数理解析研究所講究録
第 1881 巻 2014 年 107-114

107

Extinction of solutions of the fast diffusion equation
Marek Fila
Comenius University1Introduction

In this survey we consider the Cauchy problem for the fast diffusion equation:(1.1)$\{\begin{array}{ll}u_{\tau}=\nabla\cdot(u^{m-1}\nabla u) , y\in \mathbb{R}^{n}, \tau\in(0, T) ,u(y, 0)=u_{0}(y)\geq 0, y\in \mathbb{R}^{n},\end{array}$where $m<1$ and $T>0$ . It is known that for $m$ below the critical exponent$:=(n-2)/n$
all solutions with initial data in some suitable space, likewith $p:=n(1-m)/2,$vanish in finite time. We discuss results on the asymptotic behaviour of solutions near
extinction in the range$m_{c}$$L^{p}(\mathbb{R}^{n})$$m\leq m_{*}:=\underline{n-4} n>2.$

$n-2$

’

The exponent
plays an important role in [1, 2, 3, 4, 6, 7, 9].
The book [11] contains a general description of the phenomenon of extinction. It is
explained there that the size of the initial data at infinity (the tail of ) is very importantin determining both the extinction time and the extinction rates.
For $m<m_{c}$ we have explicit self-similar solutions
called generalized Barenblatt
solutions, given by the formula$m_{*}$$u_{0}$$U_{D,T}$$U_{D,T}(y, \tau) :=\frac{1}{R(\tau)^{n}}(D+\frac{\beta(1-m)}{2}|\frac{y}{R(\tau)}|^{2})^{-\frac{1}{1-m}}$

(1.2)

where
$R( \tau);=(T-\tau)^{-\beta}, \beta:=\frac{1}{n(1-m)-2}=\frac{1}{n(m_{c}-m)}=\frac{\mu}{2(n-\mu)}.$Here $T\geq 0$ (extinction time) and $D>0$ are free parameters. These solutions have adecay rate near extinction of the form.
A very interesting limit case occurs if we take $D=0$ in formula (1.2), and we find thesingular solution$\Vert u(\cdot, \tau)\Vert_{\infty}=O((T-\tau)^{n\beta})$$U_{0,T}(y, \tau):=k_{*}(T-\tau)^{\mu/2}|y|^{-\mu}, k_{*}:=(2(n-\mu))^{\mu/2}.$

whose attracting properties were studied in [6] where we obtained a continuum of extinction rates for suitable bounded data
$u_{0}.$

108
To study the behaviour of solutions near extinction one can rewrite (1.1) by introducing
the change of variables$t:= \frac{1-m}{2}\log(\frac{R(\tau)}{R(0)})$and$x:= \sqrt{\frac{\beta(1-m)}{2}}\frac{y}{R(\tau)},$with $R$ as above, and the rescaled function$v(x, t):=R(\tau)^{n}u(y, \tau)$

If

$u$

is a solution of (1.1) then$v$.

solves the equation$v_{t}=\nabla\cdot(v^{m-1}\nabla v)+\mu\nabla\cdot(xv) , t>0, x\in \mathbb{R}^{n}$

(1.3)

,

which is a nonlinear Fokker-Planck equation. The generalized Barenblatt solutions
which are stationary solutions ofare transformed into generalized Barenblatt profiles$U_{D,T}$$V_{D}$(1.3):$V_{D}(x):=(D+|x|^{2})^{\frac{1}{m-1}}, x\in \mathbb{R}^{n}$

The singular Barenblatt solution becomes$V_{0}(x)=|x|^{-\mu}, x\in \mathbb{R}^{n}\backslash \{0\}.$The critical exponent
has the property that the difference of two generahzed Barenblatt
profiles is integrable for $m\in(m_{*}, m_{c})$ , while it is not integrable for $m\leq m_{*}.$
when $D>0,$
We discuss convergence to for $m<m_{*}$ in Section 2, convergence towhen $D>0,$ $m=m_{*}$ in Section 4.
$m<m_{*}$ in Section 3, and convergence to$m_{*}$$V_{D}$$V_{0}$$V_{D}$

2Convergence to the singular Barenblatt profile

The following was shown in [6].

Theorem 2.1 Assume that$n\geq 5$

and let the initial function$u_{0}$

and$0<m<m_{*}= \frac{n-4}{n-2}$

for all$y\neq 0$

and$A|y|^{-\mu}-c_{1}|y|^{-l}\leq u_{0}(y)\leq A|y|^{-\mu}-c_{2}|y|^{-l}$$A,$ $c_{1},$ $c_{2}>0$(2.1)

be continuous, bounded, and satisfy the conditions:$0\leq u_{0}(y)\leq A|y|^{-\mu}$

for some,for$|y|\geq 1$, and$\mu+2<l\leq L:=\mu+\sqrt{2(n-\mu)}$.

(2.2)

Then the solution of problem (1.1) has complete extinction precisely at the time $T:=$, and there are positive constants
such that for $0<\tau<T$ we have$u$$(A/k_{*})^{1-m}>0$$K_{1},$ $K_{2}$$K_{1}(T-\tau)^{\theta_{l}}\leq\Vert u(\cdot, \tau)\Vert_{\infty}\leq K_{2}(T-\tau)^{\theta_{l}},$

109
where
$\theta_{l}:=\frac{n\mu-\gamma_{l}}{2(n-\mu)}>0$

,

(2.3)

$\gamma_{l}:=\frac{\mu(l-\mu-2)(n-l)}{l-\mu}.$

One of the main aims of [9] is to show that Theorem 2.1 does not hold for $l>L.$The main result from [6] can be formulated as follows.
Theorem 2.2 Let (2.1) hold. Assume that$v_{0}\geq 0$

is continuous, bounded and such thatfor$|x|^{-\mu}-c_{1}|x|^{-l}\leq v_{0}(x)\leq|x|^{-\mu}-c_{2}|x|^{-l}$$|x|\geq 1,$

where is as in (2.2) and $c_{2}>0$ . Assume also that $v_{0}(x)\leq|x|^{-\mu}$ for all $x\neq 0$ . Let
denote the solution of (1.3) with initial condition$l$$c_{1},$$v(x, O)=v_{0}(x) , x\in \mathbb{R}^{n}$.

$v$(2.4)

Then;(i) There exist$K_{1},$$K_{2}>0$such that for $t\geq 1$ we have,(2.5)$K_{1}e^{\gamma\iota^{t}}\leq\Vert v(\cdot, t)\Vert_{\infty}\leq K_{2}e^{\gamma\iota t}$

here$\gamma_{l}$is as in (2.3).

(ii) For each $r_{0}>0$ one can findholds$C_{1},$$C_{2}>0$such that for $t\geq 1$ and$|x|\geq r_{0}$$C_{1}e^{-\alpha_{l}t}\leq|x|^{-\mu}-v(x, t)\leq C_{2}e^{-\alpha_{t}t}, \alpha\iota :=(l-\mu-2)(n-l)$

the following.

(2.6)

The reason why we assume that $l>\mu+2$ is that the difference $|x|^{-\mu}-V_{D}(x)$ behaves
likeas $|x|arrow\infty$ . It was shown in [9] that the condition $\mu+2<l\leq L$ is optimal
for Theorem 2.2 (i) but not for Theorem 2.2 (ii) which holds for a larger range$|x|^{-(\mu+2)}$$l \in(\mu+2, l_{\star}) , l_{\star}:=\frac{1}{2}(n+\mu+2)$

.

(2.7)

More precisely, the following results were established in [9]:Theorem 2.3 Assume that $m<m_{*},$ $n>2$ , and(i)If$v_{0}\geq 0$

is continuous.

$v_{0}(x)<|x|^{-\mu}, x\neq 0$ ,(2.8)

and$v_{0}(x)\leq|x|^{-\mu}-c|x|^{-l}, |x|>1,$

with some as in (2.7) and $c>0$ then for any $r_{0}>0$ there exists $C(r_{0})>0$ such that thesolution of (1.3), (2.4) satisfies$l$$v(x, t)\leq|x|^{-\mu}-C(r_{0})e^{-\alpha_{l}t}|x|^{-l}, |x|\geq r_{0}, t\geq 0.$

110(ii) Assume that$v_{0}(x)\geq|x|^{-\mu}-c|x|^{-l}, |x|>1,$

with some as in (2.7) and $c>0$ . Then one can find $C>0$ such that the solution of(1.3), (2.4) satisfies$l$$v(x, t)\geq|x|^{-\mu}-Ce^{-\alpha_{l}t}|x|^{-l}, x\neq 0, t>0.$(iii) Set$\alpha_{\star}:=\alpha_{l_{\star}}=\frac{(n-\mu-2)^{2}}{4}$

If (2.8) holds then for any
solution of (1.3), (2.4) satisfies$\alpha>\alpha_{\star}$.

(2.9)

and each $r_{0}>0$ there exists $C(\alpha, r_{0})>0$ such that the$\sup_{|x|\geq r0}(|x|^{-\mu}-v(x, t))\geq Ce^{-\alpha t}, t>0.$

Theorem 2.4 Let $m<m_{*},$ $n>2$ . Assume (2.8) and $v_{0}\geq 0$ is continuous. Then forany$\gamma>\gamma_{L}:=\mu(n+2-\mu-2\sqrt{2(n-\mu)})$there exists $C(\gamma)>0$ such that the solution of (1.3), (2.4) satisfies$v(x, t)\leq C(\gamma)e^{\gamma t}, x\in \mathbb{R}^{n}, t>0.$

The fact that the optimal condition on is different for (2.5) and (2.6) is in contrast
with corresponding results for the equation $u_{t}=\Delta u+u^{p}$ , see [5, 8, 10].
$l$3Convergence to regular Barenblatt profiles$D>0$ and the rates of convergence to$D>0$ wasThe basin of attraction ofstudied in [1, 2] using certain functional inequalities of Hardy-Poincar\’e type. It wasin the range $m<m_{*}$ contains functions
established there that the basin of attraction ofsuch that$V_{D},$$V_{D},$$V_{D}$$v_{0}$$V_{D_{0}}\leq v_{0}\leq V_{D_{1}}, 0<D_{1}<D<D_{0}, |v_{0}-V_{D}|\in L^{1}(R^{n})$.

We call this set the variational basin, and for this the entropy method from [1, 2] gives
precise decay rates (the variational rates).
The main result in [7] is the following:
Theorem 3.1 Let $m<m_{*},$ $n>2$ . Assume thatas in (2.7).
(i) If$c,$$D>0$ and $\mu+2<l<l_{\star}$ , here$|v_{0}(x)-V_{D}(x)|\leq c|x|^{-l}, |x|\geq 1,$and$0<v_{0}(x)\leq V_{\delta}(x) , x\in \mathbb{R}^{n}$$l_{\star}$is111for some, then there exists $C_{1}>0$ such that the solution
condition (2.4) satisfies$\delta<D$of (1.3) with the initial$v$$\sup_{x\in \mathbb{R}^{n}}|v(x, t)-V_{D}(x)|\leq C_{1}e^{-\alpha\iota t}, t\geq 0,$

where$\alpha_{l}$is as in (2.6).

(ii) If$v_{0}(x)\leq V_{D}(x)-c|x|^{-l}, |x|\geq 1,$

and$0<v_{0}(x)\leq V_{D}(x) , x\in \mathbb{R}^{n},$then there exists $C_{2}>0$ such that the solution$v$of (1.3), (2.4) satisfies$\sup_{x\in \mathbb{R}^{n}}(V_{D}(x)-v(x, t))\geq C_{2}e^{-\alpha_{l}t}, t\geq 0.$(iii) If
$v_{0}(x)\geq V_{D}(x)+c|x|^{-l}, |x|\geq 1,$

and$v_{0}(x)\geq V_{D}(x) , x\in \mathbb{R}^{n},$then there exists $C_{3}>0$ such that the solution$v$of (1.3), (2.4) satisfies$\sup_{x\in \mathbb{R}^{n}}(v(x, t)-V_{D}(x))\geq C_{3}e^{-\alpha_{l}t}, t\geq 0.$

This result gives a sharp description of the basin of attraction of generalized Barenblatt
profiles for $m<m_{*}$ . It shows that non-integrable perturbations of
may still yield
convergence to . The condition $l>\mu+2$ is optimal since the difference of two Barenblatt
profiles is of the order $|x|^{-(\mu+2)}.$
Theorem 3.1 yields a continuum of convergence rates which depend explicitly on thetail of initial data. The rate $\alpha_{l}=(l-\mu-2)(n-l)$ converges to zero as $larrow\mu+2$ andto the maximum value(see (2.9)) as. Hereis the rate found in [1, 2] forsolutions emanating from integrable perturbations of. This fastest rate is the best
constant in a Hardy-Poincar\’e inequality (see [2]). This best constant is also the bottomof the continuous spectrum of the linearization on a suitable weighted space (see [1, 2]).
In Theorem 3.1, the assertion (i) is no longer true if. In fact, the following result
about the optimality of the range of was obtained in [7].
$V_{D}$$V_{D}$$larrow l_{\star}$$\alpha_{\star}$$\alpha_{\star}$$V_{D}$$l>l_{\star}$$l$

Theorem 3.2 Let $m<m_{*},$ $n>2$ . Assume that $D>0$ and$0<v_{0}(x)<V_{D}(x) , x\in \mathbb{R}^{n}$$or$$v_{0}(x)>V_{D}(x) , x\in \mathbb{R}^{n}.$Then for any$\epsilon>0$, there exists$C_{\epsilon}>0$such that the solution$v$of (1.3), (2.4) satisfies$\sup_{x\in \mathbb{R}^{n}}|V_{D}(x)-v(x, t)|\geq C_{\epsilon}e^{-(\alpha_{\star}+\epsilon)t}, t\geq 0$.

(3.1)

112
It follows from (3.1) that Theorem 2 (i) in [1] is optimal if $m<m_{*},$ $n>2$ . The
was discussed in [2] in terms of relative entropy which
sharpness of the rate given by
can be written as$\alpha_{\star}$$\overline{J-}[w]:=\frac{1}{1-m}\int_{\mathbb{R}^{n}}[w-1-\frac{1}{m}(w^{m}-1)]V_{D}^{m}dx, w:=\frac{v}{V_{D}}.$The statement on the sharp rate in [2] says that$\alpha=\alpha_{\star}$is the best possible rate for which$\mathcal{F}[w(\cdot, t)]\leq \mathcal{F}[w(\cdot, 0)]e^{-\alpha t}$

for some $D_{0}>D>D_{1}>0$ and $v_{0}-V_{D}$ is integrable.
holds for all $t\geq 0$ if
Theorem 3.2 implies that solutions starting from positive or negative perturbations of) at exponential rates faster than(incannot converge to$V_{D_{0}}\leq v_{0}\leq V_{D_{1}}$$V_{D}$$V_{D}$

4$e^{-\alpha_{\star}t}.$$L^{\infty}$

Critical case

The case $m=m_{*}$ was treated in [3] by functional analytic methods. suitable linearization of the non-linear Fokker-Planck equation (1.3) was viewed as the plain heat flow
on a suitable Riemannian manifold and then non-linear stability was studied by entropy
methods. One of the main results of [3] says that if $0<D_{1}<D_{0},$ $D\in[D_{1}, D_{0}]$ and$A$$V_{D_{0}}(x)\leq v_{0}(x)\leq V_{D_{1}}(x) , x\in \mathbb{R}^{n},$(4.1),of (1.3) with the initial condition $v(x, 0)=v_{0}(x)$ it holds that$|v_{0}(x)-V_{D}(x)|\leq f(|x|) , x\in \mathbb{R}^{n}, f(|\cdot|)\in L^{1}(\mathbb{R}^{n})$then for the solution

$v$

(4.2)

,$\Vert v(\cdot, t)-V_{D}\Vert_{L}\infty(\mathbb{R}^{n})\leq K(t+1)^{-\frac{1}{4}}, t\geq 0$

for some $K>0.$No lower bound for the rate was given in [3] and the question of whether the rate from
(4.2) is optimal for a class of data was posed there as an open problem together with thequestion of whether one can prove convergence, maybe with worse rates or without rates,for more general initial data. The aim in [4] is to provide some answers to these questions
by establishing optimal results on rates of convergence for a class of initial data which do
not satisfy (4.1).
Theorem 4.1 Assume that $n>2_{f}m=m_{*}= \frac{n-4}{n-2}$ and $D>0$ . Let(1.3) with the initial condition$v$$v(x, 0)=v_{0}(x) :=(|x|^{2}+D+\psi_{0}(x))^{-\frac{n-2}{2}} x\in \mathbb{R}^{n}$is continuous and nonnegative on(i) If there are $B>0$ and $\gamma\in(0,1)$ such that

where$\mathbb{R}^{n},$$\psi_{0}$be the solution of(4.3),$\psi_{0}\not\equiv 0.$$\psi_{0}(x)\leq B\ln^{-\gamma}|x|, |x|>2,$

then there exists $C>0$ such that
$V_{D}(x)(1-CV_{D}^{\frac{2}{n-2}}(x)(t+1)^{-:1}2)\leq v(x, t)\leq V_{D}(x)$,$x\in \mathbb{R}^{n},$$t\geq 0.$

113(ii) If there are $b>0$ and $\gamma\in(0,1)$ such that$\psi_{0}(x)\geq b\ln^{-\gamma}|x|, |x|>2,$

then there exists $c>0$ such that$v(O, t)\leq V_{D}(0)-c(t+1)^{-l}2, t>0.$This theorem says that if $V_{D}(x)-v_{0}(x)$ behaves likeforlarge and somethenbehaves likefor large. Hence, we obtain acontinuum of algebraic rates for initial data which do not satisfy (4.1). It is also shownin [4] that convergence tofrom below cannot occur at any rate faster than, soTheorem 4.1 (i) does not hold for $\gamma>1.$$|x|^{-n}\ln^{-\gamma}|x|$$\gamma\in(0,1)$$\Vert v(\cdot, t)-V_{D}\Vert_{L^{\infty}(\mathbb{R}^{n})}$$t^{-\gamma/2}$

$|x|$

$t$

$t^{-1/2}$$V_{D}$

Theorem 4.2 Let $n>2,$ $m=m_{\star}$ and $D>0$ , and assume thatis continuous andnonnegative on. Then there exists $c>0$ such that the solution of (1.3),(4.3) satisfies$\psi_{0}$$\mathbb{R}^{n},$$\psi_{0}\not\equiv 0$$v(0, t)\leq V_{D}(0)-c(t+1)^{-\frac{1}{2}}$$v$

for all $t>0.$

Acknowledgment. The author was supported in part by the Slovak Research andDevelopment Agency under the contract No. APVV-0134-10 and by the VEGA grant
1/0711/12.

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[2] M. Bonforte, J. Dolbeault, G. Grillo and J. L. V\’azquez, Sharp rates of decay ofsolutions to the nonlinear fast diffusion equation via functional inequalities, Proc.
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599-625.

114[8] M. Fila and M. Winkler, Rate of convergence to a singular steady state of a supercritical parabolic equation, J. Evol. Equations 8 (2008), 673-692.
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Department of Applied Mathematics and Statistics
Comenius University
84248 Bratislava
Slovakia-mail address: fila@fmph.uniba.sk$E$