Asymptotic Analysis of the Fast Diffusion Equation

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Outline of the talk Introduction Asymptotic behaviour of the Fast Diffusion Equation,m<1 The Linear Problem The Nonlinear Entropy Method

Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

Matteo Bonforte

Departamento de Matemáticas, Universidad Autónoma de Madrid,

Campus de Cantoblanco 28049 Madrid, Spain

[email protected]

http://verso.mat.uam.es/~matteo.bonforte

2020 Fields Medal Symposium

celebrating the mathematical work of Alessio Figalli The Fields Institute, Toronto, Canada

Online Event, October 19 - 23, 2020

http://www.fields.utoronto.ca/activities/20-21/fieldsmedalsym

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2010: BIRS Banff

2011: UIMP Santander

2016: CIME Cetraro

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2012: The Theory…

2016: The Application!

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2018: Rio, after the Medal…

… and after-after the Medal!

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2019: Barcelona DHC

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Outline of the talk Introduction Asymptotic behaviour of the Fast Diffusion Equation,m<1 The Linear Problem The Nonlinear Entropy Method Outline of the talk

Introduction

The Dirichlet Problem for Diffusion Equations The asymptotic behaviour of the Heat Equation

The asymptotic behaviour of the Porous Medium Equation Some properties of Solutions to Fast Diffusion Equations Asymptotic behaviour of the Fast Diffusion Equation

Time rescaling and the stationary problem Previous results

Rates of convergence

Stationary Solutions and Semilinear Elliptic Equations The Linear Problem

Linearization and Spectrum

Linear Entropy Method and Improved Poincaré Inequalities Assumption(H2)is generically true

The Nonlinear Entropy Method

Comparing linear and nonlinear quantities

Almost orthogonality and improved Poincaré inequalities Possible blow up when almost orthogonality fails Almost orthogonality improves along the nonlinear flow

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Outline of the talk Introduction Asymptotic behaviour of the Fast Diffusion Equation,m<1 The Linear Problem The Nonlinear Entropy Method The Dirichlet Problem for Diffusion Equations

The Dirichlet Problem for Diffusion Equations inΩ⊂R^N We consider, in a bounded and smooth domainΩ⊂R^Nsolutions to







∂_τu= _m¹∆ (u^m) =∇ · u^m−1∇u

, in(0,+∞)×Ω

u(0,·) =u0, inΩ

u=0, on(0,+∞)×∂Ω

Existence and uniquenessof weak solutionsfor this problem is well understood:

m=1: Heat Equation. (Fourier)

m>1: Porous Medium regime(slow diffusion, finite speed of propagation) 0<m<1: Fast Diffusion regime(extinction in finite time)

Complete references for PME/FDE: Vázquez books (2006-07).

m≤0: Ultra Fast Diffusion regime.(possible non-existence) Dirichlet problem: solutions fail to exists, Vázquez (1992).

Cauchy problem: non-existence for L¹-data, Vázquez (1992).

Optimal L^p-conditions on data for existence, Daskalopoulos-DelPino (1994-97).

Neumann/periodic: Gradient flow approach (JKO-scheme), Iacobelli (2019), Iacobelli-Patacchini-Santambrogio (2019).

Dynamical and other boundary conditions: Schimperna-Segatti-Zelik (2012-16).

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





∂_τu= _m¹∆ (u^m) =∇ · u^m−1∇u

, in(0,+∞)×Ω

u(0,·) =u0, inΩ

u=0, on(0,+∞)×∂Ω

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





∂_τu= _m¹∆ (u^m) =∇ · u^m−1∇u

, in(0,+∞)×Ω

u(0,·) =u0, inΩ

u=0, on(0,+∞)×∂Ω

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





∂_τu= _m¹∆ (u^m) =∇ · u^m−1∇u

, in(0,+∞)×Ω

u(0,·) =u0, inΩ

u=0, on(0,+∞)×∂Ω

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





∂_τu= _m¹∆ (u^m) =∇ · u^m−1∇u

, in(0,+∞)×Ω

u(0,·) =u0, inΩ

u=0, on(0,+∞)×∂Ω

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





∂_τu= _m¹∆ (u^m) =∇ · u^m−1∇u

, in(0,+∞)×Ω

u(0,·) =u0, inΩ

u=0, on(0,+∞)×∂Ω

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





∂_τu= _m¹∆ (u^m) =∇ · u^m−1∇u

, in(0,+∞)×Ω

u(0,·) =u0, inΩ

u=0, on(0,+∞)×∂Ω

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





∂_τu= _m¹∆ (u^m) =∇ · u^m−1∇u

, in(0,+∞)×Ω

u(0,·) =u0, inΩ

u=0, on(0,+∞)×∂Ω

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About Nonlinear Diffusions

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Asymptotic behaviour of Heat and Porous Medium Equations

Asymptotic behaviour of the Heat Equationut= ∆u. (Fourier∼1822) Letu0≥0 and let(λk, φk)be the eigen-elements of the Dirichlet Laplacian:

u(τ,x) =

∞

X

k=1

e^−λ^k^τˆu0,kφ_k(x) where ˆu0,k= Z

Ω

u0φ_kdx From the above formula it is quite simple to deduce that

e^λ¹^τu(τ,·)−−−−→

τ→∞ ˆu0,1φ1 in L^p(Ω),∀p∈[1,∞]

Actually, one can do better:

Relative Error Convergence

u(τ,·) e^λ¹^τˆu0,1φ1

−1 _L_∞

≤e^−(λ²^−λ¹^)τ

∞

X

k=2

e^−(λ^k^−λ²^)τ|ˆu0,k|

φk

φ1

_L∞(Ω)

| {z }

<+∞

Essential ingredients: second spectral gapλ2−λ1>0, and

boundary behaviour of eigenfunctions:|φk|.|φ1| Sharp result. Relies on representation formula, Fourier and Spectral analysis More general linear operators can be treated essentially in the same way.

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u(τ,x) =

∞

X

k=1

Ω

e^λ¹^τu(τ,·)−−−−→

τ→∞ ˆu0,1φ1 in L^p(Ω),∀p∈[1,∞]

−1 _L_∞

≤e^−(λ²^−λ¹^)τ

∞

X

k=2

e^−(λ^k^−λ²^)τ|ˆu0,k|

φk

φ1

_L∞(Ω)

| {z }

<+∞

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u(τ,x) =

∞

X

k=1

Ω

e^λ¹^τu(τ,·)−−−−→

τ→∞ ˆu0,1φ1 in L^p(Ω),∀p∈[1,∞]

−1 _L_∞

≤e^−(λ²^−λ¹^)τ

∞

X

k=2

e^−(λ^k^−λ²^)τ|ˆu0,k|

φk

φ1

_L∞(Ω)

| {z }

<+∞

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u(τ,x) =

∞

X

k=1

Ω

e^λ¹^τu(τ,·)−−−−→

τ→∞ ˆu0,1φ1 in L^p(Ω),∀p∈[1,∞]

−1 _L_∞

≤e^−(λ²^−λ¹^)τ

∞

X

k=2

e^−(λ^k^−λ²^)τ|ˆu0,k|

φk

φ1

_L∞(Ω)

| {z }

<+∞

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u(τ,x) =

∞

X

k=1

Ω

e^λ¹^τu(τ,·)−−−−→

τ→∞ ˆu0,1φ1 in L^p(Ω),∀p∈[1,∞]

−1 _L_∞

≤e^−(λ²^−λ¹^)τ

∞

X

k=2

e^−(λ^k^−λ²^)τ|ˆu0,k|

φk

φ1

_L∞(Ω)

| {z }

<+∞

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u(τ,x) =

∞

X

k=1

Ω

e^λ¹^τu(τ,·)−−−−→

τ→∞ ˆu0,1φ1 in L^p(Ω),∀p∈[1,∞]

−1 _L_∞

≤e^−(λ²^−λ¹^)τ

∞

X

k=2

e^−(λ^k^−λ²^)τ|ˆu0,k|

φk

φ1

_L∞(Ω)

| {z }

<+∞

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Outline of the talk Introduction Asymptotic behaviour of the Fast Diffusion Equation,m<1 The Linear Problem The Nonlinear Entropy Method Asymptotic behaviour of Heat and Porous Medium Equations

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Asymptotic behaviour of the Porous Medium Equationu_t= ∆u^m,m>1 The asymptotic behaviour is described in terms of the “Friendly Giant”:

U(τ,x) =S(x)τ^−1/(m−1), with U(0,x) = +∞. Here,Sis theuniquenonnegative solution to the stationary problem

(EDP) −∆S^m=cS inΩ, S=0 on∂Ω,

withc=1/(m−1)>0,sincem>1.

Logarithmic time rescaling:

t= log(τ+1) and w(t,x) =τ^1/(m−1)u(τ,x), we transform the parabolic problem into







∂tw(t,x) = ∆w^m(t,x) +cw(t,x), (t,x)∈(0,∞)×Ω, w(t,x) =0, (t,x)∈(0,∞)×∂Ω, w(0,x) =u0(x), x∈Ω.

The separation of variables solutionU becomes stationary.

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Asymptotic behaviour of the Porous Medium Equationu_t= ∆u^m,m>1 The asymptotic behaviour is described in terms of the “Friendly Giant”:

U(τ,x) =S(x)τ^−1/(m−1), with U(0,x) = +∞. Here,Sis theuniquenonnegative solution to the stationary problem

withc=1/(m−1)>0,sincem>1.

Logarithmic time rescaling:

t= log(τ+1) and w(t,x) =τ^1/(m−1)u(τ,x), we transform the parabolic problem into







∂tw(t,x) = ∆w^m(t,x) +cw(t,x), (t,x)∈(0,∞)×Ω, w(t,x) =0, (t,x)∈(0,∞)×∂Ω, w(0,x) =u0(x), x∈Ω.

The separation of variables solutionU becomes stationary.

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Asymptotic behaviour of the Porous Medium Equation (continued)

Relative Error Convergence, rescaled variables

w(t,·) S −1

L^∞(Ω)

. e^−t for allt1

Relative Error Convergence, original variables

u(τ,·) U(τ,˙)−1

L^∞(Ω)

≤ 2 m−1

t0

t0+τ for allτ ≥t0

t0∼ R

Ωu0φ1dx−(m−1)

,whereφ1is the first eigenfunction of the Laplacian.

The decay rate 1/τis sharp: it is realized by

U(τ+1,x) = S(x) (1+τ)^m−1¹

with initial datum U(1,x) =S(x).

Result obtained first by Aronson-Peletier (1981), then generalized by Vázquez (2004). New (quantitative) proof for more general (even nonlocal) diffusion equations of PME-type, by M.B.-Figalli-Sire-Vázquez (2015-18).

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w(t,·) S −1

L^∞(Ω)

. e^−t for allt1

u(τ,·) U(τ,˙)−1

L^∞(Ω)

≤ 2 m−1

t0

t0∼ R

U(τ+1,x) = S(x) (1+τ)^m−1¹

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w(t,·) S −1

L^∞(Ω)

. e^−t for allt1

u(τ,·) U(τ,˙)−1

L^∞(Ω)

≤ 2 m−1

t0

t0∼ R

U(τ+1,x) = S(x) (1+τ)^m−1¹

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w(t,·) S −1

L^∞(Ω)

. e^−t for allt1

u(τ,·) U(τ,˙)−1

L^∞(Ω)

≤ 2 m−1

t0

t0∼ R

U(τ+1,x) = S(x) (1+τ)^m−1¹

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Some properties of Solutions to FDE

Some Properties of Solutions to the FDE, 0<m<1.

The initial datum is chosen to be

0≤u0∈L^r(Ω) with r≥1 and r> N(1−m)

2 ,

hence the corresponding solution is bounded and nonnegative.

Notice thatr>1 only whenm<^N−2_N : in the very fast diffusion range.

The massR

Ωu(y, τ)dyis NOT preserved along the evolution hence solutionsextinguish in finite time

∃T =T(u0) : u(τ,·)≡0 ∀τ≥T

(Consequence of Sobolev and Poincaré inequalities).

Whenu0≥0,solutions are indeed strictly positiveinΩ×(0,T) as a consequence of parabolic (intrinsic) Harnack inequalities:

Good FDE regime: whenm∈ ^N−2_N ,1 ,

Dibenedetto, Gianazza, Kwong, Vespri (Indiana 1991, Ann.SNS 2010, LNM2012) Very FDE regime: whenm∈ 0,^N−2_N

,Bonforte-Vázquez (Adv. Math. 2010)

Bounded positive solutions are regular:

DiBenedetto-Kwong-Vespri (Indiana 1991)

They are regular up to the boundary: at leastC¹(Ω).

They are smooth in the interior:C^∞t,x(Ω×(0,T)), and even analytic!

Sharp boundary regularity recently obtained by Jin-Xiong(Preprint 2020)

The question is: what happens close to the extinction timeT?

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2 ,

The massR

∃T =T(u0) : u(τ,·)≡0 ∀τ≥T

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2 ,

The massR

∃T =T(u0) : u(τ,·)≡0 ∀τ≥T

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2 ,

The massR

∃T =T(u0) : u(τ,·)≡0 ∀τ≥T

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2 ,

The massR

∃T =T(u0) : u(τ,·)≡0 ∀τ≥T

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2 ,

The massR

∃T =T(u0) : u(τ,·)≡0 ∀τ≥T

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2 ,

The massR

∃T =T(u0) : u(τ,·)≡0 ∀τ≥T

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Outline of the talk Introduction Asymptotic behaviour of the Fast Diffusion Equation,m<1 The Linear Problem The Nonlinear Entropy Method Time rescaling and the stationary problem

Asymptotic behaviour of the Fast Diffusion Equation,m<1 The rescaled Problem and stationary solutions







uτ = ∆(u^m), u(0,·) =u0, u|∂Ω≡0,

−−−−−−−−−−→

Time-Rescaling







wt= ∆(w^m) +cw, w(0,·) =u0, w|∂Ω≡0, whereτ∈[0,T(u0)) log-rescaling

−−−−−→

“rallenty” t∈[0,∞),and w(t,x)=e

t (1−m)T u

T−Te^−t/T,x

and c:= 1

(1−m)T. or

u(τ,x) = T−τ

T _1−m¹

w(t,x) with t=Tlog

T

T−τ

. The asymptotic behaviour is related to the stationary equation

through the separate variables solution:

U(τ,x)=S(x) T−τ

T 1−m¹

−−−−−−−→

Rescaling S(x)

Here, a crucial exponent naturally arises: m_s= (N−2)/(N+2).

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





uτ = ∆(u^m), u(0,·) =u0, u|∂Ω≡0,

−−−−−−−−−−→

Time-Rescaling







−−−−−→

t (1−m)T u

T−Te^−t/T,x

and c:= 1

(1−m)T. or

u(τ,x) = T−τ

T _1−m¹

w(t,x) with t=Tlog T

T−τ

U(τ,x)=S(x) T−τ

T 1−m¹

−−−−−−−→

Rescaling S(x)

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





uτ = ∆(u^m), u(0,·) =u0, u|∂Ω≡0,

−−−−−−−−−−→

Time-Rescaling







−−−−−→

t (1−m)T u

T−Te^−t/T,x

and c:= 1

(1−m)T. or

u(τ,x) = T−τ

T _1−m¹

T−τ

U(τ,x)=S(x) T−τ

T 1−m¹

−−−−−−−→

Rescaling S(x)

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





uτ = ∆(u^m), u(0,·) =u0, u|∂Ω≡0,

−−−−−−−−−−→

Time-Rescaling







−−−−−→

t (1−m)T u

T−Te^−t/T,x

and c:= 1

(1−m)T. or

u(τ,x) = T−τ

T _1−m¹

T−τ

U(τ,x)=S(x) T−τ

T 1−m¹

−−−−−−−→

Rescaling S(x)

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Statement of the Problem

We are interested in describing the behaviour near the extinction time:

u(τ,x)

∼

???

τ→T⁻

S(x) T−τ

T 1−m¹

After rescaling, the precise question becomes:

Problem.Given a nonnegative solutionwto the (rescaled) FDE:







wt= ∆(w^m) +cw, w(0,·) =u0, w_|∂Ω≡0,

is it true thatthere exists a stationary solutionSsuch that

w(t,·)−−−→^???

t→∞ S

andin which sense? If yes, what can we say aboutconvergence rates?

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Statement of the Problem

u(τ,x)

∼

???

τ→T⁻

S(x) T−τ

T 1−m¹







wt= ∆(w^m) +cw, w(0,·) =u0, w_|∂Ω≡0,

w(t,·)−−−→^???

t→∞ S

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Statement of the Problem

u(τ,x)

∼

???

τ→T⁻

S(x) T−τ

T 1−m¹







wt= ∆(w^m) +cw, w(0,·) =u0, w_|∂Ω≡0,

w(t,·)−−−→^???

t→∞ S

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Outline of the talk Introduction Asymptotic behaviour of the Fast Diffusion Equation,m<1 The Linear Problem The Nonlinear Entropy Method Previous results

Previous results

Recall that positive bounded stationary solution exist only whenm ∈ (ms,1), with ms= (N−2)/(N+2).Hence the analysis will be resticted to this range.

(First Pioneering Result) Berryman-Holland(ARMA 1980)

Letwbe a bounded solution to the rescaled problem, andm∈(m_s,1).

Then for any sequence of timest_n→ ∞asn→ ∞, there exist a stationary solutionSsuch that

w(tn) ^W

1,2 0 (Ω)

−−−−−→

n→∞ S.

DIFFICULTY:Stationary solutionsneednotto beunique!

Different time sequences can a priori converge to different stationary solutions.

However, the asymptotic profile may be unique also when the set of stationary solutions contains more than one element.

(Uniqueness of asymptotic profile) Feiresl-Simondon(JDDE 2000)

Then there existsonestationary solutionSsuch that w(t)−−−→^C(Ω)

t→∞ S.

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Previous results

w(tn) ^W

1,2 0 (Ω)

−−−−−→

n→∞ S.

t→∞ S.

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Previous results

w(tn) ^W

1,2 0 (Ω)

−−−−−→

n→∞ S.

t→∞ S.

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Previous results

w(tn) ^W

1,2 0 (Ω)

−−−−−→

n→∞ S.

t→∞ S.

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(Convergence in Relative Error) M.B.-Grillo-Vázquez(JMPA 2012)

Letm∈(ms,1)and letube the solution to the Dirichlet problem and T=T(m,d,u0)be its extinction time. LetSbe the positive classical solution to the elliptic problem (EDP), such thatkw(t)−Sk_L^∞(Ω)→0 as t→ ∞. Then we have

lim

τ→T⁻

u(τ,·) U(τ,·)−1

_L_∞_(Ω)

= lim

t→∞

w(t,·) S −1

_L_∞_(Ω)

=0

whereU(τ,x) =S(x) [(T−τ)/T]^1/(1−m).

Equivalently, the followingimproved Global Harnack Principle (GHP) holds c(τ)⁻¹S(x) (T−τ)^1/(1−m)≤u(τ,x)≤c(τ)S(x) (T−τ)^1/(1−m). with

0<c(τ)−−−−→

τ→T⁻ 1, and S(x)dist x, ∂Ω_m¹

The GHP (with constantsc(τ)6→1) was firstly proven by DiBenedetto-Kwong-Vespri (Indiana 1991) and used as a key-tool for higher regularity estimates. It has been used more recently by Jin-Xiong to prove sharp boundary regularity (Preprint 2020).