数理解析研究所講究録
第 1779 巻 2012 年 98-110

98

Collapsing behaviour of the logarithmic
diffusion equation
Kin Ming Hui
Institute of Mathematics, Academia Smica,
Nankang, Taipei, 11529, Taiwan, R. O. C.

June 11, 2011
Abstract
I will report my result on the collapsing behaviour of the maximal solution$u(x,0)=u_{0}(x)$ in, near its extinctionof the equation $u_{t}=\Delta\log u$ in$T=\int_{R^{2}}u_{0}dx/4\pi$without using the Hamilton-Yau Hamack inequality.
time
This result extends the recent result of P. Daskalopoulos, M.A. del Pino and
N. Sesum [DP2], [DS2].
$\mathbb{R}^{2}\cross(0,T),$$\mathbb{R}^{2}$

In this report I will discuss my recent result [Hu5] on the collapsing behaviourof the maximal solution of the logarithmic diffusion equation:(0.1)$\{\begin{array}{ll}u_{t}=\Delta\log u,u>0, in \mathbb{R}^{2}\cross(0,T)u(x,0)=u_{0} in \mathbb{R}^{2}.\end{array}$(0.1) arises in many physical and geometric models. For example P.L. Lions andToscani [LT] have shown that (0.1) arises as the diffusive lmit for finite velocity
Boltzmam kinetic models and T.G. Kurtz [Ku] has proved that (0.1) is the lin$\dot{u}t-$

ing density distribution of two gases moving against each other and obeying the.
Boltzmann equation. In [G] PG. de Gennes showed that (0.1) also appears in themodel of viscous liquid film lying on a solid surface and subjectin to long range
Van der Waals interactions with fourth order term neglected.
Recently K.M. Hui [Hu3], [Hu4] (for the case $m>0$ and $m<0$ ), and J.R. Esteban,
A. Rodriguez, J.L. Vazquez [ERV] (for the case $m>0$ ) have shown that the solutionof the porous medium equation$g$$u_{t}= \Delta(\frac{u^{m}}{m})$99converges to the maximal solution of (0.1) as $marrow 0$ . In [Wl], [W2], [H], Angenent,

L. Wu and R. Hamilton showed that the equation also arises in the study of theconformally equivalent metricon under the Ricci flow
$g_{ij}=u\delta_{ij}$$\mathbb{R}^{2}$(0.2)$\frac{\partial}{\partial t}g_{ij}=-2R_{ij}$where is the Ricci curvature of the metricis given by$R_{if}$$g_{ij}$. Note that in$\mathbb{R}^{2}$the scalar curvature

$R$

(0.3)

$R=- \frac{\Delta\log u}{u}$and the Ricci curvature is given by$R_{ij}= \frac{1}{2}Rg_{ij}$

.

(0.4)

By (0.3) and (0.4) the Ricci flow equation (0.2) is equivalent to the logarithmic
diffusion equation:$u_{t}=\Delta\log u$

1.

Existence and properties of solutions

The equation (0.1) has many properties different from the heat equation such as
existence of infinite many solutions for any given initial data. There also does not$L^{1}$

exist any fundamental solution for (0.1) [Hul] which suggests that conservation ofmass does not hold. K.M. Hui [Hu2] by using approximation by Neumann solutions in bounded domains and P Daskalaopoulos and M.A.del Pino [DPl] by using
approximation by Dirichlet solutions in bounded domains proved independently
that corresponding to each$0\leq u_{0}\in L^{p}(\mathbb{R}^{2})\cap L^{1}(\mathbb{R}^{2}),p>1,2\leq f\in L^{1}(0,\infty)$

there exists a classical solution
equation,$u$of (0.1) in$\mathbb{R}^{n}\cross(0, T)$$\int_{\mathbb{R}^{2}}u(x,t)dx=\int_{\mathbb{R}^{2}}u_{0}dx-2\pi\int_{0}^{t}f(s)ds$,

satisfying the mass loss$\forall 0\leq t<T$

where $T=T(u_{0},f)>0$ given by$\int_{\mathbb{R}^{2}}u_{0}dx=2\pi\int_{0}^{T}f(s)ds$

is the extinction time for the solution . Hence the solution with flux $f$ vanishes
identically to zero at time .
$u$

$T$

100
Note that the maximal solution of (0.1) is the solution of (0.1) that corresponds
to flux $f=2$ which satisfies
VO $\leq t<T$$\int_{\mathbb{R}^{2}}u(x,t)dx=\int_{R^{2}}u_{0}dx-4\pi t$

with

.

$T= \frac{1}{4\pi}\int_{\mathbb{R}^{2}}u_{0}dx$

For any $2<f\in C(0,T)$ the solution
condition at finity:$u$with flux

satisfies the following decay$f$uniformly on [a,b]$|x| arrow\infty 1\dot{m}\frac{\log u}{\log|x|}=-f$$\forall 0<a<b<T$or equivalently$\lim^{\underline{ru_{r}}}=-f$

uniforrnly on [a,b]$\forall 0<a<b<T$.

$|x|arrow\infty u$

One would like to ask what is the asymptotic behaviour of the solution withconstant flux $f=\gamma\geq 2$ ? When $\gamma>2$, S.Y. Hsu [Hsl], [Hs2], by using the lap
number method of Matano [M], V.A. Galaktionov and L.A. Peletier [GP], provedthat if the imitial value is radially symmetric and monotone decreasing and is
the solution with flux $\gamma>2$, then there exist unique constants $\alpha>0,$ $\beta>-1/2$,$\alpha=2\beta+1$ , depending only onsuch that the rescaled function$u$$\gamma$$v(y,s)= \frac{u(y/(T-t)^{\beta},t)}{(T-t)^{a}}$,$s=-\log(T-t)$ ,for some constantconverges uniformly on every compact subset of tois radially symmetric and satisfies theaswhere$\mathbb{R}^{2}$$\phi_{\lambda,\beta}(y)$$\lambda>0$$sarrow\infty$$\phi_{\lambda,\beta}(y)=\phi_{\lambda,\beta}(|y|)$following O.D.E.:$\frac{1}{r}(\frac{r\phi’}{\phi})’+\alpha\phi+\beta r\phi’=0$

in

with
$\phi(0)=1/\lambda,\phi’(0)=0$$(0,\infty)$.

In particular for $\gamma=4$, the rescaled solution$v(x,s)= \frac{u(x,t)}{T-t}$,$s=-\log(T-t)$ ,converges unifomdy on every compact subsets of$\frac{8\lambda}{(\lambda+|x|^{2})^{2}}$$\mathbb{R}^{2}$to the function

101
as

$sarrow\infty$$f=4$ ,

for some constant $\lambda>0$ . What this said is that for solution with flux
$u(x,t) \approx\frac{8\lambda(T-t)}{(\lambda+|x|^{2})^{2}}$as$t\nearrow T$

which corresponds to contractin spheres Ricci flow solution on .
What about the asymptotic behaviour of the solution with flux $f=2$ ? J.R. King[K] by using inner and outer asymptotic expansion and matching asymptotic
method showed that if is the solution of the logarithmic diffusion equation (0.1)
with flux $f=2$ then as approaches the extinction time the vanishing behaviour
for solution is $veIy$ different from the vanishin behaviour for the case $f\equiv\gamma>2$ .
J.R. King found that for compactly supportly finite mass initial value the maximal
solution behaves like$S^{2}$

$g$

$u$

$T$

$t$

$g$

$\frac{(T-t)^{2}}{\frac{T}{2}|x|^{2}+e^{\frac{2T}{(T-t)}}}$

in the inner region $(T-t)\log|x|\leq T$ and behaves like$\frac{2t}{|x|^{2}(\log|x|)^{2}}$in the outer region $(T-t)\log|x|\geq T$ as. In [DP2] P. Daskalopoulos and
M.A. del Pino gave a rigorous proof of an extension of this formal result for radiallysymmetric initial value
satisfying the conditions,$t\nearrow T$$u_{0}(r)\geq 0$$u_{0}(x)=u_{0}(|x|)$is decreasing on $r=|x|\geq r_{1}$ for some constant $r_{1}>0$ ,$u_{0}(x)= \frac{2\mu}{|x|^{2}(\log|x|)^{2}}(1+o(1))$

as$|x|arrow\infty$,

for some constant $\mu>0$ and$R_{0}(x);=- \frac{\Delta\log u_{0}}{u_{0}}\geq-\frac{1}{\mu}$on$\mathbb{R}^{2}$.

Later P. Daskalopoulos and N. Sesum [DS2] extended this result to the case of com. However their proof of the behaviour
pactly supportedof the maximal solution in the outer region near the extinction time is very difficult
and uses the Hamilton-Yau Hamack inequality. Recently in [Hu5] I extended theirresult to the case of initial value$0\leq u_{0}\in L^{1}(\mathbb{R}^{2})\cap L^{\infty}(\mathbb{R}^{2})$$0\leq u_{0}\in L^{1}(\mathbb{R}^{2})\cap L^{\infty}(\mathbb{R}^{2})$that satisfies$u_{0}(x)=u_{0}(|x|)$is decreasing on $r=|x|\geq r_{1}$ for some constant $r_{1}>0$ ,

102
as

$u_{0}(x)= \frac{2\mu}{|x|^{2}(\log|x|)^{2}}(1+o(1))$$|x|arrow\infty$

,

and

on

$R_{0}(x):=- \frac{\Delta\log u_{0}}{u_{0}}\geq-\frac{1}{\mu}$

for some constant$\mu\geq 0$$\mathbb{R}^{2}$

with the right hand side being replaced $by-\infty$ if $\mu=0$ and$ess\inf_{\overline{B}_{r_{1}}(0)}u_{0}\geq ess\sup_{IR^{2}\backslash B_{r_{2}}(0)}u_{0}$for some constant$r_{2}>r_{1}$

.

(1.1)

Note that (1.1) is automatically satisfied if has compact support in . In [Hu5] I
proved the behaviour of the maximal solution in the outer region near the extinction
time by elementary method without using the difficult Hamilton-Yau Hamack
inequality for surfaces. I also obtained the behaviour of the maximal solution in
the inner region as the extinction time is approached.
I will now assume that satisfies the above stmctural conditions and is theandwith flux
maximal solution of (0.1) in$\mathbb{R}^{2}$$u_{0}$$u$$u_{0}$$f\equiv 2$$\mathbb{R}^{2}\cross(0,T)$.

$T= \frac{1}{4\pi}\int_{\mathbb{R}^{2}}u_{0}dx$

I will sketch of proof of [Hu5] here.

2

Inner region behaviour

By using the reflection method of D.G. Aronson and L.A. Caffarelli [AC] one has
the following lemma:
Lemma 2.1. (Lemma 1.1 of $fHu5]$) The solution$u$satisfies$u(x,t)\geq u(y,t)$

for any $t\in(0,T)$ and$x,y\in R^{2}$such that $|y|\geq|x|+2r_{2}$ .

Then for any $0<t<T$ there exists$x_{t}\in\overline{B}_{2r_{2}}$such that$u(x_{t},t)=ma^{Xu(X,t)}$.

is attained on the compact set
That is the maximum of
perform a rescaling of the solution of (0.1). Let$u(\cdot,t)$$\overline{u}(x,\tau)=\tau^{2}u(x,t)$,.

$\tau=\frac{1}{T-t},\tau>1/T$$\overline{B}_{2r_{2}}$. We will now

103Then 7 satisfies
in$\overline{u}_{\tau}=\Delta\log\overline{u}+\frac{2\overline{u}}{\tau}$$\mathbb{R}^{2}\cross(1/T,\infty)$.

Let$R_{\max}(t)= \max_{x\in \mathbb{R}^{2}}R(x,t)$and let $W(t)$ be the width function with respect to the metricas defined by
P. Daskalopoulos and R. Hamilton [DH]. We now recall a result of P. Daskalopoulos
and R. Hamilton [DH]:$g_{ij}(t)$Theorem 2.2. $(lDH])$ There ex\’ist positive constants $c>0$ and $C>0$ such that(i) $c(T-t)\leq W(t)\leq C(T-t)$(ii)$\frac{c}{(T-t)^{2}}\leq R_{\max}(t)\leq\frac{c}{(T-t)^{2}}$hold for any $0<t<T$.

By Theorem 2.2,
$c\leq$lim $sup(T-t)^{2}R_{mx}(t)\leq$ C.
$t\nearrow T$

Hence the singularity is a type II singularity. Note that
B\’enilan inequality,in$\frac{u_{t}}{u}\leq\frac{1}{t}$$\mathbb{R}^{2}\cross(0,T)$$u$satisfies the Aronson-.

As$R=- \frac{\Delta\log u}{u}=-\frac{u_{t}}{u}$,the Aronson-B\’enilan inequality is equivalent to$R \geq-\frac{1}{t}$.

So if we let$\overline{R}(x,\tau)=-\frac{\Delta\log\overline{u}}{\overline{u}}$

.

Then
in

$\frac{2}{\tau}+\frac{2}{\tau^{2}T}\geq\frac{\overline{u}_{\tau}}{u}\geq-C$$\mathbb{R}^{2}\cross(2/T,\infty)$

Theorem 2.3. (Theorem 1.3 of IHu51) For any sequence$\overline{u}_{k}(y,\tau)=\alpha_{k}\overline{u}(\alpha^{\frac{1}{k2}}y+x_{t_{k’}}\tau+\tau_{k})$,$\tau_{k}arrow\infty$$y\in \mathbb{R}^{2},\tau>-\tau_{k}+T^{-1}$where$t_{k}=T-\tau_{k}^{-1}$$\{\tau_{k}\}_{k=1}^{\infty},$.

$\forall k\in Z^{+}$

(2.1)

as

$karrow\infty$

, let

104
and

.

$\alpha_{k}=1/\overline{u}(x_{t_{k’}}\tau_{k})$on$C^{\infty}(K)$$setKThen\{\overline{u}_{k}\}_{k=1}^{\infty}hasasubsequmce\{\overline{u}_{k}\}_{i=1}^{\infty}thatconvergesuniformly\subset \mathbb{R}^{2}\cross(-\infty,\infty)asiarrow\infty t^{i}oapositivesolution$$U(y, \tau)=\frac{1}{\lambda|y|^{2}+e^{4\lambda\tau}}$of equationin$U_{\tau}=\Delta\log U$

for any compact.

$\mathbb{R}^{2}\cross(-\infty,\infty)$

with un fomly bounded non-negative scalar curvature and uniformly bounded width on
is some constant.
wherewith respect to the metric$\mathbb{R}^{2}\cross(-\infty,\infty)$$\lambda>0$$\overline{g_{ij}}(t)=U(\cdot,t)\delta_{ij}$Proof (Sketch) By definition,$\overline{u}_{k}(0,0)=1$

and$\overline{u}_{k}(y,0)\leq 1$

Let $a<b$ . By (2.1) there exist constants $M_{1}>0$ and$\overline{u}_{k}(x,\tau)\leq M_{1}$and$\forall y\in \mathbb{R}^{2}$$k_{0}\in Z^{+}$.

such that$x\in \mathbb{R}^{2},a\leq\tau\leq b,k\geq k_{0}$$|\overline{u}_{k,\tau}(x,\tau)|\leq CM_{1}$

.

(2.2)

By (2.2) one can deduce the following Hamack inequality:

For any $R_{1}>0,a<b$ , there exists a constant $C_{2}>0$ such that$\forall k\geq k_{0}$$a \leq\tau_{1}\leq b\sup_{|y|\leq R_{1}}\overline{u}_{k}(y,\tau_{1})^{9}\leq C_{2}iM\overline{u}_{k}(x,\tau_{2})a\leq\tau_{2}\underline{<}b|x|\leq R_{1}$Hence the sequence$\overline{u}_{k}$is uniformly parabolic on$\overline{B}_{R_{1}}\cross[a,b]$.

and are uniformly$hasasubsequencewhichwemayassumewithout1ossofgeneralitytobeffieHoldercontinuousinC^{2\gamma,1\gamma}(\overline{B}_{R_{1}}\cross[a,b])forany\gamma\in Z^{+}.Thenthesequence\ulcorner u_{k}\}_{k--1}^{\infty}$

for any compact set
assequence itself that converges uniformly into some positive function that satisfies the logarithmuic diffusion$C^{\infty}(K)$$karrow\infty$$U$$K\subset \mathbb{R}^{2}\cross(-\infty,\infty)$

equation. Let$\overline{R}_{k}=-\frac{\Delta\log\overline{u}_{k}}{\overline{u}_{k}}$.

Then converges uniformly on every compact subset ofto the scalar curvature$\overline{R}_{k}$$\mathbb{R}^{2}\cross$$\overline{R}=-\frac{\Delta\log U}{U}$of the metric$\overline{g_{ij}}(\tau)=U(\cdot,\tau)\delta_{ij}$. Moreover$0\leq\overline{R}(y,\tau)\leq C$$\forall(y,\tau)\in \mathbb{R}^{2}\cross(-\infty,\infty)$

.

$(-$

oo,

$\infty)$as$karrow\infty$

105
By an approximation argument the width function with respect to the metricis uniformly bounded on. Then by the result of
P Daskalopoulos and N. Sesum [DSl],$\overline{g_{ij}}(\tau)=U(\cdot,\tau)\delta_{ij}$$\mathbb{R}^{2}\cross(-\infty,\infty)$$U(y, \tau)=\frac{2}{\beta(|y-y_{0}|^{2}+\delta e^{2\beta\tau})}$for some$y_{0}\in \mathbb{R}^{2}$and constants $\beta>0,$$\delta>0$.

$Sce\overline{u}_{k}(y,0)$

attains its maximum at$y=0,$ $U(y,0)$ will attain its maximum at $y=0$ . Hence $y_{0}=0$$U(0,0)=1$$\Rightarrow$$1= \frac{2}{\beta\delta}$

.

.

Thus

.

$U(y, \tau)=\frac{1}{\lambda|y|^{2}+e^{4\lambda\tau}}$for some constant$\lambda>0$.

$\square$It can be proved thatas$\alpha_{k}arrow\infty$$karrow\infty$Hence we can change the origin in rescaling and have the following result:Lemma 2.4. (Lemma 1. 10 of $fHu5]$) Let$q_{k}(y,\tau)=\alpha_{k}\overline{u}(\alpha^{\frac{1}{k2}}y,\tau+\tau_{k})$Then$q_{k_{t}}(y,\tau)$$U(y,\tau)$asconverges uniformly in.

$C^{\infty}(K)$.

for every compact set$\tauarrow\infty$

We will now perform a change of scalin . Let$g$$\beta(\tau)=1/\overline{u}(0,\tau)$$\beta_{k}=\beta(\tau_{k})$

,

, and
$\overline{q}_{k}(y,\tau)=\beta_{k}\overline{u}(\beta^{\frac{1}{k2}}y,\tau+\tau_{k})$.

Then$\frac{\alpha_{k}}{\beta_{k}}=q_{k}(0,0)arrow U(0,0)=1$

Hence there exists$k_{0}\in Z^{+}$

as$karrow\infty$.

and constants $c_{2}>c_{1}>0$ such that$c_{1} \leq\frac{\beta_{k}}{\alpha_{k}}\leq c_{2}$Thus we have the following result.

$\forall k\geq k_{0}$.

$K\subset \mathbb{R}^{2}$to thefunction

106
Theorem 2.5. (Theorem 1. 11 of $fHu5]$)on
for any compact set

ashas a subsequence
to $U(y,\tau)$ as$\overline{q}_{k_{l}}$$K\subset \mathbb{R}^{2}\cross(-\infty,\infty)$$C^{\infty}(K)$$karrow\infty$$\overline{q}_{k}$.

that converges uniformly$\tauarrow\infty$. Moreover$\beta_{k}arrow\infty$

Lemma 2.6. (Proposition 1.17 and Proposition 1.18 of $fHu5]$)$\tauarrow\infty]jm\frac{\beta’(\tau)}{\beta(\tau)}=\tauarrow\infty]jm\frac{\log\beta(\tau)}{\tau}=2(T+\mu)$.

Let$\tilde{R}_{k}=-\frac{\Delta\log\overline{q}_{k}}{\overline{q}_{k}}$

.

Since
$\overline{R}_{k}(0,0)=\frac{\beta’(\tau_{k})}{\beta(\tau_{k})}+\frac{2}{\tau_{k}}$,then$4 \lambda=\lim_{\tauarrow\infty}\overline{R}_{k}(0,0)=2(T+\mu)$Let$\overline{q}(y,\tau)=\beta(\tau)\overline{u}(\beta(\tau)^{\frac{1}{2}}y,\tau)$.

.

We then have the following main theorem for inner region.
Theorem 2.7. (Theorem 1.21 of $fHu5l$)
to thefunction
compact set$\overline{q}(y,\tau)$converges uniformly on$C^{\infty}(K)$for any$K\subset R^{2}$$U_{\mu}(y)= \frac{1}{\frac{(T+\mu)}{2}|y|^{2}+1}$

as$Tarrow\infty$.

Corollary 2.8. (cf. IHu51) For any
that$\epsilon>0$

and $M>0$ there exist $\tau_{0}>1/T$ and $C>0$ such$\{\begin{array}{ll}|u(x,t)-\frac{(T-t)^{2}}{\lambda|x|^{2}+\beta(\tau)}|<u(0,t)\epsilon \forall|x|\leq\beta(\tau)^{\frac{1}{2}}M, \tau>\tau_{0}u(O,t)\leq C(T-t)^{2} \forall t>T-\tau_{0}^{-1}.\end{array}$where $\tau=1/(T-t)$ .
As in [DP2], [DS2], [Hu5], we now consider the cylindrical change of variables,$v(\zeta,\theta,t)=r^{2}u(r,\theta,t)$,$\zeta=\log r,r=|x|$and let,$\overline{v}(\xi, \theta,\tau)=\tau^{2}v(\tau\xi,\Theta,t)$$\tau=1/(T-t),\tau\geq 1/T$.

Then satisfies$\overline{v}$$\tau\overline{v}_{\tau}=\frac{1}{\tau}(\log^{\eta}\gamma_{\xi\xi}+\tau(\log Tv_{\theta\theta}+\xi\overline{v}_{\xi}+2\overline{v}$in$\mathbb{R}\cross[0,2\pi]\cross(1/T,\infty)$.

107
Corollary 2.9. (Lemma 1.23 of $fHu5]$) For any$| \overline{v}(\xi, \theta,\tau)-\frac{e^{2\tau\xi}}{\frac{T+\mu}{2}e^{2\tau\xi}+\beta(\tau)}|<\frac{e^{2\tau\xi}}{\beta(\tau)}\epsilon$$\epsilon>0$

there exists $\tau_{0}>1/T$ such that$\forall\xi\leq\frac{\log\beta(\tau)}{2\tau},$$\Theta\in[0,2\pi],\tau\geq\tau_{0}$.

Corollary 2.10. (Corollary 1.24 of lHu51)

as$[_{-\infty}^{-}f_{0}^{2\pi}\overline{v}(\xi, \Theta,\tau)d\Theta d\xiarrow 0$$\tauarrow\infty$

and$\lim_{\tauarrow\infty}\overline{v}(\xi_{\text{ノ}}\theta,\tau)=0$

for any$\xi^{-}<T+\mu$

uniformly on$(-\infty,\xi^{-}]\cross[0,2\pi]$.

3 Outer region behaviourLet$\xi(\tau)=(\log\beta(\tau))/2\tau$.

Lemma 3.1. (Lemma 2.1 of $fHu5]$) There exists constants $C_{1}>0,$ $C_{2}>0,$ $C_{3}>0$ and$\tau_{0}>1/T$ such that the following holds.
(i)

(ii)
(iii)

$\theta\in[0,2\pi],\tau\geq 1/T$$\overline{v}(\xi, \theta,\tau)\leq C_{1}$$\forall\xi\in \mathbb{R},$$\overline{v}(\xi, \Theta, \tau)\geq\frac{c_{2}}{\xi^{2}}$$\forall\xi\geq\xi(\tau),$$\overline{v}(\xi,\Theta,\tau)\leq\frac{C_{3}}{\xi^{2}}$$\forall\xi>0,$ $\Theta\in[0,2\pi],\tau\geq\tau_{0}$$\theta\in[0,2\pi],$ $\tau\geq\tau_{0}$.

Moreover$\xi(\tau)=T+\mu+o(1)$as$\tauarrow\infty$.

We now let$w(\xi, \theta,s)=\overline{v}(\xi, \theta, \tau)$with$s=\log\tau=-\log(T-t)$.

Then$w_{s}=e^{-s}(\log w)_{\xi\xi}+e^{s}(\log w)_{\theta\theta}+\xi w_{\xi}+2w$

in lR $\cross[0,2\pi]\cross(-\log T,\infty)$ .

The following is the main theorem for outer region.

108
Theorem 3.2. (Theorem 2.3 of IHu51) As$\tauarrow\infty$,

thefunction converges to thefunction$\overline{v}$$V(\xi)=\{\begin{array}{l}\frac{2(T+\mu)}{\xi^{2}} \forall\xi>T+\mu 0 \forall\xi<T+\mu.\end{array}$Moreover the convergence is unifom on $(-\infty,a]$ for any $a<T+\mu$ and on$\xi_{0}’>\xi_{0}>T+\mu$$[\xi_{0},\xi_{0}’]$.

Proof.(Sketch) Let$\{s_{k}\}_{k=1}^{\infty}$

be a sequence such that$s_{k}arrow\infty$as$karrow\infty$

for any

and.

$\forall\xi\in \mathbb{R},0\leq\Theta\leq 2\pi,s\geq-\log T-s_{k}$$w_{k}(\xi, \theta,s)=w(\xi, \Theta,s+s_{k})$Let$\forall b\geq\eta>T+\mu,s>-\log T-s_{k},k\in Z^{+}$,$W_{k}^{b}( \eta,s)=\int_{\eta}\int_{0}^{2n}w_{k}(\xi,\theta,s)d\Theta d\xi$$\forall\eta>T+\mu,s>-\log T-s_{k},k\in Z^{+}$$W_{k}( \eta,s)=\int_{\eta}^{\infty}\int_{0}^{2\pi}w_{k}(\xi, \theta,s)d\theta d\xi$

and let$i\in Z^{+}$
and
Since

be a monotonically increasing sequence such that $b_{i}>T+\mu$ for any$\{b_{i}\}_{i=1}^{\infty}$$b_{i}arrow\infty$as$iarrow\infty$.

$\int_{\mathbb{R}^{2}}u(x,t)dx=4\pi(T-t)$$J_{-\infty}^{\infty} \int_{0}^{2\pi}w_{k}(\xi, \theta,s)d\Theta d\xi=4\pi$,$\forall 0<t<T$$\forall s>-\log T-s_{k_{J}}k\in Z^{+}$

One can prove that there exists a fUnction di and a subsequence of
may assume without loss of generality to be itself such that.
$Z^{+}$

which we$Z^{+}$

umiformly on $[a,b]\cross[c,d]$ $b>a>T+\mu,d>c$

asunifomly on $[a,b]\cross[c,d]$ $b>a>T+\mu,d>c$as$W_{k}^{b_{i}}arrow W^{b_{t}}$for any$i\in Z^{+}$$karrow\infty$

and$W_{k}arrow W$$karrow\infty$

where$W^{b}( \eta,s)=\int_{\eta}\int_{0}^{2\pi}\overline{w}(\xi, \Theta,s)d\theta d\xi$,.

$l V(\eta,s)=J_{\eta}^{\infty}\int_{0}^{2\pi}\overline{w}(\xi, \Theta,s)d\theta d\xi$

By elementary argument one can show that$\eta\eta,=\overline{\eta}\eta$,$\forall\eta,\overline{\eta}>T+\mu,s,\overline{s}\in \mathbb{R}$

109
and
$\nu v(T+\mu,s)=4\pi$$Letting\overline{\eta}arrow T+\mu$$\forall s\in \mathbb{R}$.

,$W( \eta,s)=\frac{4\pi(T+\mu)}{\eta}$$\forall\eta>T+\mu,s\in \mathbb{R}$which will then imply the theorem after some elementary computation.

口

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