let(2.2)$u_{d}$$[0, S_{\alpha})$and.

be a solution of (1.1)-(1.3) under the condition (1.4). Then$u_{d}(x, t)\leq\zeta(t;\mathrm{m}_{\frac{\mathrm{a}}{D}}\mathrm{x}\varphi)$(2.3)$\lim_{t\uparrow}s_{\alpha}\zeta(t:\alpha)=\infty$$T_{d}\geq S$,$(x, t)\in D\cross(0, S)$,.

Further more there exists a nondecreasing function $\eta\in C((0, \infty);(0, \infty))$ such that(2.4)

$u_{d}(x, t)\geq\eta(dt)$,$(x, t)\in D\cross(0, T_{d})$.

Proof. We see (2.2) and (2.3) easily by the comparison principle. So it suffices to prove(2.4). Put(2.5)$\eta(t)=\mathrm{m}_{\frac{\mathrm{i}\mathrm{n}}{D}}v(x, t)x\in$

where

$v$

$t>0$ .

’

is asolution of$\{\begin{array}{l}v_{t}=\Delta v\mathrm{i}\mathrm{n}D\cross(0,\infty)\frac{\partial}{\partial\nu}v(x,t)=0\mathrm{o}\mathrm{n}\partial D\cross(0,\infty)v(x,0)=\varphi(x)\mathrm{i}\mathrm{n}D\end{array}$

By the maximum principle,$\eta(t)$is anondecreasing, positive, continuous function on $(0, \infty)$ ,and$u_{d}(x, t)\geq v(x, dt)\geq\eta(dt)$

So the proof of Proposition 2.1 is complete.
Let$\psi_{0}$,$\psi_{1}$,$\psi_{2}$,$\ldots$,$(x, t)\in D\cross(0, T_{d})$.

$\square$be acomplete orthonormal basis for $L^{2}(D)$ of Neumann eigenfunc-

tions with eigenvalues $0=\mu_{0}<\mu_{1}\leq\mu_{2}\leq\cdots$ , where we repeat the eigenvalues if needed

to take account their multiplicity. We remark that $\psi_{0}=1/|D|^{1/2}$ . For
denote by$P_{j}$$j\in \mathrm{N}\cup\{0\}$the projection ffom $L^{2}(D)$ to the subspace of $L^{2}(D)$ spanned by

Then
(2.6)

$\frac{\partial}{\partial t}P_{j}u_{d}=dAPjUd+P_{j}u_{d}^{p}$(2.7)$\frac{\partial}{\partial\nu}P_{j}u_{d}=0$(2.8)$P_{jd}u(x, 0)=P_{j}\varphi(x)$,in$D\cross(0, T_{d})$on$\partial D\cross(0, T_{d})$

in

$D$

.

By the standard calculations, we have the following proposition

,

, we

$\{\psi\iota\}_{l=0}^{j}$.

124
Proposition 2.2. Let $d\geq 1$ and $0<d\epsilon\leq 1$ . Let$u_{d}$

condition (1.4). Then there exist positive constantsde$a+\epsilon\leq t\leq T\mathrm{m}\mathrm{a}\mathrm{x}||u_{d}(\cdot,t)-P_{j}u_{d}(\cdot, t)||_{C^{2}(D)}\leq C_{1}($be a solution of (1.1)-(1.3) under the$C_{1}$,$C_{2}$, and$\alpha$such that$)^{-\alpha}(||u_{d}(\cdot, a)-P_{j}u_{d}(\cdot, a)||_{L^{2}(D)}$$+d^{-1}||u_{d}(\cdot, a)||_{L^{2}(D)}+d^{-1/2}||u_{d}^{p}||_{L^{2}(a,T;L^{2}(D))})$

for all $0<a<a+C_{2}\epsilon\leq T<T_{d}$ and $j=0,1$ ,$d(T-a), \min_{\overline{D}\mathrm{x}[a,T]}u_{d}$, and$\overline{D}[a,T]\max_{\cross}u_{d}$, and$C_{2}$$\ldots$. Heredepends only on $D$ ,$C_{1}$

depends only on $D$ and

$n$

$n$

,

.

Furthermore we have the following proposition, which is amain one in this section.
Proposition 2.3. Let$d\geq d_{0}$(2.12)(2.13)$d_{1}$$d_{1}$$d_{0}$

and $C=C(n, D)$ such,$||u_{d}( \cdot, t)-P_{j}u_{d}(\cdot, t)||_{C^{2}(D)}\leq C(e^{-d\mu t}+\frac{1}{d^{1/2}})$

Proof. Let
small

be a solution of (1.1)-(1.3) under the condition (1.4). Let $j\in$

and $0<\mu<\mu_{j+1}$ . Then there exist positive constants$\mathrm{N}\mathrm{U}\{0\}$that, if$u_{d}$

be aconstant such that$d_{1}\geq 1$,

and $d_{1}S\geq 4$ . Let$\frac{2}{d}\leq t\leq\frac{S}{2}$$d\geq d_{1}$.

. Taking sufficiently

if necessarily, by Proposition 2.2, we have$||u_{d}(\cdot, \tau)-P_{j}u_{d}(\cdot, \tau)||_{C^{2}(D)}|_{\tau=t/d}\leq C_{1}(||u_{d}(\cdot,\tau)-P_{j}u_{d}(\cdot, \tau)||_{L^{2}(D)}|_{\tau=(t-1)/d}$$+d^{-1}||u_{d}(\cdot, (t-1)/d)||_{L^{2}(D)}+d^{-1/2}||u_{d}^{p}||_{L^{2}((t-1)/d,t/d;L^{2}(D))})$

for all $2\leq t\leq dS/2$ . Here(2.14)$C_{1}$is aconstant depending only on , $D$ ,$n$$(x, \tau)\in\overline{D}\mathrm{x}[(t-1)/d,t/d]\min u_{d}(x, \tau)$,$(x, \tau)\in\overline{D}[(t-1)/d,t/d]\max_{\mathrm{X}}u_{d}(x, \tau)$

On the other hand, by Proposition 2.1, there exists aconstant(2.15)$C_{2}$.

such that$\eta(1)\leq\eta(t)\leq u_{d}(x, t/d)\leq\zeta(t/d;\max\varphi)\leq\zeta(S/2;\mathrm{m}_{\frac{\mathrm{a}}{D}}\mathrm{x}\varphi)\leq C_{2}F$

for all $(x,t)\in D\cross[1, dS/2]$ , where is afunction given in Proposition 2.1. By (2.13)-(2.15),$\eta$

there exists aconstant(2.16)$C_{3}$

depending only$n$and $D$ , such that$||u_{d}( \cdot, \tau)-P_{j}u_{d}(\cdot, \tau)||_{C^{2}(D)}|_{\tau=t/d}\leq C_{3}(||u_{d}(\cdot, \tau)-P_{j}u_{d}(\cdot,\tau)||_{C^{2}(D)}|_{\tau=(t-1)/d}+\frac{1}{d^{1/2}})$

125
for all$d\geq d_{1}$.

Put $v_{d}=u_{d}-P_{j}u_{d}$ . By (2.6) and (2.15), for any $0<\delta<1$ , we have$\frac{1}{2}\frac{\partial}{\partial t}\int_{D}|v_{d}|^{2}dx=\int_{D}\{d\Delta v_{d}\cdot v_{d}+(u_{d}^{p}-P_{j}u_{d}^{p})v_{d}\}dx$$\leq\int_{D}\{-d\mu_{j+1}|v_{d}|^{2}+|u_{d}^{p}-P_{j}u_{d}^{p}||v_{d}|\}dx$$\leq-d\mu\int_{D}|v_{d}|^{2}dx+C_{4}\int_{D}|u_{d}|^{2p}dx$$\leq-d\mu\int_{D}|v_{d}|^{2}dx+C_{5}$for some constants(2.17)$C_{4}$

and$C_{5}$,$0<t< \frac{S}{2}$. Therefore, there exists aconstant$C_{6}$,
such that$||u_{d}(\cdot, \tau)-P_{j}u_{d}(\cdot, \tau)||_{L^{2}(D)}^{2}|_{\tau=(t-1)/d}=||v_{d}(\cdot, \tau)||_{L^{2}(D)}^{2}|_{\tau=(t-1)/d}$$\leq e^{-2\mu(t-1)}||v_{d}(\cdot, 0)||_{L^{2}(D)}^{2}+\frac{C_{5}}{d\mu}\leq C_{6}(e^{-2\mu t}+\frac{1}{d})$for all $2\leq t\leq dS/2$ . By (2.16) and (2.17), we obtain the inequality (2.12), and the proofof Proposition 2.3 is complete.

$\square$

3. Minimum Value of the Solution at the Blow-Up Time. In this section we study
the behavior of the function
the solution$u_{d}$$u_{d}$

-P\^o

$d$

, and obtain an estimate of the minimum value ofof (1.1)-(1.3) at the blow-up time

Proposition 3.1. Let$u_{d}$

there exist constants C and(3.1)$T_{d}$.

be a solution of (1.1)-(1.3) under the condition (1.4). Then$d_{0}$

such that, if d$\geq d_{0}$,$\lim_{t\uparrow T_{d}}\mathrm{m}_{\frac{\mathrm{i}\mathrm{n}}{D}}u_{d}(x, t)x\in\geq Cd^{3/2(p-1)}$.

In order to obtain Proposition 3.1, we prove the following lemma by using Proposition

2.1.
Lemma 3.2. Let$u_{d}$

exist constants $C$ and(3.2)

be a solution of (1.1)-(1.3) under the condition (1.4). Then there$d_{0}$

such that, if$d\geq d_{0}$,$||u_{d}( \cdot, t)-P_{0}u_{d}(t)||_{L\infty(D)}\leq C(e^{-d\mu t}+\frac{1}{d^{3/2}})$,$\frac{3}{d}\leq t\leq\frac{S}{2}$,126
where $\mu=\mu_{1}/4$ .

Proof. By Proposition 2.1, there exist constants
(3.3)

Let

$C_{1}$

and$d_{1}$

such that, if$||u_{d}( \cdot, t)-P_{0}u_{d}(\cdot, t)||_{L(D)}\infty\leq C_{1}(e^{-d\mu t}+\frac{1}{d^{1/2}})$$d_{2}$

be aconstant such that$d_{2}\geq d_{1}$and $d_{2}S\geq 6$ . For$v_{d}(x, t)=u_{d}(x, t)- \overline{\varphi}-\int_{0}^{t}(P_{0}u_{d}(s))^{p}ds$,,$d\geq d_{1}$$\frac{2}{d}\leq t\leq\frac{S}{2}$

,

.

, put

$d\geq d_{2}$$g(x, t)=(u(x, t))^{p}$ $-(Poud(t))^{p}$ ,for $(x, t)\in D\cross(0, T_{d})$ . Furthermore we put$w_{d}(x, \tau)=v_{d}(x,$ $\frac{\tau}{d})-(P_{0}v_{d})(\frac{\tau}{d})$,for $(x,\tau)\in D\cross(t-1, t)$ and $1<t<dT_{d}$ . Then
(3.4)$\frac{\partial}{\partial\tau}w_{d}=\Delta w_{d}+\frac{1}{d}\tilde{g}$(3.5)$\frac{\partial}{\partial\nu}w_{d}(x, t)=0$

By$L^{\infty}$$w_{d}$

satisfies,in$D\cross(0, t)$on$\partial D\cross(0, t)$.

-estimates of the solutions of the parabolic equations, (2.15), (3.4), and (3.5), there

exist constants(3.6)$\tilde{g}(\cdot, \tau)=g(\cdot,$ $\frac{\tau}{d})-(P_{0}g)(\frac{\tau}{d})$$C_{2}$

and$C_{3}$

such that$||w_{d}(\cdot, t)||_{L\infty(D)}\leq C_{2}(||w_{d}(\cdot, t-1)||_{L^{2}(D)}+d^{-1}||\tilde{g}||_{L\infty(D\mathrm{x}(t-1,t))})$$\leq C_{2}(||w_{d}(\cdot, t-1)||_{L^{2}(D)}+2d^{-1}||g||_{L\infty(D\mathrm{x}((t-1)/d,t/d))})$$\leq C_{3}(||w_{d}(\cdot, t-1)||_{L^{2}(D)}+d^{-1}||u_{d}-P_{d}u_{d}||_{L(D\mathrm{x}((t-1)/d,t/d))}\infty)$

for all $1<t<dS/2$ . Therefore, by (3.3) and (3.6), there exists aconstant(3.7)$C_{4}$

such that$||u_{d}(\cdot, \tau)-P_{0}u_{d}(\tau)||_{L\infty(D)}|_{\tau=t/d}=||v_{d}(\cdot, \tau)-P_{0}v_{d}(\tau)||_{L\infty(D)}|_{\tau=t/d}$$\leq C_{3}(||w_{d}(\cdot, t-1)||_{L^{2}(D)}+d^{-1}||u_{d}-P_{0}u_{d}||_{L(D\mathrm{x}((t-1)/d,t/d))}\infty)$$\leq C_{4}(||w_{d}(\cdot, t-1)||_{L^{2}(D)}+\frac{1}{d}e^{-\mu t}+\frac{1}{d^{3/2}})$for all$3 \leq t\leq\frac{dS}{2}$

.

,

127
On the other hand, by (3.4) and (3.5), there exists aconstant(3.7)$C_{5}$

such that$\frac{1}{2}\frac{\partial}{\partial\tau}\int_{D}|w_{d}|^{2}dx=\int_{D}\{\Delta w_{d}\cdot w_{d}+d^{-1}\tilde{g}w_{d}\}dx$$\leq\int_{D}\{-\mu_{1}|w_{d}|^{2}+d^{-1}|\tilde{g}||w_{d}|\}dx$$\leq-\delta\mu_{1}\int_{D}|wd|^{2}dx+C_{5}d^{-2}\int_{D}|g(x, \tau/d)|^{2}dx$,

for all $0<\tau<t$ and $1<t<dS/2$ , where $\delta=1/2$ . By (2.8), (3.7), and (3.8), there exists
aconstant(3.9)$C_{6}$

such that$||w_{d}(\cdot, t-1)||_{L^{2}(D)}^{2}$$\leq e^{-2\delta\mu_{1}(t-1)}||w(\cdot, 0)||_{L^{2}(D)}^{2}+\frac{2C_{5}}{d^{2}}e^{-2\delta\mu_{1}(t-1)}\int_{0}^{t-1}e^{2\delta\mu_{1\mathit{3}}}\int_{D}|g(x,$ $\frac{s}{d})|^{2}dxds$$\leq 2C_{6}e^{-2\delta\mu_{1}(t-1)}$$+ \frac{2C_{6}}{d^{2}}e^{-2\delta\mu_{1}(t-1)}\{\int_{0}^{2}+\int_{2}^{t-1}\}e^{2\delta\mu_{1}s}\int_{D}|u_{d}^{p}$for all $3\leq t\leq dS/2$ . By (2.15), there exist constants(3.10)$e^{-2\delta\mu_{1}(t-1)} \int_{0}^{2}e^{2\delta\mu 1^{S}}\int_{D}|u_{d}^{p}(x,$$C_{7}$

(, )

and

$x$

$\frac{s}{d}$$C_{8}$$-(P_{0}u_{d})^{p}( \frac{s}{d})|^{2}dxds$

such that$\frac{s}{d})-(P_{0}u_{d})^{p}(\frac{s}{d})|^{2}dxds$$\leq C_{7}e^{-2\delta\mu_{1}(t-1)}\int_{0}^{2}e^{2\delta\mu_{1}s}ds\leq C_{8}e^{-2\delta\mu_{1}t}$By (2.15) and (3.3), there exist constants(3.11)$e^{-2\delta\mu_{1}(t-1)} \int_{2}^{t-1}e^{2\delta\mu_{1}s}\int_{D}|u_{d}^{p}(x,$$C_{9}$

and$C_{10}$.

such that$\frac{s}{d})-(P_{0}u_{d})^{p}(\frac{s}{d})|^{2}dxds$.
$\leq C_{9}e^{-2\delta\mu_{1}(t-1)}\int_{2}^{t-1}e^{2\delta\mu_{1}s}\int_{D}|u_{d}$

(, )
$x$

$\frac{s}{d}$$-(P_{0}u_{d})( \frac{s}{d})|^{2}dxds$$\leq 2C_{9}e^{-2\delta\mu_{1}(t-1)}\int_{2}^{t-1}e^{2\delta\mu_{1}s}(e^{-\mu_{1}s/2}+\frac{1}{d})ds\leq C_{10}(e^{-\mu_{1}t/2}+\frac{1}{d})$Putting $\mu=\mu_{1}/2$ , by (3.9)-(3.11), there exists aconstant(3.12)$C_{11}$.

such that$||w_{d}( \cdot., t-1)||_{L^{2}(D)}^{2}\leq C_{11}(e^{-2\mu t}+\frac{1}{d^{3}})$for all $3\leq t\leq dS/2$ . Therefore, by (3.7) and (3.12), there exists aconstant$||u_{d}( \cdot, \tau)-P_{0}u_{d}(\cdot, \tau)||_{L(D)}\infty|_{\tau=t/d}\leq C_{12}(e^{-\mu t}+\frac{1}{d^{3/2}})$$C_{12}$such that

128
for all $3\leq t\leq dS/2$ , and the proof of Lemma 3.2 is complete.

Proof of Proposition 3.1. Let$\zeta(t :\alpha)$$\square$be asolution of the ordinary differential equation(2.1), that is,(313)$\zeta(t:\alpha)=[\frac{1}{\alpha^{p-1}}-(p-1)t]-1/(p.-1)$

By Lemma 3.2, there exist constant(3.14)$C_{1}$

and$d_{1}$

such that, if$||u_{d}( \cdot, t)-P_{0}u_{d}(t)||_{L(D)}\infty|_{t=\frac{210-d}{\mu d}}\leq C_{1}\frac{1}{d^{3/2}}$$d\geq d_{1}$,,$\mu=\frac{1}{4}\mu_{1}$.

This inequality together with the comparison principle implies that
(3.15)$\zeta(t-\frac{21\mathrm{o}\mathrm{g}d}{\mu d}$:$P_{0}u_{d}( \frac{21\mathrm{o}\mathrm{g}d}{\mu d})-C_{1}\frac{1}{d^{3/2}})$$\leq ud(x, t)\leq\langle$ $(t- \frac{21\mathrm{o}\mathrm{g}d}{\mu d}$

for all $x\in D$ ,$t\geq 21\mathrm{o}d\hat{\mu d}$, and$d\geq d_{1}$:$P_{0d}u( \frac{21\mathrm{o}\mathrm{g}d}{\mu d})+C_{1}\frac{1}{d^{3/2}})$. By (3.15), we have$T_{d} \geq\frac{21\mathrm{o}\mathrm{g}d}{\mu d}+\frac{1}{p-1}[P_{0}u_{d}(\frac{21\mathrm{o}\mathrm{g}d}{\mu d})+C_{1}\frac{1}{d^{3/2}}]-(p-1)$

On the other hand, by (2.6) and (2.15), there exists aconstant,(3.16)$0<t< \frac{S}{2}$$|P_{0}u_{d}(t)- \overline{\varphi}|=\frac{1}{|D|}\int_{D}u_{d}^{p}dx\leq C_{2}t$Therefore, by (3.13), (3.14), and (3.16), there exist constants$d\geq d_{2}$$C_{2}$,$C_{3}$

such that$\overline{\varphi}\neq 0$and.

$d_{2}\geq d_{1}$such that, if,$\lim_{t\uparrow T_{d}}\mathrm{m}_{\frac{\mathrm{i}\mathrm{n}}{D}}u_{d}(x,t)x\in$$\geq\zeta(\frac{1}{p-1}\{$ $P_{0}u_{d}( \frac{21\mathrm{o}\mathrm{g}d}{\mu d})+C_{1}\frac{1}{d^{3/2}}]-(p-1)$:$P_{0}u_{d}( \frac{21\mathrm{o}\mathrm{g}d}{\mu d})-C_{1}\frac{1}{d^{3/2}})$$=[\{$ $P_{0}u_{d}( \frac{21\mathrm{o}\mathrm{g}d}{\mu d})-C_{1}\frac{1}{d^{3/2}}\}^{-(p-1)}-\{$ $P_{0}u_{d}( \frac{21\mathrm{o}\mathrm{g}d}{\mu d})+C_{1}\frac{1}{d^{3/2}}\}^{-(p-1)-1/(p-1)}]$$\geq C_{3}d^{3/2(p-1)}$,

and the proof of Proposition 3.1 is complete.

$\square$129
4. Proof of Theorem B.

Proof of Theorem . We first prove$B$$T_{d}<\infty$. By Proposition 2.1, for any $T\in(0, S)$ , wc

have$u_{d}(x, t)\geq\eta(dT)>0$,$(x, t)\in D\cross(T, T_{d})$.

This inequality together with the comparison principle implies that$u_{d}(x, t)\geq\zeta(t;\eta(dT))$,$(x, t)\in D\cross(T, T_{d})$.

Therefore we have$T_{d} \leq T+\int_{\eta(dT)}^{\infty}\frac{ds}{s^{p}}<\infty$.

Next we prove (1.8). By (3.2) and (3.16), there exist constants
(4.2)

and

$C_{1}$$d_{1}$

such that$||u_{d}(\cdot, t)-\overline{\varphi}||_{L^{\infty}(D)}\leq||u_{d}(\cdot, t)-P_{0}u_{d}(t)||_{L^{\infty}(D)}+||P_{0}u_{d}(t)-\overline{\varphi}||_{L^{\infty}(D)}$$\leq C_{1}(e^{-d\mu t}+\frac{1}{d^{3/2}}+t)$

for all$\frac{3}{d}\leq t\leq\frac{s}{2}$

(4.3)

and

$d\geq d_{1}$$||u_{d}(\cdot,$

for all$d\geq d_{2}$,. By (4.2), there exist constants$\frac{21\mathrm{o}\mathrm{g}d}{\mu d})-\overline{\varphi}||_{L^{\infty}(D)}\leq C_{2}\frac{1\mathrm{o}\mathrm{g}d}{d}$,$C_{2}$

and$d_{2}\geq d_{1}$$\mu=\frac{1}{4}\mu_{1}$such that,.

On the other hand, by the comparison principle and (4.3), we have
(4.4)$\zeta(t-\frac{21\mathrm{o}\mathrm{g}d}{\mu d};\overline{\varphi}-C_{2}\frac{1\mathrm{o}\mathrm{g}d}{d})\leq u_{d}(x, t)\leq\zeta(t-\frac{21\mathrm{o}\mathrm{g}d}{\mu d};\overline{\varphi}+C_{2}\frac{1\mathrm{o}\mathrm{g}d}{d})$for all $(x, t)\in D\cross(2\log d/\mu d, T_{d})$ . By (4.4), we have.

$\frac{1\mathrm{o}\mathrm{g}d}{\mu d}+\int_{\overline{\varphi}+C_{2}^{\underline{\mathrm{l}\circ}\mathrm{g}\underline{d}}}^{\infty}\frac{ds}{s^{p}}d\leq T_{d}\leq\frac{1\mathrm{o}\mathrm{g}d}{\mu d}+\int_{\overline{\varphi}-C_{2}^{\underline{1}_{\circ}d}}^{\infty}\doteqdot\frac{ds}{s^{p}}$Therefore there exists aconstant$C_{3}$

such that$|T_{d}- \int_{\overline{\varphi}}^{\infty}\frac{ds}{s^{p}}|\leq\frac{1\mathrm{o}\mathrm{g}d}{\mu d}+\int_{\overline{\varphi}-C_{2_{d}}^{\underline{10}\mathrm{g}\underline{d}}}^{\overline{\varphi}+C_{2}^{\underline{1}\mathrm{o}\mathrm{g}_{d}\underline{d}}}\frac{ds}{s^{p}}\leq C_{3}\frac{1\mathrm{o}\mathrm{g}d}{d}$for all$d\geq d_{2}$, and the proof of Theorem

As acorollary of Theorem$\mathrm{B}$$\mathrm{B}$, we haveis complete.

$\square$130
Corollary 4.1. Let $f(u)=e^{u}$ or(1.1)-(1.3) with the nonlinear term
$T_{d}<\infty$. Consider the Cauchy-Neumann problem
replaced by $f(u)$ . Assume the condition (1.4). Then$(u+\lambda)^{p}$$u^{p}$,$\lambda\geq 0$. Furthermore there exist constants

$C$

and

$d_{0}$

such that$|T_{d}- \int_{\overline{\varphi}}^{\infty}\frac{ds}{f(s)}|\leq C\frac{1\mathrm{o}\mathrm{g}d}{d}$for all$d\geq d_{0}$.

Remark. We remark that the results of Theorem$D$replaced by bounded smooth domains in$R^{n}$$\mathrm{B}$and Corollary 4.1 holds with the domain.

5. Proof of Theorem A. In this section we prove Theorem A. For this aim, we firstat some time $t=T$ .
prove that the solution $ud(x, t)$ is monotone in the direction$x_{n}$

Proposition 5.1. Let $ud$ be a solution of (1.1)-(1.3) under the condition (1.4). Assume$I(\varphi)>0(<0)$. Then there exist positive constants(5.1)$\frac{T}{d})<0(>0)$$\frac{\partial}{\partial x_{n}}u_{d}(x,$Proof Letand$\{\psi_{1,j}\}_{j=0}^{\infty}$$\{\psi_{2,j}\}_{j=0}^{\infty}$

$T$

,

and

$d_{0}$

such that, for all$x\in D$,.

be complete orthonormal systems of Neumann eigen-functions for the domain $D’$ and the interval $(0, 1)$ , respectively. Let
corresponding to$d\geq d\circ$$\mu k,j$

be the eigenvalue

such that $0=\mu_{k,0}<\mu_{k,1}\leq\mu_{k,2}\leq\cdots\leq\mu_{k,j}\leq\cdots$ , $k=1,2$ . In this$\psi_{k,j}$notation we repeat the eigenvalues if needed to take account their multiplicity. Then, byis acomplete orthonormal system of Neumann[1], the family of functions$\{\psi_{1,:}\psi_{2,j}\}_{i,j=0}^{\infty}$

eigenfunctions for $D$ , and the eigenvalue of$\psi_{1,0}=\frac{1}{|D’|^{1/2}}$

Let$j_{0}\in \mathrm{N}$such that$\mu_{j}>(\pi/L)^{2}$

(5.2)

,

$\psi_{2,0}=\frac{1}{L^{1/2}}$,$\ldots$is$\mu_{1,:}+\mu_{2,j}$. Furthermore we have$\psi_{2,j}(x_{n})=\sqrt{\frac{2}{L}}\cos(\frac{j\pi}{L}x_{n})$$\mu_{j\mathrm{o}}=\mu_{2,0}=(\pi/L)^{2}$

for $j=j_{0}+1$ ,$\psi_{1,:}\psi_{2,j}$. Then$\mu_{j}\leq(\pi/L)^{2}$,$j=1,2$ ,for $j=0,1$ ,$\ldots$$\ldots$, $j_{0}-1$ and. Furthermore we have,$\frac{\partial^{k}}{\partial x_{n}^{k}}P_{j_{0}}u_{d}(x,t)=\frac{(u_{d}(\cdot,t),\psi_{1,0}\psi_{2,1})_{L^{2}(D)}}{|D|^{1/2}},\frac{\partial^{k}}{\partial x_{n}^{k}}\psi_{2,1}(x_{n})$$k$Put $\mu=((\pi/L)^{2}+\mu_{j_{0}+1})/2$ . By Proposition 2.3, there exists aconstant$=1,2$ .
$C_{1}$

such that thesolution $ud$ satisfies
(5.3)

.

$||u_{d}( \cdot, \tau)-P_{j\mathrm{o}}u_{d}(\cdot, \tau)||_{C^{2}(D)}|_{\tau=t/d}\leq C_{1}(e^{-\mu t}+\frac{1}{d^{1/2}})$,$2 \leq t\leq\frac{dS}{2}$.

131
On the other hand, the function $a(t)=(u_{d}(\cdot, t),$$\frac{d}{dt}a(t)=-d(\frac{\pi}{L})^{2}a(t)+\int_{D}(u_{d}(x, t))^{p}\psi_{1,0}\psi_{2,1}dx$

By (2.15), there exists aconstant(5.4)$C_{2}$

satisfies$\psi_{1,0}\psi_{2,1})_{L^{2}(D)}$$0<t<T_{d}$ .

,such that$|a( \frac{t}{d})-e^{-(\frac{\pi}{L})^{2}}{}^{t}a(0)|.=e^{-(\frac{\pi}{L})^{2}t}\int_{0}^{t/d}\int_{D}e^{d(_{T}^{\pi})^{2}s}(u_{d}(x, s))^{p}|\psi_{1,0}\psi_{2,1}|dxds$$\leq e^{-(\frac{\pi}{L})^{2}t}\int_{0}^{t/d}e^{d(\frac{\pi}{L})^{2}s}(\int_{D}|u_{d}(x, s)|^{2p}dx)^{1/2}ds\leq\frac{C_{2}L^{2}}{d\pi^{2}}$.

for all $0<t<dS/2$ . By (5.2)-(5.4) and $a(0)>0$ , we have
(5.5)$\frac{t}{d})\leq a(\frac{t}{d})\frac{1}{|D’|^{1/2}}\frac{\partial}{\partial x_{n}}\psi_{2,1}(x)+C_{1}(e^{-\mu t}+\frac{1}{d^{1/2}})$$\frac{\partial}{\partial x_{n}}u_{d}(x,$$\leq-\frac{\sqrt{2}\pi}{L^{3/2}|D|^{1/2}},(e^{-\pi^{2}}{}^{t}a(0)-\frac{C_{2}}{d\pi^{2}})\sin(\pi x_{n})+C_{1}(e^{-\mu t}+\frac{1}{d^{1/2}})$for all $x\in D$ and $2\leq t\leq dS/2$ . By (5.5), $a(0)>0$ , and $\mu>(\pi/L)^{2}$ , there exists aconstantsuch that, for all $d\geq d_{T,1}$ ,such that, for any $T\geq T_{1}$ , there exists aconstant$d_{T,1}$$T_{1}$(5.6)$\frac{\partial}{\partial x_{n}}u_{d}(x,$$\frac{T}{d})<0$,$x=(\mathrm{x}’, x_{n})\in D$with$\min\{x_{n}, 1-x_{n}\}\geq\frac{1}{8}$.

Furthermore, by 5. $2)-(5.4)$ ,$($$\frac{\partial^{2}}{\partial x_{n}^{2}}u_{d}$(,$x$$\frac{t}{d})\leq-\frac{\pi^{2}}{L^{2}}a(\frac{t}{d})\psi_{2,1}(x)+C_{1}(e^{-\mu t}+\frac{1}{d^{1/2}})$$\leq-\frac{\sqrt{2}\pi^{2}}{L^{5/2}|D|},(e^{-\pi^{2}}{}^{t}a(0)-\frac{C_{2}}{d\pi^{2}})\cos(\pi x_{n})+C_{1}(e^{-\mu t}+\frac{1}{d^{1/2}})$for all $x=(x’, x_{n})\in D$ with $0<x_{n}\leq 1/4$ and $T\leq t\leq dS/2$ . Similarly in (5.6), there
such that, forsuch that, for any $T\geq T_{2}$ , there exists aconstantexists aconstant$d_{T,2}$$T_{2}$

all $d\geq d_{T,2}$ ,(5.7)$\frac{\partial^{2}}{\partial x_{n}^{2}}u_{d}(x,$$\frac{T}{d})<0$

Similarly, there exists aconstant,$T_{3}$$x=(x’, x_{n})\in D$such that, for any

with$T\geq T_{3}$$0<x_{n} \leq\frac{1}{4}$.

, there exists aconstantsuch that, for all $d\geq d_{T,3}$ ,(5.4)$\frac{\partial^{2}}{\partial x_{n}^{2}}u_{d}(x,$$\frac{T}{d})>0$,$x=(x’,x_{n})\in D$with$\frac{3}{4}\leq x_{n}<1$,$d_{T,3}$

132
for all$0<\lambda\leq\lambda_{4}$. By (5.6)-(5.8), there exist constants$\frac{\partial}{\partial x_{n}}u_{d}$for all$d\geq d_{1}$(,$x$$\frac{T}{d})<0$

,

$T$

and

$d_{1}$

such that$x\in D$, and the proof of Proposition 5.1 is complete.

$\square$

We are ready to complete the proof of Theorem A. We prove Theorem Aby applying
the arguments of [5] and [8] together with Propositions 3.1 and 5.1.

Proof of Theorem . We first assume$A$exist constants

$T$

and

$d_{1}$

such that,$I(\varphi)>0$$v=\partial u_{d}/\partial x_{n}$(5.9)$d\geq d_{1}$, where $\Gamma_{1}=D’\cross\{0, L\}$ and$\frac{\partial}{\partial x_{n}}u_{d}(x, t)=v(x, t)<0$

satisfies’$\{\begin{array}{l}v_{t}=d\Delta v+pu_{d}^{p-1}vv(x,t)=0\frac{\partial}{\partial\nu}v(x,t)=0v(x,T/d)\leq 0\end{array}$for all, and prove (1.6). By Proposition 5.1, there$\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{n}DD,\cross(T/d, T_{d})\Gamma_{1}\cross(T/d,T_{d}’)\Gamma_{2}\cross(T/d,T_{d}),$$\Gamma_{2}=\partial D’\cross(0, L)$in$D\cross(0,T)$

Assume that $a=(a’, a_{n})\in B_{d}(\varphi)\cap(\overline{D’}\cross(0,1))$ . Let$T_{*}$. By the maximum principle,

and$\Gamma_{2}\cross(0, T)$.

be aconstant to be chosen latersuch that $T/d\leq T_{*}<T_{d}$ . Put $Q\equiv D’\cross(b, c)\cross(T_{*}, T_{d})$ , where , $c\in(0, L)$ such that$b$$b<a_{n}<c$ and $c-b\geq L/2$

. Put

$J(x’, x_{n}, t)= \frac{\partial}{\partial x_{n}}u_{d}(x, t)+\epsilon\zeta(x_{n})(u_{d}(x, t))^{q}$

where $1<q<p$ and$\epsilon>0$,$\zeta(s)=\sin(\frac{\pi(s-b)}{c-b})$,

is apositive constant to be chosen later. Then we have

(5.10) $J_{t}-d\Delta J-r(x, t)J=-\epsilon\zeta K(x, t)-\epsilon q(q-1)u_{d}^{q-2}|\nabla u_{d}|^{2}\leq-\epsilon\zeta K(x, t)$

in

$Q$

where
(5.11) $r(x, t)=-2dq\epsilon\zeta’u_{d}^{q-1}+pu_{d}^{p-1}$ ,$K(x, t)=(p-q)u_{d}^{p+q-1}+d\zeta^{-1}\zeta’u_{d}^{q}-2dq\epsilon\zeta’u^{2q-1}$

On the other hand,$\zeta^{-1}\zeta’=-(\frac{\pi}{c-b})^{2}\geq-(\frac{2\pi}{L})^{2}$.

133
By Propositions 2.1 and 3.1, there exist constants $T_{1}\in(T/d, T_{d})$ and(5.12)

for all,$p \frac{-q}{2}(u_{d}(x, t))^{p+q-1}\geq d(\frac{2\pi}{L})^{2}(u_{d}(x, t))^{q}$$d\geq d_{2}$$d_{2}\geq d_{1}$such that$(x,t)\in D\cross(T_{1}, T_{d})$. Furthermore we take asufficiently small so that

(5.13)$\epsilon$$(x, t)\in D\cross(T_{1}, T_{d})$$\frac{p-q}{2}(u_{d}(x, t))^{p+q-1}\geq 2dq\epsilon|\zeta’|u^{2q-1}$

Taking $T_{*}=T_{1}$ and$\{$$d\geq d_{2}$.

, by (5.10)-(5.13), we have$J_{t}\leq d\Delta J+r(x, t)J$in

$J(x, t)<0$

onon$\frac{\partial}{\partial\nu}J(x, t)=0$

By (5.9), taking asufficiently small$\epsilon$$Q$,$D’\cross\{b, c\}\cross(T_{*}, T_{d})$,$\partial D’\cross(b, c)\cross(T_{*}, T_{d})$.

if necessary, we have $J(x, T_{*})<0$ , $x\in D’\cross(b, c)$ .

By the maximum principle, we have
(5.14)$J(x, t)\leq 0$

for$(x, t)\in\overline{D’}\cross(b, c)\cross(T_{*}, T_{d})$By $a=(a’, a_{n})\in B(\varphi)$ and $a_{n}\in(b, c)$ , there exist asequence
positive constant$\delta$.

$\{(a_{k}’, akn, tk)\}_{k=1}^{\infty}$

and asuch that$\lim_{karrow\infty}(a_{k}’, a_{kn’ k}t)=(a’, a_{n}, T_{d})$,$\{(a_{k}’, a_{kn}+\delta)\}_{k=1}^{\infty}\subset\overline{D’}\cross(b, c)$$\lim_{karrow\infty}u(a_{k}’, a_{kn}, t_{k})=\infty$,.

By (5.9),$\lim_{karrow\infty}u_{d}(a_{k}’, a_{kn}+\delta, t_{k})=\infty$,

and by (5.14),$\int_{u_{d}(a_{k},a_{kn},t_{k})}^{u_{d}(a_{k}’,a_{k_{n}}+\delta,t_{k})},\frac{ds}{s^{q}}\leq-\epsilon\int_{a_{kn}}^{a_{kn}+\delta}\zeta(s)ds$.

By $q>1$ , we take the limit as $karrow\infty$ to have.

$0 \leq-\epsilon\int_{a_{n}}^{a_{n}+\delta}\zeta(s)ds<0$This contradiction shows$d\geq d_{2}$. Furthermore, if

we have$a\not\in B(\varphi)$. Therefore we have$a\in(\overline{D’}\cross\{L\})\cap B(\varphi)$$(\overline{D’}\cross\{L\})\cap B(\varphi)=\emptyset$

for all, by (5.5),$d\geq d_{2}$$(\overline{D’}\cross(0,1))\cap B(\varphi)=\emptyset$$(\overline{D’}\cross(0,1))\cap B(\varphi)\neq\emptyset$

for all. Therefore, and the proof of (1.6) is complete. By thesimilar argument as in the proof of (1.6), we have (1.7), and the proof of Theorem Ais
complete.

$\square$

By Theorem$\mathrm{A}$, we have the following results

134
Corollary 5.2. Let $n\geq 1$ . Consider the Cauchy-Neumann problem (1.1)-(1.3), where$D= \prod_{i=1}^{n}(0, L_{i})$Let$\varphi$,

be a nonnegative continuous function on$\int_{D}\varphi\cos(\frac{\pi}{L_{\dot{l}}}x_{i})dx>0$

Chen there exists a positive constant$i=0,1$ ,$L_{i}>0$$d_{0}$$\overline{D}$$\ldots$

,

$n$

.

such that,$i=1,2$ ,$\ldots$, .
$n$such that, for any $d\geq d_{0}$ ,$B_{d}(\varphi)$

consists of$a$single point such that
$B_{d}(\varphi)=\{(0, \ldots, 0)\}\subset\partial D$.

Remark. Applying the results of [5] together with Proposition 5.1, we may prove Corollary

5.2 for the case n $=1$ without Proposition 3.1.
Corollary 5.3. Theorems
placed by$e^{u}$

and$(u+\lambda)^{p}$

$A$

(A

, 5.1 and Corollary 5.2 hold with the nonlinear term$\geq 0$$u^{p}$re-), respectively.

REFERENCES[1]

R. Banuelos and K. Burdzy, On the “Hot Spot Conjecture” of J. Rauch, Jour. Func. Anal. 164 (1999),
1-33.
[2] L. A. Caffarelli and A. Friedman, Blow-up of solutions of nonlinear heat equations, J. Math. Anal.
Appl. 129 (1988), 409-419.
[3] Y. G. Chen, Blow-up solutions of a semilinear parabolic equations with the Neumann and Robinboundary conditions, J. Fac. Sci. Univ. Tokyo 37 (1990), 537-574.
[4] Y. G. Chen and T. Suzuki, Single-point blow-up for semilinear heat equations in a non-radial domain,
Proc. Japan Acad. Ser. AMath. Sci. 64 (1988), 57-60.
[5] X. Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite point blow-up in onedimensional semilinear heat equations, Jour. Diff. Eqns 78 (1989), 160190.
[6] C. Fermanian Kammerer, F. Merle, and H. Zaag, Stability of the blow-up profile of non-linear heat
equations from the dynamical system point of view, Math. Ann. 317 (2000), 347-387.
[7] A. Friedman and A. A. Lacey, The blow-up time for solutions of nonlinear heat equations with small, SIAM J. Math. Anal. 18 (1987), 711-721.
[8] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ.
Math. Jour. 34 (1985), 425-447.
[9] Y. Giga and R. V. Kohn, Nondegeneracy of blow-up for semilinear heat equations, Comm. Pure Appl.
Math. XLII (1989), 845-884.
[10] B. Kawohl, Rearrangements and Conveity of Level Sets in PDE,, Springer Lecture Notes in Math.
1150, 1985
$d\dot{l}ffilS:on$

135[11] F. Merle, Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure
Appl. Math. 45 (1992), 263-300.
[12] F. Merle, Compactness and single-point blowup of positive solutions on bounded domains, Proc. Roy.

Soc. Edinburgh Sect. A127 (1997), 47-65.
$+|u|^{p-1}u$ , Duke Math. J.
[13] F. Merle, Stability of the blow-up profile for equations of the type86 (1997), 143-195.
[14] N. Mizoguchi, Location of blowup points of solutions for a semilinear parabolic equation, preprint.
[15] N. Mizoguchi and E. Yanagida, Blowup and life span of solutions for a semilinear parabolic equation,$U_{t}=\mathrm{A}u$

SIAM J. Math. Anal. 29 (1998), 1434-1446.
[16] N. Mizoguchi and E. Yanagida, Life span of solutions with large initial data in a semilinear parabolic
equation (to appear in Indiana Univ. Math. J.).
[17] N. Mizoguchi and E. Yanagida, Life span of solutions for a semilinear parabolic problem with small
diffusion, preprint.
[18] J. Rauch, Five problems: An introduction to the qualitative theory of partial differential equations,in Partial Differential Equations and Related Topics, Springer Lecture Notes in Math. 446, 1975.
[19] J. J. L. Velazquez, Estimates on the (n $-1)$ -dirnensional Hausdorff measure of the blow-up set for
semilinear heat equation, Indiana Univ. Math. Jour. 42 (1993), 445-476.
[20] F. B. Weissler, Single point blow-up for a general semilinear heat equation, Indiana Univ. Math. Jour.
34 (1983), 881-913$a$