Smoothing Effect, Positivity and Harnack Inequalities for...

Fast Diffusion Problems and Local Weak Solutions Upper Estimates Lower Estimates Harnack Inequalities Panorama and open problems Short review of related works Fast p-Laplacian.

Smoothing Effect, Positivityand Harnack Inequalities

for Very Fast Evolution Equations

Matteo Bonforte

Departamento de Matemáticas,Universidad Autónoma de Madrid,

Campus de Cantoblanco28049 Madrid, Spain

email: [email protected](Joint work with Juan Luis Vázquez

)FRONTIERS OF MATHEMATICS AND APPLICATIONS

SANTANDER, 9-13 AUGUST 2010


Fast Diffusion Equation m < 1

Fast Diffusion Equation

ut =1m

∆(um) , in Q = (0, T)× Ω ⊆ (0,+∞)× Rd ,

u(0, ·) = u0 , u0 ∈ Lrloc(Ω) and m < 1

Finite Extinction Time (FET)

∃ T > 0 such that u(t, x) = 0 , ∀ x ∈ Ω and ∀ t ≥ T

The homogeneous Dirichlet problem on bounded domains Ω always extinguishin finite time T when 0 ≤ m < 1, and when m < 0, there is the effect ofimmediate extinction, T = 0.

The Cauchy problem in Rd extinguish in finite time when the initial datumbelongs to Lrc (Rd), with rc = d(1− m)/2, for any m < 1, but rc > 1 ifm > mc.

Conservation of mass does not allow extinction in finite time, e.g. homogenousNeumann problem, large solutions.


Local Weak Solutions for Fast Diffusion Equation

u ∈ C(

0, T; L2loc(Ω)

)and |u|m ∈ L2

loc

(0, T; W1,2

loc (Ω))

such that, for every open bounded subdomain [t1, t2]× K ⊂ (0, T]× Ω, we have∫K

u(t2)ϕ(t2) dx−∫

Ku(t1)ϕ(t1) dx +

∫ t2

t1

∫K

(uϕt +∇um · ∇ϕ) dx dt = 0,

for any test function ϕ ∈ W1,2loc

(0, T; L2(K)

)∩ L2

loc

(0, T; W1,2

0 (K))

.

Comparison principle does not hold for local weak solution, no boundary datais specified

Bounded local weak solutions are continuous in Q = [0, T]× Ω.Hölder continuity: DiBenedetto et al. 1988, 1992Our results do not depend neither on an explicit modulus of continuity noron the Hölder continuity.Indeed our Harnack inequalities imply Hölder continuity.

Using continuity, guaranteed by our sharp local smoothing effects (upperbounds), we can prove a local comparison argument that allows us to get sharplower bounds and as a consequence new forms of Harnack inequalities.


Local comparison argument

Restriction of any continuous local weak solution

(RDP)

∂tu = ∆um , in (0, T)× BR0 ,u(0, ·) = u0χBR0

in BR0 ,

u(t, x) = uloc(t, x) in (0, T)× ∂BR0

uloc is the continuous local weak solution under consideration and u ≡ uloc

Minimal Dirichlet Problem

(mDP)

∂tu = ∆um , in (0, T)× BR0 ,u(0, ·) = u0χBR

in BR0 , 0 < 2R < R0 ,

u(t, x) = 0 for any (t, x) ∈ (0, T)× ∂BR0

We can conclude that u(t, x) ≤ u(t, x) on [0, T)× BR0 .Minimal Life Time: Tm(u(0)) ≤ T(u0).

Maximal Dirichlet Problem - Large Solutions

(MDP)

∂tu = ∆u m , in (0, T)× BR0 ,u(0, ·) = u0χBR0

in BR0 ,

u(t, x) = +∞ in (0, T)× ∂BR0

We can conclude that u(t, x) ≤ u(t, x) on [0, T)× BR0


Theorem. (Local Smoothing effects)

Let r ≥ 1 if m > mc = (d − 2)/d or r > rc = d(1− m)/2 if m ≤ mc. Let u be alocal weak solution to the FDE in the cylinder (0, T)× Ω ⊆ (0,+∞)× Rd. Thenthere are positive constants C1, C2 such that for any 0 < R < dist(x0, ∂Ω) we have

supx∈BR/2

u(t, x) ≤ C1

tdϑr

[∫BR

|u0(x)|r dx]2ϑr

+ C2

[ tR2

] 11−m

.

where ϑr = 1/(2r − d(1− m)) = 1/2(r − rc), and the constants Ci depend on m, dand r. We give explicit expression for Ci.

We recover the well known smoothing effect in Rd by letting R→ +∞This result holds for any local weak solution, thus providing (Hölder) continuity of anylocal weak solutions (c.f. DiBenedetto et al.)This result holds for large solutions⇒ existence and interior boundedness.

Small improvements:Supremum in [ε, t]× BR in the r.h.s , for any ε ∈ (0, t).Radii: R/2 can be improved up to any R0 < R = dist(x0, ∂Ω)The result extends to m ≤ 0 and to more general operators of the formut = ∇ · a(t, x, u,∇u).

Similar results previously proved for mc < m < 1, by Herrero - Pierre, DiBenedetto -Gianazza - Vespri, Daskalopulos - Kenig [...]


Theorem. (Asymptotic behaviour of the Large-FDE problem)

Let r ≥ 1 if m > mc = (d − 2)/d or r > rc = d(1− m)/2 if m ≤ mc. Consider thelarge-FDE problem on a bounded domain Ω ⊂ Rd

∂tu = ∆u m , in (0, T)× Ω ,u(0, ·) = u0 in Ω ,u(t, x) = +∞ in (0, T)× ∂Ω

then there exists a continuous local weak solution u defined in the cylinder[0,∞)× Ω ⊆ (0,+∞)× Rd. Moreover there are positive constants C1, C2 such that

C2

[t

dist(x, ∂Ω)2

] 11−m

≤ u(t, x) ≤ C1

tdϑr

[∫Ω

|u0(x)|r dx]2ϑr

+ C2

[t

dist(x, ∂Ω)2

] 11−m

.

where ϑr = 1/(2r − d(1− m)) = 1/2(r − rc), and the constants Ci depend on m, dand r. We give explicit expression for Ci.

This is the the sharp asymptotic behaviour of the Large-FDE problem.The upper estimates (LSE) imply existence and interior boundedness for theLarge-FDE problem.The result extends to m ≤ 0 and to more general operators of the formut = ∇ · a(t, x, u,∇u).


Theorem (Aronson-Caffarelli type Estimates)

Let 0 < m < 1 and let u be a local weak solution to the FDE over (0, T)× Ω. Let x0

be a point in Ω such that B5R ⊂ Ω. Then the following inequality holds for all0 < t < T

R−d∫

BR(x0)

u0(x) dx ≤ C1 R−2/(1−m) t1

1−m + C2 T1

1−m R−2 t−m

1−m um(t, x0).

with C1 and C2 given positive constants depending only on d.If moreover u0 ∈ Lrc

loc(Ω)we have

R−d‖u0‖L1(BR(x0)) ≤ C1 R−2/(1−m) t1

1−m + C3 ‖u0‖Lrc (BR(x0))R−2 t−m

1−m um(t, x0).

These estimates are the analogous of the celebrated Aronson-Caffarelliestimates, valid for the slow diffusion case, m > 1, namely

R−d∫

BR(x0)

u0(x) dx ≤ C1 R2/(m−1) t−1

m−1 + C2 R−dtd/2u1+(d(m−1)/2)(t, x0).

In the case m > 1 the AC estimates define the so-called waiting time

tc = c(m, d)‖u0‖1−mL1(BR(x0))

R2+d(m−1).

namely a time that we have to wait in order that positivity takes place, in viewof the slow diffusion.


Theorem (Local Positivity Estimates)

Let 0 < m < 1 and let u be a local weak solution to the FDE over (0, T)× Ω. Let x0

be a point in Ω such that B6R(x0) ⊂ Ω , and let 2R < R0 ≤ dist(x0, ∂Ω). There existsa time t∗ ∈ (0, T] such that for all t ∈ (0, t∗] ⊆ (0, T]

um(t, x0) ≥ C′1 R2−d‖u0‖L1(BR)T−1

1−m tm

1−m .

where C′1 > 0 depends only on d and the critical time

t∗ := kd (R0 − 2R)2 Vol(BR0 \ BR

)m−1‖u0‖1−mL1(BR)

≤ T

where kd > 0 depends only on d. If moreover u0 ∈ Lrloc(Ω), r ≥ maxrc, 1

u(t, x0) ≥ C′1

[R

dr ‖u0‖L1(BR)

Rd ‖u0‖Lr(BR)

] 1m [ t

R2

] 11−m

.

The role of the critical time is different from the slow diffusion casePositivity without dependence on T or Tm for general initial data is false ingeneral. We provide a counterexample and also DiBenedetto-Gianazza-Vespri.The assumption u0 ∈ Lr

loc, r > rc is necessary to avoid Tm in the estimates whenm < mc. Since rc > 1 iff m < mc, upper estimates on the minimal life time Tm

in terms of L1-norm, are not possible.


The Good Fast Diffusion Range mc < m < 1

(i) Sharp upper and lower estimates for the extinction time for the Dirichlet problem on any ballBR of the form:

c1‖u0‖1−mL1(BR/3)

R2−d(1−m) ≤ T ≤ c2‖u0‖1−mL1(BR)

R2−d(1−m).

(ii) In that range of m our lower estimates imply the lower Harnack inequalities of DiBenedettoet al., in the form

u(t, x0) ≥ cm,d

[ tR2

] 11−m

for any x ∈ BR and any 0 < t < t∗ := kd (R0 − 2R)2 Vol(BR0 \ BR

)m−1‖u0‖1−mL1(BR)

.

Our upper and lower estimates (+work) imply Global Harnack Principles:ON DOMAINS Ω. E. DiBenedetto, Y. C. Kwong and V. Vespri (1992).For any ε ∈ (0, T) there exist constants c,C depending only upon d, m, ‖u0‖1+m,diam(Ω), ∂Ω and ε, such that for all (t, x) ∈ (0, T)× Ω, t > ε

c dist(x, ∂Ω)1/m (T − t)1/(1−m) ≤ u(t, x) ≤ C dist(x, ∂Ω)1/m (T − t)1/(1−m)

ON THE WHOLE SPACE Rd . M. B. - J. L. Vázquez, (2006)Let u0 ∈ L1(Rd), u0 ≥ 0 and u0(x) |x|2/(1−m) ≤ A , for any |x| ≥ R0. Then, for anyε > 0 there exist constants τ1, τ2, M1 and M2, such that

B(t − τ1, x; M1) ≤ u(t, x) ≤ B(t + τ2, x; M2), ∀(t, x) ∈ (ε,∞)× Rd

whereB(t, x; M) =

[b1

(Mm−1t

) 22−d(1−m)

+ b2|x|2] −1

(1−m)t

11−m , bi = bi(m, d).


Main Ingredients for Positivity

The Flux Lemma for Minimal Dirichlet Problem

If u is a positive smooth solution of the mDP in (0, T]× BR0 with extinction timeT > 0. Then, the following estimate holds true

k0 (R0 − 2R)2∫

BR0

u(s, x) dx ≤∫ T

s

∫A0

um dx dt,

for any 0 ≤ s ≤ T , and any 0 < 2R < R0, and for a suitable constant k0 = k0(d).

Estimates for Extinction Time

t∗ = k0 (R0 − 2R)2

[ ∫BR

u0 dx

Vol(BR0 \ BR

)]1−m

≤ T ≤ 8 [d(1− m)− 2]S22

(d − 2)2(1− m)‖u0‖1−m

rc

Local Aleksandrov Principle for Minimal Dirichlet Problem

For any t > 0, 0 < 2R < R0 one has u(t, x0) ≥ u(t, x2) for any t > 0 and for anyx2 ∈ BR0 (x0) \ B2R0 (x0). Hence,

u(t, x0) ≥ |BR0 (x0) \ B2R0 (x0)|−1∫

BR0 (x0)\B2R0 (x0)

u(t, x) dx


Local Smoothing Effect + Lower Estimates⇒ Harnack Inequalities

Local Smoothing

supx∈BR/2

u(t, x) ≤ C1

‖u0‖2rϑrLr(BR)

tdϑr

+C2

[ tR2

] 11−m

Local Positivity

infx∈BR

u(t, x) ≥ C3

[R

dr ‖u0‖L1(BR)

Rd ‖u0‖Lr(BR)

] 1m

×[ t

R2

] 11−m

Harnack Inequalities

supx∈BR/2

u(t, x) ≤ C1

‖u0‖2rϑrLr(BR)

tdϑr+ C4

[R

dr ‖u0‖Lr(BR)

Rd ‖u0‖L1(BR)

] 1m

infx∈BR

u(t, x)

infx∈BR(x0)

u(t0 ± θ, x

)≥ H u

(t0, x0

)where ϑq = 1/(2q− d(1− m)) for any q > 0. Moreover θ = δu

(t0, x0

)1−mR2 andH = H(u0, t0, t∗,R) are explicitly calculated.


The intriguing realm of Intrinsic Harnack Inequalities

Classical forms of Harnack inequalities do not hold for the nonlinear diffusion

The intrinsic geometry enters the game: for the good fast diffusion range wehave the result of DiBenedetto and collaborators (1994, 2007).There exist positive constants c and δ depending only on m, d, such that for all(t0, x0) ∈ Q = (0, T)× Ω and all cylinders of the type

I8R(t0, x0) =(

t0 − c u(t0, x0)1−m(8R)2, t0 + c u(t0, x0)

1−m(8R)2)×B8R(x0) ⊂ Q ,

we havec u(t0, x0) ≤ inf

x∈BR(x0)u(t, x)

for all times t0 − δ u(t0, x0)1−m R2 < t < t0 + δ u(t0, x0)

1−m R2. The constants δand c tend to zero as m→ 1 or as m→ mc .

In the linear case, i.e. m→ 1, only forward Harnack inequalities hold:

c u(t0, x0) ≤ infx∈BR(x0)

u(t0 + ϑ, x)

for suitable times ϑ = δR2, where c, δ > 0 depends only on d.


Intrinsic Harnack Inequalities of Forward-Backward-Elliptic typeLet 0 < m < 1 and consider a local nonnegative weak solution u of the FDE definedin a cylinder Q = (0, T)× Ω, taking initial data u(0, x) = u0(x) in Lr

loc(Ω)

, withr = 1 if mc < m < 1 or r > rc if 0 < m ≤ mc. Also, let x0 be a point in Ω and let6R ≤ dist(x0, ∂Ω). There exists constants h1 , h2 depending only on m, d, p, suchthat, for any ε ∈ [0, 1] the following inequality holds

infx∈BR(x0)

u(t ± θ, x

)≥ h1 ε

2pϑp1−m

‖u(t0)‖L1(BR)Rdp

‖u(t0)‖Lr(BR)Rd

2pϑp+ 1m

u(t, x0)

for any

t0 + εt∗(t0) < t ± θ < t0 + t∗(t0) , t∗(t0) = h2 R2−d(1−m)‖u(t0)‖1−mL1(BR(x0))

.

The above estimate is completely of local type, since it involves only local quantities.The fact that the intrinsic cylinder [t0 + εt∗(t0), t0 + t∗(t0)]× BR0 (x0) is contained in(0, T]× Ω is a consequence, not as an hypothesis.We have shown that the size of the intrinsic cylinders is proportional to a ratio of local Lp

norms. Note that in the supercritical range it simplifies and only depends on the local L1

norm.


In the supercritical range mc < m < 1, we can let r = 1 to get

infx∈BR(x0)

u(t ± θ, x

)≥ h1 ε

2pϑp1−m u

(t, x0)

for any t0 + εt∗(t0) < t ± θ < t0 + t∗(t0).We recover the above mentioned results of DiBenedetto et al.Joining the upper and lower estimates for the Cauchy Problem, we obtain the GlobalHarnack principle.

In the subcritical range 0 < m ≤ mc, the Harnack estimates cannot have a universalconstant independent of u0, (counterexamples).

The quantity ε represents an arbitrary small waiting time, that is needed in order for theregularization to take place and to allow quantitative intrinsic Harnack inequalities.

Backward Harnack inequalities are a bit surprising, but they reflect a typical feature of thefast diffusion processes, that is the extinction phenomena, namely

infx∈BR(x0)

u(t − θ, x

)≥ h1 ε

2pϑp1−m


‖u(t0)‖Lp(BR)Rd

2pϑp+ 1m

u(t, x0)

for any t0 + εt∗(t0) < t− θ < t0 + t∗(t0) This inequality is compatible with the fact thatthe solution extinguish at some later time, remaining strictly positive before. Thisbackward inequality is typical of singular equation and can not hold for the degenerate-porous media- case m > 1, neither for the linear heat equation case, m = 1.The same remark applies for the Elliptic Harnack inequality, that is when θ = 0.


We provide a form of Harnack inequalities of forward, backward and elliptic type,avoiding the intrinsic framework, and the waiting time ε ∈ [0, 1].

An alternative form of Harnack InequalitiesUnder the running assumptions, there exists positive constants C1, C2 and h2

depending only on m, d and p such that

supx∈BR

u(t, x) ≤ C1

tdϑp‖u(t0)‖2pϑp

p + C2

‖u(t0)‖Lp(BR)Rd


1m

infx∈BR

u(t ± ϑ, x)

for any

0 ≤ t0 < t ± ϑ < t0 + t∗(t0) ≤ T , t∗(t0) = h2 R2−d(1−m)‖u(t0)‖1−mL1(BR(x0))

,

where ϑp = 1/(2p− d(1− m)).

This form of Harnack inequalities implies Hölder continuity.


THE SIZE OF INTRINSIC CYLINDERS. The new critical time t∗,

t∗ :=kd

2(R0 − 2R)2

Vol(BR0 \ BR

)1−m

[∫BR

u(t0) dx]1−m

gives a quantitative estimate on the maximum size of the intrinsic cylinders:(t0 − δu

(t0, x0

)1−mR2, t0 + δu(t0, x0

)1−mR2)×BR(x0) ⊆ (t0−t∗, t0+t∗)×BR(x0)

in the supercritical fast diffusion range this time can be chosen a priori just interms of the initial datum, but in the subcritical range its size changes with time;the diffusion is so fast that the initial information is lost after some time, whichis represented by t∗.

Never forget that a large class of solutions extinguish in finite time.


(I) Good Fast Diffusion Range: m ∈ (mc, 1) and p ≥ 1. Local smoothing effectholds, also Reverse smoothing effect. Thus Intrinsic Harnack Inequalities ofvarious types: Forward in time for small times; Elliptic, backward and forwardHarnack Inequalities for intermediate times. For times close to the extinctiontime, in case extinction occurs, Elliptic Harnack inequalities hold up toextinction. The new results allow to recover the older, with a different proof.

(II) Very Fast Diffusion Range: m ∈ (0,mc) with local integrability exponentp ≥ pc > 1 . Local smoothing effect and the Lower estimates hold. For anypositive time we have the Aronson-Caffarelli estimates.For for small intrinsiccylinders, we have the Intrinsic Parabolic Harnack inequalities ofElliptic-Backward-Forward type. An open problem is to pass from local toglobal estimates in this very fast diffusion range.


(c) Critical case: m = mc and p > pc = 1. The local upper and lower estimates of zone (II)apply, and also Harnack inequalities. Work in progress: how to pass from local to globalestimates, lower bounds with super-exponential time decay.

(III) Negative exponent range: m ≤ 0 with p > pc. The Local Smoothing effect still valid, theonly known local upper bound. NO positivity result is known in this range, we cannottreat this case. Solutions of the homogeneous Dirichlet problem on bounded domainsvanish instantaneously. Open problem: find positivity and a posteriori Harnackinequalities.

(IV) Negative range: m ≤ 0 with p < pc. NO Smoothing Effect is true, u0 not in Lp withp > pc. Sols of the Cauchy problem with data in Lp(Rd) will vanish instantaneously andalso sols of homogeneous Dirichlet problem. Open problem: positivity. NO Harnackinequalities since solution may not be neither locally bounded.

(V) Very Fast Diffusion Range m ∈ (0,mc) with small integrability exponent p ∈ [1, pc]. NOsmoothing effect (Backward effects), since initial data are not in Lp with p > pc. Lowerestimates are as in (II). NO Harnack inequalities since solution may not be neither locallybounded.


Short review of related worksBertsch et al. (1990): pressure equation v = u1−m they cover the whole fast diffusionrange 0 < m < 1 in terms of viscosity solutions.DiBenedetto, Kwong and Vespri (1991): Dirichlet problem on a bounded domain.Analiticity of positive solutions and Elliptic-Forward Intrinsic Harnack inequalities in thegood range mc < m < 1. Global Harnack Principle in the wider range ms < m < 1.DiBenedetto and Kwong (1992): Intrinsic Harnack Inequalities of Forward type in thegood fast diffusion range mc < m < 1.The power ms = (d − 2)/(d + 2): Del Pino and Saez (2001), asymptotics of theevolutionary Yamabe problem, Elliptic Harnack inequalities.Bonforte, Vázquez (2006), in the good FDE range mc < m < 1. Quantitative Elliptic andForward Harnack Inequalities, and Global Harnack Principle for Cauchy problem on Rd .DiBenedetto, Gianazza and Vespri, (2007). In the good FDE range mc < m < 1:Harnack inequalities of Forward, Elliptic and Backward type for a a class of singularoperartors of the form ut = ∇ · a(t, x, u,∇u).M. Bonforte, J. L. Vázquez, Positivity, local smoothing, and Harnack inequalities for veryfast diffusion equations, Advances in Math. 223 (2010), 529–578.DiBenedetto, Gianazza and Vespri, (Preprint, 2009). Harnack Type Estimates and HölderContinuity for Non-Negative Solutions to Certain Sub-Critically Singular ParabolicPartial Differential Equations. Ask Vincenzo details ;-)Problem. None of the above results considers the problem of finding suitable Harnackinequalities when the time approaches the finite extinction time. For the Dirichletproblem on domains this is has been done in a recent paper by B.-Grillo-Vázquez.


The Fast p-Laplacian 1 < p < 2 (M.B, R. Iagar, J. L. Vázquez)

M. Bonforte, R. G. Iagar, J. L. Vázquez, Local smoothing effects, positivity, and Harnackinequalities for the fast p -Laplacian equation, To appear in Advances in Math. (2010).

Fast p-Laplacian Equation

ut = ∆p(u) = ∇ ·

(∣∣∇u∣∣p−2∇u

), in Q = (0, T)× Ω ⊆ (0,+∞)× Rd ,

u(0, ·) = u0 , u0 ∈ Lrloc(Ω) and 1 < p < 2

Local Smoothing, Positivity, and Intrinsic Harnack Inequalities similar to the FDE.

Nice local energy inequality

Let u be a continuous local weak solution of the fast p-Laplacian equation over the domainΩ ⊆ Rd , with 1 < p < 2, and let 0 ≤ ϕ ∈ W2,2

0 (Ω) be any admissible test function. Then thefollowing inequality holds:

ddt

∫Ω|∇u|pϕ dx ≤ −

pd

∫Ω

(∆pu)2ϕ dx +p2

∫Ω|∇u|2(p−1)∆ϕ dx,

in the sense of distributions in D′ (0, T). This inequality holds also in the case p = 1.

The above inequality implies that continuous local weak solution are indeed strong solutions;we prove a bit more: ut ∈ L2

loc ⊂ L1loc. This result answers a question posed by P. Lindqvist

about finding an estimate to prove that ut ∈ L2loc.


THE ENDThank you very much!!

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