††thanks: ∗ Corresponding author: J. Dolbeault

# Constructive stability results in interpolation inequalities and explicit
improvements of decay rates of fast diffusion equations

<EMAIL_ADDRESS><EMAIL_ADDRESS>bruno.nazaret@univ-
paris1.fr<EMAIL_ADDRESS>
###### Abstract

We provide a scheme of a recent stability result for a family of Gagliardo-
Nirenberg-Sobolev (GNS) inequalities, which is equivalent to an improved
entropy – entropy production inequality associated with an appropriate fast
diffusion equation (FDE) written in self-similar variables. This result can be
rephrased as an improved decay rate of the entropy of the solution of (FDE)
for well prepared initial data. There is a family of Caffarelli-Kohn-Nirenberg
(CKN) inequalities which has a very similar structure. When the exponents are
in a range for which the optimal functions for (CKN) are radially symmetric,
we investigate how the methods for (GNS) can be extended to (CKN). In
particular, we prove that the solutions of the evolution equation associated
to (CKN) also satisfy an improved decay rate of the entropy, after an explicit
delay. However, the improved rate is obtained without assuming that initial
data are well prepared, which is a major difference with the (GNS) case.

Matteo Bonforte

Departamento de Matemáticas

Universidad Autónoma de Madrid, and ICMAT

Campus de Cantoblanco, 28049 Madrid, Spain

Jean Dolbeault

Ceremade, UMR CNRS n∘ 7534

Université Paris-Dauphine, PSL Research University

Place de Lattre de Tassigny, 75775 Paris Cedex 16, France

Bruno Nazaret

SAMM (EA 4543), FP2M (FR CNRS 2036)

Université Paris 1

90, rue de Tolbiac, 75634 Paris Cedex 13

Nikita Simonov

LaMME, UMR CNRS n∘ 8071

Université d’Évry Val d’Essonne

23, boulevard de France, 91037 Évry, France

_It is with great pleasure that we dedicate this paper to Juan Luis Vázquez
on the occasion of his 75th birthday._

MSC 2020. Primary: 26D10, 46E35, 35K55; Secondary: 35B40, 49K20.

Keywords. Gagliardo-Nirenberg-Sobolev inequality, Caffarelli-Kohn-Nirenberg
inequality, stability, entropy methods, fast diffusion equation, Harnack
Principle, self-similar solutions, Hardy-Poincaré inequalities, spectral gap,
rates of convergence.

##  1 Introduction and main results

Let us start with some Gagliardo-Nirenberg-Sobolev inequalities (without
weights) and related flow issues in Section 1.1 before extending the results
to Caffarelli-Kohn-Nirenberg inequalities (with weights) and related flows in
Section 1.2.

###  1.1 Gagliardo-Nirenberg-Sobolev inequalities and related flows

The _Gagliardo-Nirenberg-Sobolev inequality_

$\left\|{\nabla
f}\right\|_{2}^{\theta}\,\left\|{f}\right\|_{p+1}^{1-\theta}\geq\mathcal{C}_{\mathrm{GNS}}\,\left\|{f}\right\|_{2\,p}$
(1)

holds on the space of the functions $f\in\mathrm{L}^{p+1}(\mathbb{R}^{d})$
with $\nabla f\in\mathrm{L}^{2}(\mathbb{R}^{d})$, with exponents given by

$\theta=\tfrac{d\,(p-1)}{(d+2-p\,(d-2))\,p}\,,\quad
p\in(1,+\infty)\;\mbox{if}\;d=1\;\mbox{or}\;2\,,\quad
p\in(1,p_{\star}]\;\mbox{if}\;d\geq 3\,,\quad p_{\star}=\tfrac{d}{d-2}\,.$

According to [27, 17], equality in (1) is achieved if and only if $f$ is equal
to

$\mathsf{g}(x)=\big{(}1+|x|^{2}\big{)}^{-\frac{1}{p-1}}\quad\forall\,x\in{\mathbb{R}}^{d}\,,$
(2)

up to a multiplication by a constant, a translation and a scaling. We denote
by $\mathfrak{M}$ the manifold of optimal functions for (1). This inequality
has a number of interesting limit cases: Sobolev’s inequality if $d\geq 3$ and
$p=p_{\star}$, the Euclidean Onofri inequality if $d=2$ in the limit as
$p\to+\infty$, and the scale invariant Euclidean logarithmic Sobolev
inequality in the limit as $p\to 1_{+}$. Let us define the _deficit
functional_

$\delta[f]:=(p-1)^{2}\,\left\|{\nabla
f}\right\|_{2}^{2}+4\,\tfrac{d-p\,(d-2)}{p+1}\,\left\|{f}\right\|_{p+1}^{p+1}-\mathcal{K}_{\mathrm{GNS}}\,\left\|{f}\right\|_{2\,p}^{2\,p\,\chi}$

with $\chi=\frac{d+2-p\,(d-2)}{d-p\,(d-4)}$ and $\mathcal{K}_{\mathrm{GNS}}$
chosen so that $\delta[\mathsf{g}]=0$. Up to a scaling, the inequality
$\delta[f]\geq 0$ is equivalent to (1), as was shown in [17].

In the critical case $p=p_{\star}$, $d\geq 3$, optimal functions in (1) are
known as the _Aubin-Talenti functions_ and this result goes back to [1, 33,
30]. Later, in [9], H. Brezis and E.H. Lieb asked the next natural question:
which distance to $\mathfrak{M}$ is controlled by the deficit $\delta$ ? Soon
after, an answer was given in [2] by G. Bianchi and H. Egnell: there is a
positive constant $\mathcal{C}_{\mathrm{BE}}$ such that

$\tfrac{1}{(p-1)^{2}}\,\delta[f]=\|{\nabla
f}\|_{2}^{2}-\mathsf{S}_{d}\,\|{f}\|_{2^{*}}^{2}\geq\mathcal{C}_{\mathrm{BE}}\,\inf_{\varphi\in\mathfrak{M}}\,\|{\nabla
f-\nabla\varphi}\|_{2}^{2}\,,$

where $\mathsf{S}_{d}$ is the optimal constant in Sobolev’s inequality. This
striking result had anyway an important drawback: $\mathcal{C}_{\mathrm{BE}}$
is obtained by a _non-constructive_ method. Various extensions and
improvements as, _e.g._ , in [16, 26] have been obtained, as well as similar
results for (1) in [11, 31], in the subcritical range: we refer to [6] for a
review of the literature. New _constructive stability results_ were recently
obtained:

######  Theorem 1.

[6, Corollary 5.4 and Theorem 6.1] Let $d\geq 3$ and $p\in(1,p_{\star}]$, or
$d=1$, $2$ and $p\in(1,+\infty)$. For any
$f\in\mathrm{L}^{2p}(\mathbb{R}^{d})$ with $\nabla
f\in\mathrm{L}^{2}(\mathbb{R}^{d})$ and

$A:=\sup_{r>0}r^{\frac{d-p\,(d-4)}{p-1}}\int_{|x|>r}|f|^{2p}\,\mathrm{d}x<\infty\,,$

we have the estimate

$\delta[f]\geq\mathcal{C}\,\inf_{\varphi\in\mathfrak{M}}\int_{{\mathbb{R}}^{d}}{\big{|}(p-1)\,\nabla
f+f^{p}\,\nabla\varphi^{1-p}\big{|}^{2}}\,\mathrm{d}x$ (3)

for some explicit $\mathcal{C}>0$ which depends only on $d$, $p$,
$\left\|{f}\right\|_{2p}$ and $A$.

As a function of $f$ or, to be precise, as a function of the mass
$\left\|{f}\right\|_{2p}$ and $A$, $\mathcal{C}$ takes positive values on
$\mathfrak{M}$ even if it is not uniformly bounded away from $0$ on
$\mathfrak{M}$. The distance to $\mathfrak{M}$ is measured by a _Fisher
information_ functional and the strategy of the proof involves _entropy_
methods. Inequality (3) is equivalent to an improved _entropy – entropy
production inequality_ , which relates the Fisher information with a _relative
entropy_ , or _free energy_ , defined for $m\in[m_{1},1)$ with $m_{1}:=1-1/d$
by

$\mathcal{F}[v]:=\frac{1}{m-1}\int_{{\mathbb{R}}^{d}}{\left(v^{m}-\mathcal{B}^{m}-m\,\mathcal{B}^{m-1}\,(v-\mathcal{B})\right)}\,\mathrm{d}x$
(4)

where

$\mathcal{B}(x):=\big{(}1+|x|^{2}\big{)}^{\frac{1}{m-1}}\quad\forall\,x\in{\mathbb{R}}^{d}\,.$
(5)

The functional $\mathcal{F}$ enters in the study of nonlinear evolution
equations as follows. Let us consider the _fast diffusion equation_

$\frac{\partial v}{\partial t}+\nabla\cdot\left(v\,\nabla
v^{m-1}\right)=2\,\nabla\cdot(x\,v)\,,\quad v(t=0,\cdot)=v_{0}\,.$ (6)

By a standard computation, a solution $v$ of (6) is such that

$\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{F}[v(t,\cdot)]=-\,\mathcal{I}[v(t,\cdot)]\;\mbox{where}\;\mathcal{I}[v]:=\frac{m}{1-m}\int_{{\mathbb{R}}^{d}}{v\,\big{|}\nabla
v^{m-1}-\nabla\mathcal{B}^{m-1}\big{|}^{2}}\,\mathrm{d}x$ (7)

where $\mathcal{I}[v]$ is the _relative Fisher information_ with respect to
$\mathcal{B}$. The exponents $m$ in (6) and $p$ in (1) are related by the
condition $p=1/(2\,m-1)$. In practice, for $m>m_{1}$ (subcritical range), the
result of Theorem 1 can be rephrased as a result on _decay rates_ for the
solutions of (6).

######  Corollary 2.

[6, Corollary 5.2] Let $d\geq 1$ and $m\in(m_{1},1)$. If $v$ is a solution of
(6) with nonnegative initial datum $v_{0}\in\mathrm{L}^{1}({\mathbb{R}}^{d})$
such that
$\int_{{\mathbb{R}}^{d}}{v_{0}}\,\mathrm{d}x=\int_{{\mathbb{R}}^{d}}{\mathcal{B}}\,\mathrm{d}x$,
$\int_{{\mathbb{R}}^{d}}{x\,v_{0}}\,\mathrm{d}x=0$ and

$A[v_{0}]:=\sup_{r>0}r^{\frac{2}{1-m}-d}\int_{|x|>r}v_{0}\,\mathrm{d}x<\infty\,,$

then we have

$\mathcal{F}[v(t,.)]\leq\mathcal{F}[v_{0}]\,e^{-\,(4+\zeta)\,t}\quad\forall\,t\geq
0$ (8)

for some positive constant $\zeta$ which depends explicitly only on $m$, $d$,
and $A[v_{0}]$.

The integral $\int_{{\mathbb{R}}^{d}}{x\,v_{0}}\,\mathrm{d}x$ is finite
because $A[v_{0}]<+\infty$. Note that if the center of mass of $v_{0}$ is
finite, then
$\int_{{\mathbb{R}}^{d}}{x\,v(t,x)}\,\mathrm{d}x=\int_{{\mathbb{R}}^{d}}{x\,v_{0}}\,\mathrm{d}x\,e^{-2t}$
for any $t\geq 0$.

Let us consider the _optimized free energy functional_

$\mathcal{F}_{\star}[v]:=\inf_{B\in\mathfrak{B}}\frac{1}{m-1}\int_{{\mathbb{R}}^{d}}{\left(v^{m}-B^{m}-m\,B^{m-1}\,(v-B)\right)}\,\mathrm{d}x$

where $\mathfrak{B}$ is the set of all _Barenblatt_ functions obtained from
$\mathcal{B}$ using a multiplication by a constant, translations and scalings.

######  Corollary 3.

Let $m\in[m_{1},1)$ if $d\geq 2$, $m\in(1/2,1)$ if $d=1$ and consider $\zeta$
as in Corollary 2. If $v$ is a solution of (6) with nonnegative initial datum
$v_{0}\in\mathrm{L}^{1}({\mathbb{R}}^{d})$ such that $A[v_{0}]$ is finite,
then

$\mathcal{F}_{\star}[v(t,.)]\leq\mathcal{F}_{\star}[v_{0}]\,e^{-\,(4+\zeta)\,t}\quad\forall\,t\geq
0\,.$ (9)

In the subcritical range, this result is a straightforward consequence of
Corollary 2, but it is new in the critical case $m=m_{1}$ corresponding to
$p=p_{\star}$.

The _entropy – entropy production inequality_ relates rates of convergence for
the solutions of (6) with (1), but (1) can also be invoked in the context of
the standard _fast diffusion equation_

$\frac{\partial u}{\partial t}=\Delta u^{m}\,,\quad u(t=0,\cdot)=u_{0}\,.$
(10)

Assume that $m\in[m_{1},1)$ if $d\geq 2$ and $m\in(1/2,1)$ if $d=1$. Using the
_Rényi entropy powers_ formalism, we learn from [6, Lemma 2.1] that a solution
$u$ of (10) with initial datum
$u_{0}\in\mathrm{L}^{1}_{+}\\!\left({\mathbb{R}}^{d},(1+|x|^{2})\,\mathrm{d}x\right)$
such that $u_{0}^{m}\in\mathrm{L}^{1}({\mathbb{R}}^{d})$ satisfies

$\int_{{\mathbb{R}}^{d}}{u^{m}(t,x)}\,\mathrm{d}x\geq\left(\left(\int_{{\mathbb{R}}^{d}}{u_{0}^{m}}\,\mathrm{d}x\right)^{\frac{m-m_{c}}{1-m}}+\tfrac{(1-m)\,C_{0}}{m-m_{c}}\,t\right)^{\frac{1-m}{m-m_{c}}}\quad\forall\,t\geq
0\,,$ (11)

for some constant $C_{0}$ which explicitly involves
$\mathcal{C}_{\mathrm{GNS}}$ and where $m_{c}:=(d-2)/d$. Equality in (11)
holds for any $t\geq 0$ if and only if $u_{0}\in\mathfrak{B}$. At $t=0$, (11)
is an equality for any $u_{0}$ and we can recover the Gagliardo-Nirenberg-
Sobolev inequality (1) written with the optimal constant by differentiating
with respect to $t$ the growth estimate (11) at $t=0$. A more readable
estimate is obtained by considering the _optimized free energy functional_
$\mathcal{F}_{\star}$ applied to (10).

######  Corollary 4.

Assume that $d\geq 1$, $m\in[m_{1},1)$ and consider a solution of (10) with
initial datum
$u_{0}\in\mathrm{L}^{1}_{+}\\!\left({\mathbb{R}}^{d},(1+|x|^{2})\,\mathrm{d}x\right)$
such that $u_{0}^{m}\in\mathrm{L}^{1}({\mathbb{R}}^{d})$. With
$\kappa=\zeta/d$ and $\zeta$ as in Corollary 3, we have

$\mathcal{F}_{\star}[u(t,.)]\leq\mathcal{F}_{\star}[u_{0}]\,\big{(}1+d\,(m-m_{c})\,t\big{)}^{-\frac{m+\kappa}{m-m_{c}}}\quad\forall\,t\geq
0\,.$

This new result is remarkable. While the best matching function
$B(t,\cdot)\in\mathfrak{B}$ is such that
$\int_{{\mathbb{R}}^{d}}{B^{m}(t,x)}\,\mathrm{d}x\sim
t^{(1-m)/(m-m_{c})}\to+\infty$ as $t\to+\infty$, according to (11), it turns
out that $\mathcal{F}[u(t,.)]$, which involves
$\int_{{\mathbb{R}}^{d}}{\left(u^{m}(t,x)-B^{m}(t,x)\right)}\,\mathrm{d}x$,
decays to $0$ at an algebraic rate. As a consequence, we have
$\lim_{t\to+\infty}\left\|{u(t,.)-B(t,.)}\right\|_{1}=0$, with an explicit
rate, by the Csiszár-Kullback-Pinsker inequality. The case $\kappa=0$ is a
consequence of (1) and the improvement $\kappa>0$ is a consequence of Theorem
1.

###  1.2 Caffarelli-Kohn-Nirenberg inequalities and related flows

So far the results are simple consequences of the method of [6]. We are now
going to extend them to a larger class of inequalities. On ${\mathbb{R}}^{d}$
with $d\geq 1$, let us consider the _Caffarelli-Kohn-Nirenberg interpolation
inequalities_

$\left\|{f}\right\|_{2p,\gamma}\leq{\mathsf{C}}_{\beta,\gamma,p}\,\left\|{\nabla
f}\right\|_{2,\beta}^{\theta}\,\left\|{f}\right\|_{p+1,\gamma}^{1-\theta}$
(12)

with optimal constant ${\mathsf{C}}_{\beta,\gamma,p}$, parameters $\beta$,
$\gamma$ and $p$ such that

$\gamma-2<\beta<\frac{d-2}{d}\,\gamma\,,\quad\gamma\in(-\infty,d)\,,\quad
p\in\left(1,p_{\star}\right]\quad\mbox{with}\quad
p_{\star}:=\frac{d-\gamma}{d-\beta-2}\,,$ (13)

and an exponent

$\theta=\frac{(d-\gamma)\,(p-1)}{p\,\big{(}d+\beta+2-2\,\gamma-p\,(d-\beta-2)\big{)}}$

which is determined by the scaling invariance. This formula for $\theta$
extends to (12) the expression for (1), which corresponds to the special case
$\beta=\gamma=0$. Here $\mathrm{L}^{q,\gamma}({\mathbb{R}}^{d})$ and
$\mathrm{L}^{q}({\mathbb{R}}^{d})$ respectively denote the spaces of all
measurable functions $f$ such that

$\left\|{f}\right\|_{q,\gamma}:=\left(\int_{{\mathbb{R}}^{d}}{|f|^{q}\,|x|^{-\gamma}}\,\mathrm{d}x\right)^{1/q}\quad\mbox{and}\quad\left\|{f}\right\|_{q}:=\left\|{f}\right\|_{q,0}$

are finite. Inequality (12) holds in the space
$\mathrm{H}^{p}_{\beta,\gamma}({\mathbb{R}}^{d})$ of functions
$f\in\mathrm{L}^{p+1,\gamma}({\mathbb{R}}^{d})$ such that $\nabla
f\in\mathrm{L}^{2,\beta}({\mathbb{R}}^{d})$, defined as the completion of the
space $\mathcal{D}({\mathbb{R}}^{d}\setminus\\{0\\})$ of the smooth functions
on ${\mathbb{R}}^{d}$ with compact support in
${\mathbb{R}}^{d}\setminus\\{0\\}$, with respect to the norm given by
$f\mapsto(p_{\star}-p)\,\left\|{f}\right\|_{p+1,\gamma}^{2}+\left\|{\nabla
f}\right\|_{2,\beta}^{2}$. Since the weights are locally integrable, these
spaces can also be defined as the completion of the space
$\mathcal{D}({\mathbb{R}}^{d})$. The limitation $p\leq p_{\star}$ in (13)
amounts, for a given $p>1$ to a restriction to the admissible set of
parameters $(\beta,\gamma)$, namely $\beta\geq d-2+(\gamma-d)/p$. On the other
hand, we notice that

$\beta<\frac{d-2}{d}\,\gamma\quad\Longleftrightarrow\quad
p_{\star}<\frac{d}{d-2}\,.$

Inequality (12) belongs to a family of inequalities introduced by L.
Caffarelli, R. Kohn and L. Nirenberg in [10] and also earlier by V.P. Il’in in
[28]. The range of admissible parameters $(\beta,\gamma)$ is limited by (13)
to a cone in the quadrant $\beta<d-2$ and $\gamma<d$ (see Fig. 1), but the
inequality also holds in a cone in the quadrant $\beta>d-2$ and $\gamma>d$
using the property of _inversion symmetry_ : see [4, Section 2.1] for details.

A central issue in Caffarelli-Kohn-Nirenberg inequalities (12) is to decide
whether the equality case is achieved among radial functions or not when
$d\geq 2$. We summarize this alternative by _symmetry_ versus _symmetry
breaking_. Symmetry in (12) means that the equality case is achieved by the
(generalized) _Aubin-Talenti type functions_

$\mathsf{g}(x)=\big{(}1+|x|^{\sigma}\big{)}^{-\frac{1}{p-1}}\quad\forall\,x\in{\mathbb{R}}^{d}\,,\quad\mbox{with}\quad\sigma:=2+\beta-\gamma\,.$
(14)

This definition of Aubin-Talenti type functions extends the one in (2). In the
critical case $p=p_{\star}$, $\theta=1$, V. Felli and M. Schneider proved in
[25] that _symmetry breaking_ holds if

$\gamma<0\quad\mbox{and}\quad\beta_{\rm
FS}(\gamma)<\beta<\frac{d-2}{d}\,\gamma\,,$

where

$\beta_{\rm FS}(\gamma):=d-2-\sqrt{(\gamma-d)^{2}-4\,(d-1)}\,.$

Reciprocally, if

$\gamma<d\,,\quad\mbox{and}\quad\gamma-2<\beta<\frac{d-2}{d}\,\gamma\quad\mbox{and}\quad\beta\leq\beta_{\rm
FS}(\gamma)\,,$

then _symmetry_ holds according to [21]. The results are exactly the same in
the subcritical case $p\in\left(1,p_{\star}\right)$ as was shown in [4,
Theorem 2] and in [23, Theorem 1.1]. See Fig. 1.

$d=4,\;p=2$$d=4,\;p=6/5$$\gamma$$\gamma$$\beta$$\beta$ Figure 1:  _In
dimension $d=4$, the critical exponent $p=p_{\star}=d/(d-2)=2$ corresponds to
the left figure, while $p=6/5$ is subcritical and corresponds to the right
figure. The half cone of admissible regions of the parameters $(\beta,\gamma)$
appear in grey, with symmetry breaking in dark grey and symmetry in light grey
(the symmetry region is bounded if and only if $p<p_{\star}$)._

More is known. In the limit case $\beta=(d-2)\,\gamma/d$ corresponding to
$p=p_{\star}=d/(d-2)$, we learn from [15] that the optimal constant is
achieved among radial functions, but that optimal functions exist only for
$\beta\geq 0$. The limit case as $\beta\to(\gamma-2)_{+}$, that is, $p\to
1_{+}$, gives rise to a family of logarithmic Hardy inequalities which has
been studied in [18, 19]. We refer to [21, 23] for more details on earlier
contributions and to [22] for a general overview. Stability results _à la_
Bianchi-Egnell, with no constructive estimates, appeared in [36] in the
critical case of (12), as an extension of the results of [2, 26]. So far there
are no constructive stability results for (12).

Exactly as in the case of Inequalities (1), it is interesting to consider on
${\mathbb{R}}^{d}$ the nonlinear flow defined by

$\frac{\partial u}{\partial
t}+|x|^{\gamma}\,\nabla\left(|x|^{-\beta}\,u\,\nabla u^{m-1}\right)=0\,,\quad
u(t=0,\cdot)=u_{0}\,,$ (15)

which generalizes (10) to $(\beta,\gamma)\neq(0,0)$ for $m\in[m_{1},1)$ with a
generalized critical exponent defined by
$m_{1}:=1-\frac{2+\beta-\gamma}{2\,(d-\gamma)}$. The computation

$\frac{\mathrm{d}}{\mathrm{d}t}\int_{{\mathbb{R}}^{d}}{u\,|x|^{-\gamma}}\,\mathrm{d}x=0\,,\,\,\frac{\mathrm{d}}{\mathrm{d}t}\int_{{\mathbb{R}}^{d}}{u^{m}\,|x|^{-\gamma}}\,\mathrm{d}x=\frac{m^{2}}{1-m}\int_{{\mathbb{R}}^{d}}{u\,\left|\nabla
u^{m-1}\right|^{2}\,|x|^{-\beta}}\,\mathrm{d}x$ (16)

enters in the formalism of the _generalized Rényi entropy powers_. With
$f=u^{m-1/2}$ and $p=1/(2\,m-1)$ so that $u=f^{2p}$, we notice that

$\textstyle\int_{{\mathbb{R}}^{d}}{u\,|x|^{-\gamma}}\,\mathrm{d}x=\left\|{f}\right\|_{2p,\gamma}^{2p}\,,\quad\int_{{\mathbb{R}}^{d}}{u^{m}\,|x|^{-\gamma}}\,\mathrm{d}x=\left\|{f}\right\|_{p+1,\gamma}^{p+1}\\\
\textstyle\mbox{and}\quad\int_{{\mathbb{R}}^{d}}{u\,|\nabla
u^{m-1/2}|^{2}\,|x|^{-\beta}}\,\mathrm{d}x=\frac{4\,(1-m)^{2}}{(2\,m-1)^{2}}\,\left\|{\nabla
f}\right\|_{2,\beta}^{2}\,.$

As in the non-weighted case, $m=m_{1}$ corresponds to $p=p_{\star}$. Taking
into account (12) and (16), any solution of (15) satisfies

$\frac{\mathrm{d}}{\mathrm{d}t}\int_{{\mathbb{R}}^{d}}{u^{m}\,|x|^{-\gamma}}\,\mathrm{d}x\leq
C\left(\int_{{\mathbb{R}}^{d}}{u^{m}\,|x|^{-\gamma}}\,\mathrm{d}x\right)^{-\frac{1-\theta}{\theta(p+1)}}$

for some numerical constant $C$ involving ${\mathsf{C}}_{\beta,\gamma,p}$.
Altogether, this proves a growth estimate similar to (11). As for (1), this
estimate is in fact equivalent to (12).

Again, more readable estimates are achieved using self-similar variables and
relative entropies. Equation (15) can be rewritten in these variables as

$|x|^{-\gamma}\,\frac{\partial v}{\partial
t}+\nabla\cdot\left(|x|^{-\beta}\,v\,\nabla
v^{m-1}\right)=\sigma\,\nabla\cdot\left(x\,|x|^{-\gamma}\,v\right)\,.$ (17)

The counterpart of Corollary 2 is an improved decay rate of the free energy
$\mathcal{F}$ now defined as

$\mathcal{F}[v]=\frac{2\,p}{1-p}\int_{{\mathbb{R}}^{d}}{\left(v^{\frac{p+1}{2\,p}}-\mathsf{g}^{p+1}-\frac{p+1}{2\,p}\mathsf{g}^{1-p}\left(v-\mathsf{g}^{2p}\right)\right)|x|^{-\gamma}}\,\mathrm{d}x\,.$
(18)

where $\mathsf{g}^{2p}$ is now a stationary solution to (17) and
$m=(p+1)/(2\,p)$.

######  Theorem 5.

Let $d\geq 1$. Assume that $(\beta,\gamma)\neq(0,0)$ satisfies

$\gamma<d\,,\quad\gamma-2<\beta<\frac{d-2}{d}\,\gamma\quad\mbox{and}\quad\beta<\beta_{\rm
FS}(\gamma)\,,$ (19)

let $\alpha=1+(\beta-\gamma)/2$ and assume that $m\in[m_{1},1)$. If $v$ solves
(17) with a nonnegative initial datum
$v_{0}\in\mathrm{L}^{1,\gamma}({\mathbb{R}}^{d})$ such that
$\int_{{\mathbb{R}}^{d}}{v_{0}(x)\,|x|^{-\gamma}}\,\mathrm{d}x=\int_{{\mathbb{R}}^{d}}{\mathsf{g}(x)^{2p}\,|x|^{-\gamma}}\,\mathrm{d}x$
and

$A[v_{0}]:=\sup_{R>0}R^{\frac{2+\beta-\gamma}{1-m}-(d-\gamma)}\,\int_{|x|>R}v_{0}(x)\,|x|^{-\gamma}\,\mathrm{d}x\,<\infty\,,$

then there are some $\zeta>0$ and some $T>0$ which depend explicitly only on
$m$, $d$, $\left\|{v_{0}}\right\|_{1,\gamma}$ and $A[v_{0}]$ such that

$\mathcal{F}[v(t,.)]\leq\mathcal{F}[v_{0}]\,e^{-\,(4\,\alpha^{2}+\zeta)\,t}\quad\forall\,t\geq
2\,T\,.$ (20)

Under the restriction that $A[v_{0}]$ is finite (with a definition for $A$
which generalizes the one of Corollary 2), Inequality (20) provides us with an
improved rate of convergence since the optimal rate of convergence (without
the restriction $A[v_{0}]<\infty$) for functions in
$\mathrm{L}^{1,\gamma}({\mathbb{R}}^{d})$ is $4\,\alpha^{2}$, see Section 3.2.
It is remarkable that no other condition is needed in the case
$(\beta,\gamma)\neq(0,0)$, which is a major difference with Corollary 2 where
the conditions
$\int_{{\mathbb{R}}^{d}}{v_{0}}\,\mathrm{d}x=\int_{{\mathbb{R}}^{d}}{\mathcal{B}}\,\mathrm{d}x$
and $\int_{{\mathbb{R}}^{d}}{x\,v_{0}}\,\mathrm{d}x=0$ have to be assumed.
Although somewhat hidden, these conditions are also present in Corollaries 3
and 4 as we use the _optimized free energy functional_ $\mathcal{F}_{\star}$
and the optimization with respect to $B\in\mathfrak{B}$ induces a similar
normalization condition.

In Theorem 5, $T$ is a _threshold time_ which is similar to the threshold time
in the non-weighted case and determines an _asymptotic time layer_
$[T,+\infty)$. We actually prove that

$\mathcal{F}[v(t,.)]\leq\mathcal{F}[v(T,.)]\,e^{-\,(4\,\alpha^{2}+2\,\zeta)\,(t-T)}\quad\forall\,t\geq
T\,.$ (21)

Compared with Corollary 2, we have no improved decay estimate on the _initial
time layer_ $[0,T]$. The asymptotic decay rate of $\mathcal{F}[v(t,.)]$ as
$t\to+\infty$ is known from [5, 4]. Based on a fully quantitative regularity
theory, the main progress here is that we give constructive estimates of $T$
and an improved decay rate on $[T,+\infty)$.

###  1.3 Simplifying assumptions and outline of the paper

In this paper, we present some results in the spirit of [6] and the key ideas
of the proofs, with several simplifications:

  * –

we do not track the exact dependence of the constants on the parameters,

  * –

we do not distinguish in the estimates quantities which depend on the relative
entropy and those coming from $A[v]$ defined as in Theorem 5, on the basis of
the following result:

######  Lemma 6.

Let $d\geq 1$. Assume that the parameters $\beta$, $\gamma$ and $m$ are as in
Theorem 5 and take $\sigma=2+\beta-\gamma$. There are two explicit positive
numerical constants $c_{1}$ and $c_{2}$ such that, for any nonnegative
function
$v\in\mathrm{L}^{1,\gamma}\big{(}{\mathbb{R}}^{d},(1+|x|^{\sigma})\,dx\big{)}$
such that $A[v]$ is finite, we have

$\displaystyle\int_{{\mathbb{R}}^{d}}{|x|^{\sigma-\gamma}\,v}\,\mathrm{d}x\leq\int_{{\mathbb{R}}^{d}}{v\,|x|^{-\gamma}}\,\mathrm{d}x+c_{1}\,A[v]\,,$
$\displaystyle\left(\int_{{\mathbb{R}}^{d}}{v^{m}\,|x|^{-\gamma}}\,\mathrm{d}x\right)^{\kern-3.0pt\frac{1}{m}}\leq
c_{2}\left(\int_{{\mathbb{R}}^{d}}{v\,|x|^{-\gamma}}\,\mathrm{d}x\right)^{\kern-3.0pt1-(d-\gamma)\,\frac{1-m}{\sigma\,m}}\left(\int_{{\mathbb{R}}^{d}}{|x|^{\sigma-\gamma}\,v}\,\mathrm{d}x\right)^{\kern-3.0pt(d-\gamma)\,\frac{1-m}{\sigma\,m}}.$

As a consequence, the relative entropy as defined in (18) is controlled as
soon as $v$ has finite mass and $A[v]$ is finite. For a proof of Lemma 6, it
is easy to adapt the result of [6, Proposition 7.3] for the first inequality
and use the Carlson-Levin estimate for the second inequality: see [12, 29] and
[14, Lemma 5] for the proof of a similar result.

This paper is organized as follows. Section 2 is devoted to the regularization
properties of the evolution equations (10) and (15). On the basis of [7], the
results on the _relative uniform convergence_ and on _threshold time_
$t_{\star}$ of [6] for (10) are extended to (15). With less details than in
[6], all intermediate estimates are stated but only the differences with [6]
are emphasized: see Theorem 7 for the main result of the section. Entropy
methods and _improved entropy – entropy production estimates_ are applied in
Section 3 to prove Theorem 5. The fact that no additional constraint has to be
imposed to get the improved decay rates for the solutions to (15), a major
difference with the standard fast diffusion equation (10), is commented there.
Section 4 is devoted to a summary of the strategy for proving the stability
results for Gagliardo-Nirenberg-Sobolev inequalities, see Theorem 1, whose
detailed proof can be found in [6]. Our goal here is to explain that the
improved decay rates can be extended to the initial time layer using a
differential inequality based on the _carré du champ_ method. Such an estimate
is missing in the case of the evolution equation (15) associated with the
Caffarelli-Kohn-Nirenberg inequalities, but a similar property is expected:
this motivates the conjecture of Section 5.

##  2 A threshold time for the convergence in relative error

Equation (15) admits a family of self-similar solutions, that we call
_Barenblatt solutions_ as a straightforward generalization of the non-weighted
case, as in [5]. These solutions can be written as

$B(t,x):=R(t)^{-d+\gamma}\,\mathsf{g}^{2p}\left(x/R(t)\right)$ (22)

where $\mathsf{g}$ is defined by (14), $p=1/(2\,m-1)$, $R(t)=c\,t^{1/\xi}$ for
some constant $c>0$ that depends on $m$, $d$, $\beta$ and $\gamma$, and

$\xi=2+\beta-\gamma-(d-\gamma)\,(1-m)\,.$ (23)

In order to fix notations, let us define

$\mathcal{M}:=\int_{{\mathbb{R}}^{d}}{\mathsf{g}^{2p}}\,\mathrm{d}x\,.$ (24)

The purpose of this section is to prove that Barenblatt solutions attract all
solutions of (15). Notice that the case $(\beta,\gamma)=(0,0)$ is covered.

###  2.1 Convergence in relative error

The basin of attraction of the set of Barenblatt solutions in the strong
topology of the _uniform convergence in relative error_ has been defined in
[34], and characterized in [7]. It is a key point to estimate the
stabilization rates, see [3, 5]. The result goes as follows.

######  Theorem 7.

Let $d\geq 2$, $m\in[m_{1},1)$ and assume that $\beta$ and $\gamma$ satisfy
(13). Let $u$ be a solution to (15) corresponding to a nonnegative initial
datum $u_{0}\in\mathrm{L}^{1,\gamma}({\mathbb{R}}^{d})$ such that
$\int_{{\mathbb{R}}^{d}}{u_{0}\,|x|^{-\gamma}}\,\mathrm{d}x=\mathcal{M}$ and

$A[u_{0}]=\sup_{R>0}R^{\frac{2+\beta-\gamma}{1-m}-(d-\gamma)}\,\int_{|x|>R}u_{0}(x)\,|x|^{-\gamma}\,\mathrm{d}x<\infty\,.$

Then there exists an explicit $\varepsilon_{\star}$ such that for any
$\varepsilon\in(0,\varepsilon_{\star})$

$\sup_{x\in{\mathbb{R}}^{d}}\Big{|}\frac{u(t,x)}{B(t,x)}-1\Big{|}\leq\varepsilon\,\quad\forall\,t\geq
t_{\star}:=\mathscr{C}_{\star}\,\varepsilon^{-\mathrm{a}}\,.$ (25)

Here $\varepsilon_{\star}$ and $\mathrm{a}>0$ are numerical constants which
depend only on $d$, $m$, $\beta$ and $\gamma$ while $\mathscr{C}_{\star}$
depends also on $A[u_{0}]$.

###  2.2 Global Harnack Principle

######  Proposition 8.

[8, Theorem 2.1] Under the assumptions of Theorem 7, there exist positive
constants $\overline{t}$ and $\overline{M}$ such that any solution $u$ to (15)
satisfies

$u(t,x)\leq
B_{\overline{M}}\big{(}t+\overline{t},x\big{)}\,\quad\forall(t,x)\in\left[2\,\overline{t}\right.,\left.+\infty\right]\times{\mathbb{R}}^{d}\,.$
(26)

where
$B_{M}(t,x):=(M/\mathcal{M})^{\sigma/\xi}\,B\big{(}t,(M/\mathcal{M})^{(1-m)/\xi}\,x\big{)}$.

######  Proposition 9.

[8, Theorem 3.1] Under the assumptions of Theorem 7, there exist positive
constants $\underline{t}$ and $\underline{M}$ such that any solution $u$ to
(15) satisfies

$u(t,x)\geq
B_{\underline{M}}\big{(}t-\underline{t},x\big{)}\,\quad\forall(t,x)\in\left[2\,\underline{t}\right.,\left.+\infty\right]\times{\mathbb{R}}^{d}\,.$
(27)

where $B_{M}(t,x)$ is as in Proposition 8.

The proof of Propositions 8 and 9 can be found in [8], with another proof in
[6, Propositions 4.6 and 4.7] which is better adapted to our purposes.
Combining the results of Propositions 8 and 9 we obtain a precise control of
the solution $u(t,x)$ which is called in the literature a  _global Harnack
principle_ , see [35]. Let us stress that the quantities $\overline{t}$,
$\underline{t}$, $\overline{M}$ and $\underline{M}$ can be explicitly computed
and their value can be found in [8]. We write below their dependencies with
respect to the main parameters. In particular, we can chose
$\overline{M}=\kappa_{1}\,\mathcal{M}$ and
$\underline{M}=\kappa_{2}\,\mathcal{M}$ for some positive constants
$\kappa_{1}$ and $\kappa_{2}$ which depend only on $d$, $m$, $\gamma$, and
$\beta$.

###  2.3 Convergence in relative error and the threshold time

Let us define

$\overline{\varepsilon}:=\left(\overline{M}/\mathcal{M}\right)^{\sigma/\xi}-1\,,\quad\underline{\varepsilon}:=1-\left(\underline{M}/\mathcal{M}\right)^{\sigma/\xi}\,,\quad\mbox{and}\quad\varepsilon_{m}:=\min\left\\{\tfrac{1}{2}\,,\overline{\varepsilon}\,,\underline{\varepsilon}\right\\}$

where $\overline{M}$ and $\underline{M}$ are as in Propositions 8 and 9.
Integrating inequalities (27) and (26) over the whole space
${\mathbb{R}}^{d}$, we deduce that $\overline{M}>\mathcal{M}$ and
$\underline{M}<\mathcal{M}$. As a consequence we obtain that
$\overline{\varepsilon}$, $\underline{\varepsilon}$ and $\varepsilon_{m}$ are
positive, and $\varepsilon_{m}$ depends only on $d$, $m$, $\beta$ and
$\gamma$.

#### The outer estimate

Here we compare a solution $u(t,x)$ with a Barenblatt profile with same mass
$\mathcal{M}$ as in [6, Section 4.4.1], outside a large ball in $x$ and for
large values of $t$.

######  Corollary 10.

Under the assumptions of Theorem 7 and for any
$\varepsilon\in\left(0,\varepsilon_{m}\right)$ there exist $\rho(\varepsilon)$
and $T(\varepsilon)$ for which any solution $u$ to (15) satisfies

$\left(1-\varepsilon\right)\,B(t,x)\leq
u(t,x)\leq\left(1+\varepsilon\right)\,B(t,x)\;\mbox{if}\;|x|\geq
R(t)\,\rho(\varepsilon)\;\mbox{and}\;t\geq T(\varepsilon)\,.$ (28)

Furthermore, there exist positive constants $\overline{C}$ and $\underline{C}$
such that, for all $x\in{\mathbb{R}}^{d}$,

$\underline{C}\,B(t,x)\leq u(t,x)\leq\overline{C}\,B(t,x)\quad\forall\,t\geq
4\,T(\varepsilon)\,.$ (29)

An explicit expression of $T(\varepsilon)$ and $\rho(\varepsilon)$ can be
computed and is not detailed here: see [6, Section 4.4.1] for similar
computations. We only remark that
$\rho(\varepsilon)=O(1/\sqrt{\varepsilon}\,)$ and
$T(\varepsilon)=\left(1+A\right)^{1-m}O(1/\varepsilon)$ as
$\varepsilon\rightarrow 0$.

###### Proof.

The proof is based on Propositions 8 and 9 and on the comparison (for $t$
large enough) of $B_{M}(t\pm c,x)$ with $B(t,x)$, where $c$ can be either
$\overline{t}$ or $\underline{t}$ and $M$ either $\overline{M}$ or
$\underline{M}$. We observe that the quotient $B_{M}(t\pm c,x)/B(t,x)$ can be
written as

$\frac{B_{M}(t\pm c,x)}{B(t,x)}=\left(\frac{\lambda(t\pm
c)}{\lambda(t)}\right)^{\gamma-d}\,\left(\frac{1+\lambda(t)^{\sigma}\,|x|^{\sigma}}{(M/\mathcal{M})^{\frac{\sigma(1-m)}{\xi}}+\lambda(t\pm
c)^{\sigma}\,|x|^{\sigma}}\right)^{\frac{1}{1-m}}\,,$

where $\lambda(t)=R(t)^{-1}$. Inequality (28) follows from the fact that
$\lambda(t\pm c)\sim\lambda(t)$ as $t\rightarrow\infty$. Inequality (29)
follows from a similar analysis performed directly on the same quotient
$B_{M}(t\pm c,x)/B(t,x)$. ∎

#### The inner estimate

We consider what happens inside a ball as in [6, Section 4.4.2].

######  Corollary 11.

Under the assumptions of Theorem 7 and for any
$\varepsilon\in\left(0,\varepsilon_{m}\right)$ and for any $t\geq
4\,T(\varepsilon)$, there exists a constant $\mathrm{K}>0$ and an exponent
$\vartheta>0$ such that any solution $u$ to (15) satisfies

$\Big{|}\frac{u(t,x)}{B(t,x)}-1\Big{|}\leq\frac{\mathrm{K}}{\varepsilon^{\frac{1}{1-m}}}\left(\frac{1}{t}+\frac{\sqrt{\mathcal{F}[u_{0}]}}{R(t)}\right)^{\vartheta}\,\quad\mbox{if}\quad|x|\leq
2\,\rho(\varepsilon)\,R(t)$ (30)

The constants $\mathrm{K}$ and $\vartheta$ are numerical constants, which
depend only on $d$, $m$, $\beta$ and $\gamma$. Their explicit values follow
from the proof as in [6].

###### Proof.

The proof follows the proof of [6, Proposition 4.1] and further properties of
parabolic regularity which are detailed in [6]. We have to estimate the term
$|u(t,x)/B(t,x)-1|$ in the ball of radius $2\,\rho(\varepsilon)\,R(t)$. This
can be done by interpolating its $\mathrm{L}^{\infty}$ norm between
$\mathrm{L}^{p}$ and $C^{\mu}$ semi-norms, defined on a bounded open domain
$\Omega$ as

$\lfloor
u\rfloor_{C^{\mu}\left(\Omega\right)}:=\sup_{\begin{subarray}{c}x,y\in\Omega\\\
x\neq y\end{subarray}}\frac{|u(x)-u(y)|}{|x-y|^{\mu}}\,,$

using the interpolation inequality (see [6, Section 3.1.4])

$\left\|{u}\right\|_{\mathrm{L}^{\infty}(B_{R}(x))}\,\leq\,C\left(\lfloor
u\rfloor_{C^{\mu}(B_{2R}(x))}^{\frac{(d-\gamma)}{(d-\gamma)+p\,\mu}}\,\|u\|_{\mathrm{L}^{p,\gamma}(B_{2R}(x))}^{\frac{p\,\mu}{(d-\gamma)+p\,\mu}}+R^{-\frac{d-\gamma}{p}}\,\|u\|_{\mathrm{L}^{p,\gamma}(B_{2R}(x))}\right)\,.$
(31)

Here $C$ is a positive constant which depends on $d$, $\gamma$, $\mu$ and $p$
and

$\textstyle\|u\|_{\mathrm{L}^{p,\gamma}(B_{R}(x_{0}))}:=\left(\int_{B_{R}(x_{0})}|u|^{p}\,|x|^{-\gamma}\,\mathrm{d}x\right)^{1/p}\,.$

In order to estimate the $C^{\mu}$ semi-norm of $u(t,x)$ and $B(t,x)$ on the
domain $|x|\leq\,2\,\rho(\varepsilon)R(t)$ with $t\geq 4\,T(\varepsilon)$, we
have to deal with the time dependence of those functions and the domain
itself. In order to simplify the analysis, we introduce the scaling
$\hat{u}_{\tau,\kappa}(t,x):=\kappa^{\sigma/(1-m)}\,\tau^{(d-\gamma)/\xi}\,u\big{(}\tau\,t,\kappa\,\tau^{1/\xi}\,x\big{)}$
for some positive $\kappa$ and $\tau>0$. If $u$ is a solution to (15), then so
is $\hat{u}_{\tau,\kappa}$, for any $\tau,\kappa>0$. The Barenblatt profile is
transformed by the previous scaling as
$\hat{B}_{\tau,\kappa}(t,x)=B_{\kappa^{\xi/(1-m)}\,\mathcal{M}}\big{(}t,x\big{)}$
where $B_{M}(t,x)$ is as in Proposition 8. The advantage comes from the
identity

$\frac{u(t,x)-B(t,x)}{B(t,x)}=\frac{\hat{u}_{t,1}\big{(}1,t^{-1/\xi}\,x\big{)}-\hat{B}_{t,1}\big{(}1,t^{-1/\xi}\,x\big{)}}{\hat{B}_{t,1}\big{(}1,t^{-1/\xi}\,x\big{)}}\,,$
(32)

as the domain $|x|\leq 2\,\rho(\varepsilon)\,R(t)$ is included in $|y|\leq
2\,Z\,\rho(\varepsilon)$ where $y=x\,t^{-1/\xi}$ for $Z$ large enough. So, to
estimate the $C^{\mu}$ semi-norm of the quotient $u(t,x)/B(t,x)$, it is enough
to consider the right-hand side of (32) on a domain which is now independent
of the time.

The denominator of the right-hand side of (32) can be estimated from below by
a direct computation, while we use the parabolic regularity theory developed
in [7, 32] to bound the numerator $|\hat{u}_{t,1}(1,y)-\hat{B}_{t,1}(1,y)|$.
We remark that the $C^{\mu}$-norm of $\hat{B}_{t,1}$ can be estimated by a
direct computation, while for $\hat{u}_{t,1}$ we use the fact that it is a
solution to a linear equation

$\frac{\partial u}{\partial
t}=|x|^{\gamma}\,\nabla\cdot\left(|x|^{-\beta}\,A(t,x)\,\nabla u\right)$ (33)

where the coefficient $A(t,x)=m\,u^{m-1}(t,x)$. To obtain an estimate which is
independent of $\varepsilon$, we estimate directly the $C^{\mu}$-norm of
$\hat{u}_{t,1}$ on the whole space ${\mathbb{R}}^{d}$. We apply a standard
trick in regularity theory: we cover ${\mathbb{R}}^{d}$ with subdomains of
type $B_{k}(0)\setminus B_{k/2}(0)$. In order to estimate the norm of
$\hat{u}_{t,1}$, we apply to the rescaled function $\hat{u}_{\tau,k}$ the
identity

$\lfloor\hat{u}_{t,k}(1,\cdot)\rfloor_{C^{\mu}\left(B_{1}(0)\setminus
B_{1/2}(0)\right)}=k^{\frac{\sigma}{1-m}+\mu}\,\lfloor\hat{u}_{t,1}(1,\cdot)\rfloor_{C^{\mu}\left(B_{k}(0)\setminus
B_{k/2}(0)\right)}\quad\forall\,t>0\,.$

By (29), the function $\hat{u}_{\tau,k}$ solves (33) with a bounded and
bounded away from zero coefficient $A(t,x)$. Therefore, by [7, Proposition
4.2], there exists a constant $\overline{c}_{1}$, which depends only on $d$
and $n$, such that

$\begin{split}\lfloor\hat{u}_{t,k}(1,\cdot)\rfloor_{C^{\mu}\left(B_{1}(0)\setminus
B_{1/2}(0)\right)}&\leq\overline{c}_{1}\,\|\hat{u}_{t,k}\|_{\mathrm{L}^{\infty}\left((\frac{1}{2},4)\times
B_{1}(0)\setminus B_{1/2}(0)\right)}\\\
&\leq\overline{c}_{1}\,k^{\frac{\sigma}{1-m}}\,\|\hat{u}_{t,1}\|_{\mathrm{L}^{\infty}\left((\frac{1}{2},4)\times{\mathbb{R}}^{d}\right)}\,.\end{split}$

By setting $t_{0}=0$ and letting $R\rightarrow\infty$ in [7, Inequality
(2.1)], we obtain the estimate
$\|\hat{u}_{t,1}\|_{\mathrm{L}^{\infty}\left((\frac{1}{2},4)\times{\mathbb{R}}^{d}\right)}\leq\overline{c}_{2}\,\mathcal{M}$,
for some positive constant $\overline{c}_{2}$ which depends only on $d$, $m$,
$\beta$ and $\gamma$. Combining the above estimates, we find that

$\begin{split}\lfloor\hat{u}_{t,1}(1,\cdot)\rfloor_{C^{\mu}\left({\mathbb{R}}^{d}\right)}&\leq\lfloor\hat{u}_{t,1}(1,\cdot)\rfloor_{C^{\mu}\left(B_{1}(0)\right)}+\sum_{k=1}^{\infty}\lfloor\hat{u}_{t,1}(1,\cdot)\rfloor_{C^{\mu}\left(B_{2^{k+1}}(0)\setminus
B_{2^{k}}(0)\right)}\\\
&\leq\overline{c}_{1}\,\overline{c}_{2}\,\frac{2^{\mu}}{2^{\mu}-1}\,\mathcal{M}\,.\end{split}$
(34)

In order to use Inequality (31), we need an estimate of
$\left\|{u(t,x)-B(t,x)}\right\|_{1,\gamma}$. To do so, we use the Csiszár-
Kullback-Pinsker inequality which allows us to control the evolution of
$\left\|{u(t,x)-B(t+\overline{\tau},x)}\right\|_{1,\gamma}$ where
$\overline{\tau}>0$ is a time-shift needed for the definition of the relative
entropy. Up to a scaling, $\overline{\tau}$ can be defined to be such that
$B(\overline{\tau},x)=\mathsf{g}^{2p}(x)$. It is then convenient to use the
triangle inequality as follows

$\left\|{u(t,x)-B(t,x)}\right\|_{1,\gamma}\leq\left\|{u(t,x)-B(t+\overline{\tau},x)}\right\|_{1,\gamma}+\left\|{B(t+\overline{\tau},x)-B(t,x)}\right\|_{1,\gamma}\,.$
(35)

By a simple although lengthy computation, we have that

$\left\|{B(t+\overline{\tau},x)-B(t,x)}\right\|_{1,\gamma}\leq\frac{\overline{c}}{t}\quad\forall\,t\geq
T(\varepsilon)\,.$ (36)

The Csiszár-Kullback-Pinsker inequality as in [6, Lemma 2.12] implies that
there exists an explicit constant $\mathsf{C}$, which depends only on $d$,
$m$, $\beta$ and $\gamma$, such that

$\left\|{u(t,x)-B(t+\overline{\tau},x)}\right\|_{1,\gamma}\leq\mathsf{C}\,\frac{\sqrt{\mathcal{F}[u_{0}]}}{R(t)}\,.$
(37)

Thanks to (28) it is sufficient to estimate the right-hand side of (32) in the
domain $|y|\leq 2\,Z\,\rho(\varepsilon)$ for $Z$ large enough. Combining (31)
with $p=1$, applied to the difference $\hat{u}_{t,1}-\hat{B}_{t,1}$ together
with the estimates (35), (36) and (37), we obtain for all $|y|\leq
2\,Z\,\rho(\varepsilon)$ and $t\geq T(\varepsilon)$

$\Big{|}\hat{u}_{t,1}\big{(}1,y\big{)}-\hat{B}_{t,1}\big{(}1,y\big{)}\Big{|}\leq\max\\{\overline{c},\mathsf{C}\\}\left(\frac{1}{t}+\frac{\sqrt{\mathcal{F}[u_{0}]}}{R(t)}\right)^{\vartheta}$

which allows to estimate the numerator of the right-hand side in (32), and
allows to conclude the proof of (30) by estimating the denominator of the
right-hand side in (32) by a direct computation. ∎

###  2.4 Proof of Theorem 7

###### Proof.

From Corollary 10 we deduce that

$\Big{|}\frac{u(t,x)}{B(t,x)}-1\Big{|}<\varepsilon$ (38)

for $t\geq T(\varepsilon)$ and $|x|>\rho(\varepsilon)\,R(t)$, where
$T(\varepsilon)=(1+A)^{1-m}\,O(1/\varepsilon)$. From Corollary 11, we deduce
that inequality (38) holds if $t>4\,T(\varepsilon)$, $|x|\leq
2\,\rho(\varepsilon)\,R(t)$ and $t>0$ is such that

$\frac{\mathrm{K}}{\varepsilon^{\frac{1}{1-m}}}\left(\frac{1}{t}+\frac{\sqrt{\mathcal{F}[u_{0}]}}{R(t)}\right)^{\vartheta}\,<\varepsilon\,.$

Since $R(t)\leq\left(Ct\right)^{\frac{1}{\xi}}$, for some positive constant
$C=C(d,m,\gamma,\beta)$, the last condition is satisfied if

$t\geq\frac{1+\mathcal{F}[u_{0}]^{\frac{\xi}{2}}}{\varepsilon^{\mathrm{a}}}\quad\mbox{where}\quad\mathrm{a}:=\frac{\vartheta}{\xi}\,\frac{2-m}{1-m}$
(39)

Combining the above estimate (39) with
$T(\varepsilon)=(1+A)^{1-m}\,O(1/\varepsilon)$, by elementary computations,
one finds that there exist a computable constant $\mathsf{C}$ which depends on
$d$, $m$, $\gamma$, $\beta$, $\mathcal{F}[u_{0}]$ and $A[u_{0}]$ for which
(25) holds for
$t\geq\,\mathsf{C}(A[u_{0}],\mathcal{F}[u_{0}])\,\varepsilon^{-\mathrm{a}}$.
The dependence on $\mathcal{F}[u_{0}]$ is eliminated using Lemma 6, although
more accurate estimates are obtained if the dependence on $\mathcal{F}[u_{0}]$
is kept as in [6]. This completes the proof of Theorem 7. ∎

##  3 Improved entropy – entropy production estimates

We prove Theorem 5 in Section 3.4 using an artificial dimension, entropy
methods on the time interval $(t_{\star},+\infty)$ with $t_{\star}$ given by
Theorem 7 and spectral gap estimates, which are exposed respectively in
Sections 3.1, 3.2 and 3.3.

###  3.1 An artificial dimension

Inequality (12) can be recast as an interpolation inequality with same weight
in all integrals which, in terms of scaling properties, amounts to introduce
an _artificial dimension_. To a function
$f\in\mathrm{H}^{p}_{\beta,\gamma}({\mathbb{R}}^{d})$, let us associate the
function $F\in\mathrm{H}^{p}_{\nu,\nu}({\mathbb{R}}^{d})$ with $\nu:=d-n<0$
such that

$f(x)=F\left(|x|^{\alpha-1}\,x\right)\quad\forall\,x\in{\mathbb{R}}^{d}\,,$
(40)

where

$\alpha=1+\frac{\beta-\gamma}{2}\quad\mbox{and}\quad
n=2\,\frac{d-\gamma}{\beta+2-\gamma}\,.$

Notice that $p_{\star}=n/(n-2)$. In spherical coordinates, with $r=|x|$ and
$\omega=x/r$ for any $x\in{\mathbb{R}}^{d}\setminus\\{0\\}$, let us define the
derivation operator

$\mathsf{D}_{\alpha}\kern 0.5ptU:=\left(\alpha\,\frac{\partial U}{\partial
r},\frac{1}{r}\,\nabla_{\kern-2.0pt\omega}U\right)\,.$

With $\alpha>0$ and $p\in(1,p_{\star}]$, we can rewrite (12) as

$\left\|{U}\right\|_{2p,\nu}\leq\mathsf{K}_{\alpha,n,p}\,\left\|{\mathsf{D}_{\alpha}\kern
0.5ptU}\right\|_{2,\nu}^{\theta}\,\left\|{U}\right\|_{p+1,\nu}^{1-\theta}\quad\forall\,U\in\mathrm{H}^{p}_{\nu,\nu}({\mathbb{R}}^{d})\,,$
(41)

for some optimal constant $\mathsf{K}_{\alpha,n,p}$ which is explicitly
related with the optimal constant in (12): see [4, Proposition 6]. Inequality
(41) can be interpreted as a Gagliardo-Nirenberg-Sobolev inequality in the
artificial dimension $n$. As $\alpha\neq 1$ unless $\beta=\gamma$, notice that
symmetry issues in (41) are in no way simpler than in (12). A remarkable point
is that the Aubin-Talenti type function as defined by (14) is transformed into
the more standard function

$x\mapsto\left(1+|x|^{2}\right)^{\frac{1}{1-p}}\,.$

We refer to [4, Section 2.3] and [21, Section 3.1] for further details.

Through the transformation $u(t,x)=U\left(t,|x|^{\alpha-1}\,x\right)$, a
solution $u$ of (15) is transformed into a solution of

$\frac{\partial U}{\partial t}={\mathcal{L}_{\alpha}\,}U^{m}$ (42)

where $\mathsf{D}_{\alpha}^{*}\kern 0.5pt$ denotes the adjoint of
$\mathsf{D}_{\alpha}\kern 0.5pt$ on
$\mathrm{L}^{2}({\mathbb{R}}^{d},|x|^{-\nu}\,dx)$ and
${\mathcal{L}_{\alpha}\,}=-\,\mathsf{D}_{\alpha}^{*}\,\mathsf{D}_{\alpha}\kern
0.5pt$ is an elliptic self-adjoint differential operator given in spherical
coordinates by

${\mathcal{L}_{\alpha}\,}u=\alpha^{2}\left(u^{\prime\prime}+\frac{n-1}{r}\,u^{\prime}\right)+\frac{1}{r^{2}}\,\Delta_{\omega}\,u\,.$

Functions obtained from $B(t,x)$ by reduction to the artificial dimension $n$
are self-similar Barenblatt solutions of (42). If $\mathcal{U}$ is a solution
of (42) with initial datum $\mathcal{B}$ given by (5), then
$\mathcal{U}(t,x)=R(t)^{-n}\,\mathcal{B}\big{(}x/R(t)\big{)}$ if and only if

$\frac{\mathrm{d}R}{\mathrm{d}t}=\alpha^{2}\,R^{\,n\,(1-m)-1}\,.$

If we additionally assume that $R(0)=1$, this can be solved as

$R(t):=(1+\alpha^{2}\,\xi\,t)^{1/\xi}$

and $\xi=n\,(m-m_{c})$ and $m_{c}=(n-2)/n$ as in (23). Up to a time shift,
this definition of $R(t)$ generalizes the definition of Section 2 to the case
$\alpha\neq 1$. Other self-similar Barenblatt solutions of (42) have same
scaling properties and initial data given by $\mathcal{B}$, up to a
multiplication by a constant and a scaling.

###  3.2 Flow and entropies in self-similar variables

Self-similar solutions suggest to rewrite (42) in the corresponding scales
using the _self-similar change of variables_

$U(t,x)=\frac{\lambda^{d}}{R(t)^{d}}\,V\left(\frac{1}{2}\,\log
R(t),\frac{\lambda\,x}{R(t)}\right)$ (43)

where $\lambda^{n\,(m-m_{c})}=(1-m)/(2\,m)$. Hence if $U$ solves (42), then
$V$ solves

$\frac{\partial V}{\partial t}=\mathsf{D}_{\alpha}^{*}\kern
0.5pt\cdot\left(V\left(\mathsf{D}_{\alpha}\kern
0.5ptV^{m-1}-2\,x\right)\right)\,,\quad V(t=0,\cdot)=V_{0}$ (44)

with nonnegative initial datum
$V_{0}=\lambda^{-d}\,U_{0}(\cdot/\lambda)\in\mathrm{L}^{1}({\mathbb{R}}^{d})$.
Using homogeneity and scaling properties of (44), there is no restriction to
fix $\int_{{\mathbb{R}}^{d}}{V_{0}\,|x|^{-\nu}}\,\mathrm{d}x=\mathcal{M}$,
with $\mathcal{M}$ defined by (24). As a consequence, we shall assume from now
on that

$\int_{{\mathbb{R}}^{d}}{V(t,\cdot)\,|x|^{-\nu}}\,\mathrm{d}x=\mathcal{M}\quad\forall\,t\geq
0$

without loss of generality.

The _free energy_ (or _relative entropy_) and the _Fisher information_ (or
_relative entropy production_) are defined respectively by

$\mathcal{F}[V]:=\frac{1}{m-1}\int_{{\mathbb{R}}^{d}}{\left(V^{m}-\mathcal{B}^{m}-m\,\mathcal{B}^{m-1}\,(V-\mathcal{B})\right)|x|^{-\nu}}\,\mathrm{d}x$

and, as a generalization of definition (7) of the Fisher information,

$\mathcal{I}[V]:=\frac{m}{1-m}\int_{{\mathbb{R}}^{d}}{V\,\left|\mathsf{D}_{\alpha}\kern
0.5ptV^{m-1}-\mathsf{D}_{\alpha}\kern
0.5pt\mathcal{B}^{m-1}\right|^{2}\,|x|^{-\nu}}\,\mathrm{d}x\,.$

With $V=|f|^{2\,p}$, $p=1/(2\,m-1)$, Inequality (41) is equivalent to the
_entropy – entropy production inequality_

$\mathcal{I}[V]\geq 4\,\alpha^{2}\,\mathcal{F}[V]$ (45)

_in the symmetry range_ for (12). If $V$ solves (44), it is a straightforward
computation to check that

$\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{F}[V(t,\cdot)]=-\,\mathcal{I}[V(t,\cdot)]$
(46)

after one integration by parts (which has to be justified: see [20]), and as a
consequence, we obtain that

$\mathcal{F}[V(t,\cdot)]\leq\mathcal{F}[V_{0}]\,e^{-\,4\,\alpha^{2}\,t}\quad\forall\,t\geq
0\,.$ (47)

###  3.3 An improved spectral gap

We consider the _linearized free energy_ and the _linearized Fisher
information_ given respectively by

$\mathsf{F}[h]:=\frac{m}{2}\int_{{\mathbb{R}}^{d}}{|h|^{2}\,\mathcal{B}^{2-m}\,|x|^{-\nu}}\,\mathrm{d}x\quad\mbox{and}\quad\mathsf{I}[h]:=m\,(1-m)\int_{{\mathbb{R}}^{d}}{|\mathsf{D}_{\alpha}\kern
0.5pth|^{2}\,\mathcal{B}\,|x|^{-\nu}}\,\mathrm{d}x\,.$

These quadratic forms are obtained as

$\mathsf{F}[h]=\lim_{\varepsilon\to
0}\varepsilon^{-2}\,\mathcal{F}\big{[}\mathcal{B}+\varepsilon\,\mathcal{B}^{2-m}\,h\big{]}\quad\mbox{and}\quad\mathsf{I}[h]=\lim_{\varepsilon\to
0}\varepsilon^{-2}\,\mathcal{I}\big{[}\mathcal{B}+\varepsilon\,\mathcal{B}^{2-m}\,h\big{]}\,.$

The following result is taken from [4, Proposition 4].

######  Proposition 12.

Let $d\geq 2$, $\alpha\in(0,+\infty)$, $\nu=d-n<0$ and $\delta=1/(1-m)\geq n$.
Then the _Hardy-Poincaré inequality_

$\int_{{\mathbb{R}}^{d}}{|\mathsf{D}_{\alpha}\kern
0.5pth|^{2}\,\mathcal{B}\,|x|^{-\nu}}\,\mathrm{d}x\geq\Lambda\int_{{\mathbb{R}}^{d}}{|h|^{2}\,\mathcal{B}^{2-m}\,|x|^{-\nu}}\,\mathrm{d}x$

holds for any
$h\in\mathrm{L}^{2}({\mathbb{R}}^{d},\mathcal{B}^{2-m}\,|x|^{-\nu}\,dx)$ such
that $\int_{{\mathbb{R}}^{d}}{h\,\mathcal{B}^{2-m}}\,\mathrm{d}x=0$, with an
optimal constant $\Lambda$ given by

$\Lambda=\left\\{\begin{array}[]{rl}2\,\alpha^{2}\,(2\,\delta-n)&\mbox{if}\quad
0<\alpha^{2}\leq\frac{(d-1)\,\delta^{2}}{n\,(2\,\delta-n)\,(\delta-1)}\,,\\\\[6.0pt]
2\,\alpha^{2}\,\delta\,\eta&\mbox{if}\quad\alpha^{2}>\frac{(d-1)\,\delta^{2}}{n\,(2\,\delta-n)\,(\delta-1)}\,,\end{array}\right.$

where $\eta$ is given by

$\eta=\sqrt{\tfrac{d-1}{\alpha^{2}}+\big{(}\tfrac{n-2}{2}\big{)}^{2}}-\tfrac{n-2}{2}=\tfrac{2}{2+\beta-\gamma}\sqrt{d-1+\big{(}\tfrac{d-2-\beta}{2}\big{)}^{2}}-\tfrac{d-2-\beta}{2+\beta-\gamma}\,.$

###  3.4 Proof of Theorem 5

The constant in the entropy – entropy production inequality (45) can be
improved for a solution to (44) if time is large enough and the initial datum
$v_{0}$ is such that $A[v_{0}]<\infty$.

######  Lemma 13.

Assume that $(\beta,\gamma)\neq(0,0)$ satisfies (19). Let $\alpha$ and $\nu$
be as in Section 3.1 and define
$\zeta:=\left(2\,(1-m)\,\Lambda-4\,\alpha^{2}\right)/4>0$ with $\Lambda$ as in
Proposition 12. If $V$ is a solution to (44) with an initial datum $V_{0}$
such that

$\sup_{R>0}R^{\frac{n(m-m_{c})}{1-m}}\int_{|x|>R}V_{0}(x)\,|x|^{-\nu}\,\mathrm{d}x<\infty$

where $m_{c}=(n-2)/n$ as in (23), then there exists $T>0$ such that

$\left(4\,\alpha^{2}+2\,\zeta\right)\mathcal{F}[V(t,\cdot)]\leq\,\mathcal{I}[V(t,\cdot)]\quad\forall\,t\geq
T\,.$ (48)

###### Proof.

By applying the change of variables (40) and (43), solutions to (17) are
tranformed in solutions to (44) and in particular the shifted time-dependent
Barenblatt profile $B(t+\overline{\tau},x)$ is transformed into the stationary
solution $\mathcal{B}$ defined in (5). As in Section 2, $\overline{\tau}>0$ is
such that $B(\overline{\tau},x)=\mathsf{g}^{2p}(x)$ and $B$ is as in (22).
Since

$\left\|\frac{B(t,x)}{B(t+\overline{\tau},x)}\right\|_{\mathrm{L}^{\infty}({\mathbb{R}}^{d})}\leq
c_{1}\quad\mbox{and}\quad\left\|\frac{B(t+\overline{\tau},x)-B(t,x)}{B(t+\overline{\tau},x)}\right\|_{\mathrm{L}^{\infty}({\mathbb{R}}^{d})}\leq\frac{c_{2}}{t}$

for $t>0$ large enough as a consequence of Theorem 7, we find that for any
$\varepsilon>0$ small enough there exists an explicit $T=T(\varepsilon)>0$
such that

$\sup_{x\in{\mathbb{R}}^{d}}\left|\frac{V(t,x)}{\mathcal{B}(x)}-1\right|\leq\varepsilon\quad\forall\,t\geq
T\,.$

According to [5, Lemma 18], one can deduce that

$\mathsf{I}[h(t,\cdot)]\leq\frac{(1+\varepsilon)^{3-2m}}{(1-\varepsilon)}\,\mathcal{I}[V(t,\cdot)]+\,\varepsilon\,s_{\varepsilon}\,\mathsf{F}[h(t,\cdot)]\,,$

where $h(t,\cdot)=V(t,\cdot)\,\mathcal{B}^{m-2}-\mathcal{B}^{m-1}$ and
$s_{\varepsilon}$ is a positive function of $\varepsilon$ such that
$\lim_{\varepsilon\to 0}s_{\varepsilon}>0$. As in the proof of [5, Lemma 19],
using Proposition 12 and the fact that
$(1+\varepsilon)^{m-2}\leq\mathcal{F}[V(t,\cdot)]/\mathsf{F}[h(t,\cdot)]\leq(1-\varepsilon)^{m-2}$
by [5, Lemma 15], we obtain

$\big{(}2\,(1-m)\,\Lambda-\,\rho_{\varepsilon}\,\varepsilon\big{)}\,\mathcal{F}[V(t,\cdot)]\leq\,\mathcal{I}[V(t,\cdot)]\,,$

where $\rho_{\varepsilon}$ is bounded and stays bounded as
$\varepsilon\rightarrow 0$. A discussion has to be made depending on the cases
in Proposition 12. When
$2\,(1-m)\,\Lambda=4\,\alpha^{2}\,(1-m)\,(2\,\delta-n)$, it follows from
$m>m_{1}$ that $(1-m)\,(2\,\delta-n)>1$ and we find
$2\,(1-m)\,\Lambda-\,\rho_{\varepsilon}\,\varepsilon>4\,\alpha^{2}+\zeta$ for
$\varepsilon$ small enough. When $2\,(1-m)\,\Lambda=4\,\alpha^{2}\,\eta$, the
result follows because the condition $\eta>1$ is equivalent to (19). ∎

###### Proof of Theorem 5.

As a consequence of (48) and (46), we have

$\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{F}[V(t,\cdot)]\leq-\,(4\,\alpha^{2}+\zeta)\,\mathcal{F}[V(t,\cdot)]\,.$

Inequalities (20) and (21) are consequences of Grönwall’s lemma. ∎

##  4 A stability result for GNS

In this section, we deal with the non-weighted case $(\beta,\gamma)=(0,0)$.
Detailed proof are given in [6]. Here we simply outline the main steps of the
proofs and emphasize the differences with the weighted case studied in Section
3.

###  4.1 A quotient estimate

For any function $v\neq\mathcal{B}$ such that
$\int_{{\mathbb{R}}^{d}}{v}\,\mathrm{d}x=\mathcal{M}$, with $\mathcal{M}$
defined by (24), which is smooth enough and sufficiently decaying at infinity
let us consider the quotient

$\mathcal{Q}[v]:=\frac{\mathcal{I}[v]}{\mathcal{F}[v]}$

where $\mathcal{F}[v]$ and $\mathcal{I}[v]$ are defined respectively by (4)
and (7).

######  Lemma 14.

Assume that $v$ solves (6). Then we have

$\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{Q}[v(t,\cdot)]\leq\mathcal{Q}\,(\mathcal{Q}-4)\quad\forall\,t\geq
0\,.$

Equation (6) corresponds to the fast diffusion equation without weights,
_i.e._ , $\beta=\gamma=0$ and the result follows from

$\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{I}[v(t,\cdot)]\leq-\,4\,\mathcal{I}[v(t,\cdot)]\,,$

which itself arises from the _carré du champ_ method adapted to nonlinear
flows. See for instance [20] and references therein. With
$(\beta,\gamma)\neq(0,0)$, such an estimate is so far formal.

###  4.2 Scheme of the proof of Theorem 1

In the absence of weights the result follows by considering the improved
spectral gap obtained for the flow (6). We shall only sketch the main steps of
the proof and the interested reader may find the whole proof in [6, Chp. 5].
We first consider the subcritical case $1<p<p^{\star}$.

Step 0: normalization. We notice that the deficit functional $\delta[\cdot]$
defined in (1.1) is invariant by translations so that, without loss of
generality, we assume that $|f|^{2p}$ has zero center of mass. As well, there
is no harm to assume
$\int_{{\mathbb{R}}^{d}}{|f|^{2p}}\,\mathrm{d}x=\int_{{\mathbb{R}}^{d}}{\mathsf{g}^{2p}}\,\mathrm{d}x$
where $\mathsf{g}$ is as in (2), since the general case can be recover by
scalings, see [6, Chapter 5]. We learn from [6, Lemma 1.12] that

$\frac{p+1}{p-1}\,\delta[f]=\mathcal{I}[v]-4\,\mathcal{F}[v]$

with the notation $v=|f|^{2p}$ and $p=1/(2\,m-1)$. Notice that translations
are not allowed if $(\beta,\gamma)\neq(0,0)$, but that
$\int_{{\mathbb{R}}^{d}}{x\,|f|^{2p}}\,\mathrm{d}x=0$ is not required in that
case.

Step 1: improved entropy – entropy production inequality in the asymptotic
time layer. Let $v(t)$ be the solution the Cauchy problem (6) with initial
datum $v(0)=|f|^{2p}$. By Theorem 7 we know that, for $\varepsilon>0$ small
enough, the relative error satisfies
$|(v(t,\cdot)-\mathsf{g}^{2p})/\mathsf{g}^{2p}|<\varepsilon$ for any
$t>t_{\star}$ where $t_{\star}$ is as in (25). So, we are in the position of
using Lemma 13: under the center of mass condition, the improved entropy –
entropy production inequality

$\mathcal{I}[v(t,\cdot)]\geq(4+\zeta)\,\mathcal{F}[v(t,\cdot)]\quad\forall\,t\geq
t_{\star}$ (49)

holds with $\zeta=2\,d\,(m-m_{1})$. The computation of the precise value of
$\zeta$ can be found in [6, Proposition 2.10]. The additional constraint
$\int_{{\mathbb{R}}^{d}}{x\,v_{0}}\,\mathrm{d}x=0$ is needed to have
$\zeta>0$.

Step 2: improved entropy – entropy production inequality in the initial time
layer. By integrating the differential inequality of Lemma 14 backwards on
$[0,t_{\star}]$, under the assumption that $Q(v(t_{\star}))\geq 4+\zeta$, one
finds

$\mathcal{I}[v(t,\cdot)]\geq(4+\mu)\,\mathcal{F}[v(t,\cdot)]\quad\forall\,t\in[0,t_{\star}]\quad\mbox{where}\quad\mu=\frac{4\,\zeta\,e^{-4\,t_{\star}}}{4+\zeta-\zeta\,e^{-4\,t_{\star}}}\,.$

As a consequence, the improvement obtained in the asymptotic time layer
$[t_{\star},+\infty)$ can be transferred to the initial time layer
$[0,t_{\star}]$ and up to the initial datum, with a smaller improvement of the
constant. By multiplying the improved inequality by $4/(4+\mu)$, we obtain

$\mathcal{I}[v]-4\,\mathcal{F}[v]\geq\frac{\mu}{4+\mu}\,\mathcal{I}[v]$

with $v=v(t,\cdot)$ for any $t\geq 0$ and, as a special case, for $v=v_{0}$.
Notice that the constant $\mu/(4+\mu)$ can be estimated explicitly since the
dependence of $t_{\star}$ is given by (25).

Step 3: rescaling and proof of Inequality (3). So far we have proven, with the
above notation, that

$\frac{p+1}{p-1}\,\delta[f]\geq\mathcal{C}\,\mathcal{I}[v]\,,$

where $\mathcal{C}=4/(4+\mu)$. To obtain inequality (3) we simply observe that

$\mathcal{I}[v]\geq\inf_{\varphi\in\mathfrak{M}}\int_{{\mathbb{R}}^{d}}{\big{|}(p-1)\,\nabla
f+f^{p}\,\nabla\varphi^{1-p}\big{|}^{2}}\,\mathrm{d}x$

where the infimum is taken on the manifold of all optimal functions for (1).
To obtain the result in its general form it is enough to rescale and go back
to the original variables.

Step 4: the critical case $p=p^{\star}$. Due to the lack of an improved
spectral gap in Step 1 under the previous scheme, with mass and center of mass
constraints, a second moment constraint is also needed. This amounts to change
the evolution equation to a new one which allows to control the evolution of
the second moment as well: in practice we need to rescale Equation (6) as done
in [6, Chapter 5]. Entropy and entropy production should be optimized, _i.e._
, considered with respect to the _best matching Barenblatt_ profile which is
$\mathsf{g}^{2p}$ up to a time-dependent rescaling.This rescaling amounts to
an additional time shift. Explicit estimates of the time shift requires some
more work based on a system of ODEs and a phase portrait analysis. For more
details see [6, Chapter 6].

###  4.3 Proof of Corollaries 2 and 3

###### Proof of Corollary 2.

The entropy of the initial datum $\mathcal{F}[v_{0}]$ is finite under the
current assumptions as a consequence of Lemma 6. The proof of inequality (8)
follows from identity (46) by combining a Grönwall argument with inequality
(49). In this way we obtain

$\mathcal{F}[v(t,\cdot)]\leq\mathcal{F}[v_{0}]\,e^{-\,(4+\zeta)\,t}\quad\forall\,t\geq
0\,.$ (50)

∎

###### Proof of Corollary 3.

Let us consider the sub-critical case $m_{1}<m<1$. We can obtain inequality
(47) from (50) as follows. First, we can rescale the initial datum in such a
way that $\mathcal{F}[v_{0}]=\mathcal{F}_{\star}[v_{0}]$. Then it is enough to
take the infimum over all the Barenblatt profiles in the left-hand side of
(50).

In the critical case $m=m_{1}$, we have to deal with an additional time-
dependent scaling in order to control the evolution of the second moment. For
more details see [6, Chapter 5]. ∎

##  5 A conjecture on stability for Caffarelli-Kohn-Nirenberg inequalities

In Gagliardo-Nirenberg-Sobolev inequalities (1) and in Caffarelli-Kohn-
Nirenberg inequalities (12), the invariance under scalings plays an important
role, as well as in the corresponding fast diffusion equations (10) and (15).
This explains why entropy methods are so efficient to get sharp results for
the best constants as discussed in [20]. A key feature is the _carré du champ_
method, which has been rigorously implemented for (10) in [13], in the context
of parabolic equations. The regularity and decay estimates needed to justify
the _carré du champ_ method in the context of the fast diffusion flow
associated with (12) are so far missing, although some partial estimates are
known from [20, 24]. This is why the symmetry results in [21] were based on an
elliptic version of the method, which formally also applies to parabolic
equation (15). Proving that all necessary integrations by parts can be
justified would establish the following conjecture:

_For some $\zeta>0$, Inequality (20) holds for any $t\geq 0$._

In other words, this means that in Theorem 5, one can take $T=0$. Indeed, the
_carré du champ_ estimate would allow us to extend the estimate on the
asymptotic time layer $(T,+\infty)$ to the initial time layer $(0,T)$ and find
a smaller but still constructive value for $\zeta$ depending only on the
initial datum, by the same scheme as in [6]. A straightforward consequence
would be an improved entropy – entropy production inequality that would
provide us with a _stability result with a constructive estimate_ for
Caffarelli-Kohn-Nirenberg inequalities (12) similar to the result of Theorem
1. With this method, the stability would be measured by a relative Fisher
information as for (1). Such a result is to be expected as, in the critical
case of (12), a stability result _without_ constructive estimate has already
been established by F. Seuffert in [31].

## Acknowledgments

M.B. has been supported by the Projects MTM2017-85757-P and
PID2020-113596GB-I00 (Ministry of Science and Innovation, Spain) and the
Spanish Ministry of Science and Innovation, through the “Severo Ochoa
Programme for Centres of Excellence in R&D” (CEX2019-000904-S) and by the E.U.
H2020 MSCA programme, grant agreement 777822. J.D. has been supported by the
Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR).
N.S. has been supported by the Spanish Ministry of Science and Innovation,
through the FPI-grant BES-2015-072962, associated to the project
MTM2014-52240-P (Spain) and by the E.U. H2020 MSCA programme, grant agreement
777822, by the Project EFI (ANR-17-CE40-0030) of the French National Research
Agency (ANR), by the DIM Math-Innov of the Region Île-de-France and by the
“Fondation Mathématique Jacques Hadamard”. The authors thank an anonymous
referee for his detailed reading, comments and suggestions.
© 2022 by the authors. This paper may be reproduced, in its entirety, for non-
commercial purposes.

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