Large time behavior for nonlinear higher order convection-diffusion equations

Large time behavior for nonlinear higher order convection–diffusion equations

Comportement asymptotique pour des equations de convection–diffusion nonlinéaires d’ordre supérieur

Mahmoud Qafsaoui

Ecole Supérieure des Techniques Aéronautiques et de Construction Automobile, 34, rue Victor Hugo, 92300 Levallois Perret, France Received 11 January 2005; received in revised form 5 September 2005; accepted 22 September 2005

Available online 9 February 2006

Abstract

We study the large time asymptotic behavior, inL^p(1p∞), of higher derivativesD^γu(t)of solutions of the nonlinear equation

u_t+Tu=a· ∇^θ(ψ (u)) onRⁿ×(0,∞),

u(0)=u₀∈L¹(Rⁿ), (1)

where the integersnandθare bigger than or equal to 1,ais a constant vector inR^pwithp=_θ+n−1 n−1

=^(θ_θ!⁺_(nⁿ₋⁻₁₎¹⁾_!^!. The functionψ

is a nonlinearity such thatψ∈C^θ(R)andψ (0)=0, andT is a higher order elliptic operator with nonsmooth bounded measurable coefficients onRⁿ. We also establish faster decay whenu₀∈L¹(Rⁿ)∩L^∞(Rⁿ).

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

Nous étudions le comportement asymptotique, dansL^p(1p∞), des dérivées d’ordre supérieurD^γu(t)des solutions de l’équation nonlinéaire (1), oùn∈N^∗,θ∈N^∗etaest un vecteur constant deR^pavecp=_θ+n−1

n−1

. La fonctionψest nonlinéaire vérifiantψ∈C^θ(R)etψ (0)=0, etT est un opérateur elliptique d’ordre supérieur à coefficients peu réguliers dansRⁿ. Nous étudions également le cas particulier oùu₀∈L¹(Rⁿ)∩L^∞(Rⁿ).

©2006 Elsevier Masson SAS. All rights reserved.

MSC:35B40; 35K25; 35K57; 35G05; 35A08

Keywords:Generalized convection–diffusion equations; Higher order operators; Heat kernel; Asymptotic behavior;L^p-estimates

Mots-clés :Équations de convection–diffusion généralisées ; Noyau de la chaleur ; Opérateurs d’ordre supérieur ; Comportement asymptotique ; Estimations-L^p

E-mail address:[email protected] (M. Qafsaoui).

0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2005.09.008

1. Introduction

Our aim is to study the asymptotic behavior of higher derivatives of solutions of the Cauchy problem for the generalized convection–diffusion equation (1) using sufficiently smooth nonlinearitiesψ. A typical example of (1) is given by

u_t+

^mA^m

u=a· ∇^θ ψ (u)

onRⁿ×(0,∞), u(0)=u₀∈L¹

Rⁿ ,

whereAis a bounded measurable positive function independent of timet.

Let us start by mentioning some works which inspired ours. Escobedo and Zuazua studied in [3] the large time behavior of solutions of the Cauchy problem for the convection–diffusion equation (1) withT = −,ψ (u)= |u|^q−1u andθ=1. They proved, for q >1, the existence and the uniqueness of a classical solutionu∈C([0,∞);L¹(Rⁿ)) such thatu∈C((0,∞);W^2,p(Rⁿ))∩C¹((0,∞);L^p(Rⁿ))for everyp∈(1,∞). The argument used relies essentially on the classical Banach fixed point theorem. They also showed that, fort large, this solution behaves like the heat kernelK_t which can be regarded as the fundamental solution of the heat equation with the Dirac mass as initial data.

More precisely, under the assumptionq >1+1/n, ifMdenotes the mass ofu₀(M=

Rⁿu₀(x)dx) then the solution usatisfies for allp∈ [1,∞],

t→+∞lim t^n2(1−p1)u(t )−MK_t

p=0.

They also obtained a faster decay in the particular case when the initial datau₀belongs toL¹(Rⁿ;1+ |x|)∩L^q(Rⁿ).

The techniques used rely on standard heat kernel estimates on the integral representation of the solution. Subsequently, they finished their work by an extension to the more general equationu_t−u=a· ∇(ψ (u))whereψis an arbitrary sufficiently smooth nonlinearity.

In [2], Biller, Karch and Woyczy´nski studied the large time behavior of solutions of the Lévy conservation laws ut +Lu+ ∇ ·ψ (u)=0 with initial datau0, where ψ is a nonlinearity and(−L)is the generator of a positivity- preserving symmetric Lévy semigroup onL¹(Rⁿ). In particular, they showed that in the case where the symbolAof the operatorLsatisfiesA(ζ )∼ |ζ|^ιfor|ζ|<1 (0< ι <2) andA(ζ )∼ |ζ|²for|ζ|>1, and under the assumptions ψ∈C²withψ(0)=0 andu0∈L¹(Rⁿ)∩L^∞(Rⁿ), the following holds

t→+∞lim t^nι(1−p1)u(t )−e^−tLu₀

p=0 for everyp∈ [1,∞]

as for the corresponding linear equation, and ifF :=_∞

0

Rⁿψ (u(y, s))dyds,

t→+∞lim t^nι(1−

1

p)+¹ιu(t )−e^−tLu0+F·

∇e^−tL

p=0 for everyp∈(1,∞]

resulting from the presence of the nonlinear term.

In [4], we considered the equation u_t+Ltu=a∇u onRⁿ×(0,∞),

u(0)=u₀∈L¹ Rⁿ

, (2)

whereLt =L^∗₀bL0 (see below for the definition of L0) andb is a positive bounded function such that b(x, t )= b(x+at ). We showed that the derivativesD^γuof order less than or equal to 2m−1 of the solution to (2) have an asymptotic behavior similar to the one of the corresponding derivatives of the heat kernel with speeda,

tlim→∞t^{4mn(1−p1)+|}

γ|

4mD^γ_xu(x, t )−MD^γ_xK_t(x+at )

p=0, (3)

for allp∈ [1,∞]. The method used to derive this result was the simple change of variablesv(x, t ):=u(x−at, t ) which reduces the study to the heat equation

v_t+L0v=0 onRⁿ×(0,∞), v(0)=v₀=u₀.

We have shown (Proposition 5, [4]) that the solutionvsatisfies

tlim→∞t^{4mn(1−1p)+|}

γ|

4mD_x^γv−MD_x^γK_t

p=0, (4)

for allp∈ [1,∞]and allγ∈Nⁿsuch that|γ|2m−1.

Taking into account this linear transformation, the principal key to obtain (4) (and then (3)) was the following estimates ([4], Theorem 7),

D_x^γK(t )∗u₀−MD_x^γK(t )

p

⎧⎨

⎩

Ct^{−4mn(1−p1)−|}

γ|

4m−4m¹ u0L¹(Rⁿ;|x|) if|γ|<2m−1, Ct^{−4mn(1−p1)−|}

γ|

4m−_8m^ν u_0L¹_(Rⁿ_;|x|^ν/2₎ if|γ| =2m−1

(5)

valid foru0∈L¹(Rⁿ;1+ |x|),ν∈(0,1), for allt >0 and allp∈ [1,∞]. The exact value ofνis given in [1,4]. The result is then extended to the caseu0∈L¹(Rⁿ)by a simple density argument.

It is worth mentioning that in the same paper, we pointed out that (3) also holds for equations of type (2) associated to higher order operators of the form

|α|=|β|=mD^αa_αβD^β with bounded uniformly continuous (BUC), vanish mean oscillation (VMO) or bounded mean oscillation (BMO) coefficientsa_αβ. For the last case, a small BMO-norm for the coefficients is required.

In this paper, we deal with the general equation (1). In order to make our presentation clear, we divide our study into two parts. The first one concerns the asymptotic behavior when u₀ belongs toL¹(Rⁿ). More precisely, under some assumptions onψ(respectivelyψ), the solution of (1) verifies for allp∈ [1,∞]and allt >0,

tlim→∞t^{4mn(1−1p)+|γ|4m}D_x^γu(x, t )−MD_x^γK_t(x)

p=0,

for allγ∈Nⁿsuch that|γ| +θ2m−1 (respectively|γ| +θ=2m). The techniques used are different of those used in [4] and are in the same spirit as those appearing in, e.g., [3,2].

In the second part, we consider the caseu₀∈L¹(Rⁿ)∩L^∞(Rⁿ)and we establish a faster decay of ordert^−1/4m under the condition|ψ (t )|Ct^{1+(4m−θ )/n}ξ(t ), whereξ is a continuous and nondecreasing function which vanishes whentgoes to 0 (see Proposition 2.2). We also deal with the asymptotics due to the nonlinearityψ,

t^{4mn(1−p1)+|}

γ|+θ

4m

D_x^γu(t )−D_x^γe^−tTu₀+

D

ψ u(y, s)

dyds

a∇^θ

D^γ_xe^−tT

p

→0

whent→ ∞and whereD= [0,∞] ×Rⁿ. This is valid under assumptionsp >1,|γ| +θ2m−1 and|ψ (t )|Ct^q forq >1+(4m)/n.

Now, after describing the problem and before stating our results, let us supply a few notations which will be used throughout this paper. A part of them was used above.

For a multi-index λ=(λ1, . . . , λn)∈Nⁿ, we set |λ| =λ1+ · · · +λn. For anyx =(x1, . . . , xn)∈Rⁿ and any multi-indexλ∈Nⁿ, we haveD_x^λ= ^∂λ1

∂x₁^λ1 · · · ^∂λn

∂x^λnn

and we will often writeD^λinstead ofD_x^λwhen there is no risk of confusion. By∇^muwe denote the vector(D^λu)_|λ|=m.

We shall use the classical definition for the Sobolev spaceW^m,p,m∈Zandp∈ [1,∞]. In particular, the notation H^mstands forW^m,2. Norms inL^p-spaces will be denoted by · p. We shall also use the weighted spaceL¹(Rⁿ;1+

|x|)= {f ∈L¹(Rⁿ),

Rⁿ|f (x)|(1+ |x|)dx <∞}equipped with the normfL¹(Rⁿ;|x|)=

Rⁿ|f (x)||x|dx.

To complete our notation,Cwill denote a generic constant whose value may change from line to line.

In accordance with the notation above, we give some properties related to the class of higher order elliptic operators studied here. More details can be found in [1] and [4]. For the reader’s convenience, we recall the construction of the class of operators and the related properties useful for our study.

Letm∈N^∗anda_αβ(x)be bounded measurable functions onRⁿwhereα, β∈Nⁿare of lengthm. Set Q(u, v)=

Rⁿ

|α|=|β|=m

a_αβ(x)D^βu(x)D^αv(x)dx

for allu, v∈H^m(Rⁿ). The formQis continuous onH^m(Rⁿ)and then by a variation on the Lax–Milgram lemma, there exists a unique operatorL:H^m(Rⁿ)→H^−m(Rⁿ)linear and continuous such that for allu, v∈H^m(Rⁿ), we

haveLu, v =Q(u, v). We writeL=(−1)^m

|α|=|β|=mD^α(aαβD^β). Here,, stands for the usual scalar product onL².

We suppose that the class of operatorsLis elliptic in the sense of the Gårding inequality (i.e., there exists a constant δ >0 such that for allu∈H^m(Rⁿ),Q(u, u)δ∇^mu²₂). Then, as a consequence of this inequality, the operatorL restricted to the domain{u∈H^m(Rⁿ)|Lu∈L²(Rⁿ)}is maximal accretive and(−L)is the generator of a contraction semigroup e^{−t L}onL²(Rⁿ).

Now, let us come to our class of operators T appearing in (1). They are defined by T =L^∗₀AL₀ where A∈ L^∞(Rⁿ,R)is such thatAσ >0,L^∗₀is the adjoint ofL₀and

L₀=(−1)^m

|α|=|β|=m

a_αβD^α+β

is a particular case of operatorsLdefined above and is associated to thepositive constantscoefficientsaαβ.

ByK_t(x, y)∈D(Rⁿ×Rⁿ), we denote the distributional kernel of the semigroup e^−tT and we refer to it as to the heat kernel ofT. We have shown in ([1], Proposition 51) the following useful estimates on the heat kernel.

Proposition 1.1.There exist constantsCandc >0such that for allt∈(0,∞)and allx∈Rⁿ, we have D_x^λK_t(x, y)Ct^−n+|λ|4m Gm,c

|x−y| t^1/4m

, (G)

for any multi-indexλ∈Nⁿsuch that|λ|2m−1, and whereGm,δ(y)=exp(−δy^4m4m−1)forδ >0.

Note that details on the restriction|λ|2m−1 can be found in ([1], Proposition 51 and/or [4], Proposition 2).

Also, the reader interested in full informations on the properties of the semigroup e^−tT (and then of the heat kernel K_t(x, y)) can see [1] and [5]. In particular, let us mention the following useful result which will be used to obtain estimates on the time derivatives of the heat kernel (see, for example, [1] Lemma 33).

Lemma 1.1.If the kernel ofe^−tT satisfies the estimates(G), then the same estimates hold for the kernel oft_dt^de^−tT. Also, we have established in [1] thatD^λe^−tT mapsL²intoL^∞with estimates

D^λe^−tT

2,∞Ct^{−8mn−|λ|4m},

where · p,qdenotes the norm fromL^p intoL^q. Since the same holds for the adjoint operatorT^∗, the duality then entails the ultracontractivity propertyD^λe^−tT :L¹→L^∞with estimates

D^λe^−tT

1,∞Ct^−n+|λ|4m .

We have also shown the Hölder regularityD^λe^−tT :L¹→ ˙C^0,ν for|λ| =2m−1 andν∈(0,1), whereC˙^0,ν is the homogeneous Hölder space.

In [5], a part of our study has concerned the extension to the derivatives taken simultaneously with respect toxand yand we have obtained

D^λe^−tTD^μ

1,∞Ct^{−n+|λ4m|+|μ|} with the corresponding estimates on the kernel

D_x^λD_y^μK_t(x, y)Ct^{−n+|λ4m|+|μ|}Gm,c

|x−y| t^1/4m

for all|λ|,|μ|2m−1, and D^λe^−tTD^μ

L¹→ ˙C^0,ν Ct^{−n+|λ|+|4mμ|+ν}

when|λ| =2m−1,|μ|2m−1 or|λ|2m−1,|μ| =2m−1.

From now on,K_t(x)=K(t, x)stands forK_t(x,0).

2. Main results 2.1. Cauchy problem

Regarding the existence, uniqueness and regularity results, the classical Banach fixed point argument (see for example [3,2,6]) yields

Proposition 2.1.There exists a unique solutionu∈C([0,∞);L¹(Rⁿ))to(1)such thatu∈C((0,∞);W^4m,p(Rⁿ))∩ C¹((0,∞);L^p(Rⁿ))for allp∈(1,∞). This solution satisfies the conservation of mass property:

Rⁿ

u(x, t )dx=

Rⁿ

u0(x)dx for allt0,

and theL¹-contraction property:

u(t )

1u₀1 for allt0.

To show the conservation integral property, we integrate (1) with respect tox and we obtain d

dt

Rⁿ

u(x, t )dx+

Rⁿ

Tu(x, t )dx=0

since

Rⁿa· ∇^θ(ψ (u(x, t )))dx=0. On the other hand,

Rⁿ

Tu(x, t )dx=L^∗₀w(0, t )=P(0)w(0, t )=0,

wherefˆdenotes the Fourier transform offinRⁿ,P(ζ )=

jζ_j^αi+βiandw=AL0u. Therefore,_dt^d

Rⁿu(x, t )dx=0.

The proof of theL¹-contraction property is a simple adaptation of the classical argument used in ([3], Proposi- tion 1).

2.2. Asymptotics with initial datau₀∈L¹(Rⁿ)

Lemma 2.1.Letu₀∈L¹(Rⁿ). Then, for allp∈ [1,∞]there existsC=C(p, n, m) >0such that the solutionuof(1) satisfies for allt >0,

u(t )

pCt^{−4mn(1−p1)}u01.

The proof of this result is a simple adaptation of the argument used in ([2], Theorems 3.2–3.3 and Corol- lary 3.2) since e^−tT2,∞Ct^−8mn ande^−tT1,∞Ct^−4mn. The later implies u(t )2ct^−8mn u₀1 and then u(t )_∞c(t /2)^−8mn u(t /2)2ct^−4mn u₀1. Eventually, by interpolation and theL¹-contraction property, we ob- tainu(t )pu(t )₁^p1u(t )¹_∞^−p1 ct^{−4mn(1−1p)}u0¹₁^−p1u0₁^1p=ct^{−4mn(1−1p)}u01(cis a generic constant).

Theorem 2.1(GradientL^p-estimates). Supposeu₀∈L¹(Rⁿ),m2and ψ (t )C|t|^1+4m−θn for allt∈ {s∈R| |s|1

. (6)

Then, for allp∈ [1,∞]there existsC=C(p, n, m) >0such that the solutionuof (1)satisfies for allt >0and all γ∈Nⁿsuch that|γ|1,

D_x^γu(t )

pCt^{−4mn(1−p1)−4m|γ|}K0 (7)

provided|γ| +θ2m−1and whereK0=max(u01, u0⁽ⁿ₁^{+4m−θ )/n}). If in addition,θ >1and ψ(t )C|t|^4mn−θ for allt∈

s∈R| |s|1

, (8)

then(7)also holds when|γ| +θ=2mwithK0=max(u01,u0^(4m₁ ^{−θ )/n}).

Proof. The solutionuof (1) satisfies the integral equation u(t )=K(t )∗u0+

t 0

K(t−s)∗a· ∇^θ ψ

u(s) ds and then

D^γu(t )=D^γK(t )∗u₀+ t 0

D^γK(t−s)∗a· ∇^θ ψ

u(s)

ds, (9)

where∗is the convolution symbol with respect to the space variablex.

Assume that|γ| +θ2m−1. It follows from (9) that D^γu(t )=D^γK(t )∗u₀+

t 0

a· ∇^θ

D^γK(t−s)

∗ψ u(s)

ds. (10)

Hence, using Young inequality yields D^γu(t )

pD^γK(t )∗u₀

p+ t 0

a· ∇^θ

D^γK(t−s)

∗ψu(s)

pds

D^γK(t )

pu01+ t /2 0

a· ∇^θ

D^γK(t−s)

pψu(s)

1ds

+ t t /2

a· ∇^θ

D^γK(t−s)

1ψu(s)

pds.

On the one hand,(G)implies that D^λ_xK(t )

pC_p,mt^{−4mn(1−1p)−|}

λ|

4m (11)

for allp∈ [1,∞] and allλ∈Nⁿ such that|λ|2m−1. On the other hand, by using successively (11), (6) and Lemma 2.1 we get

t /2 0

a· ∇^θ

D^γK(t−s)

pψu(s)

1dsC|a| t /2 0

(t−s)^{−4mn(1−p1)−|γ|+θ4m} u(s)^{(n+4m−θ )/n}

(n+4m−θ )/nds

C|a|u₀⁽ⁿ₁^{+4m−θ )/n} t /2 0

(t−s)^{−4mn(1−1p)−|}

γ|+θ

4m s^{−(1−4mθ )}ds

C|a|u₀⁽ⁿ₁^{+4m−θ )/n} t

2 ₋ⁿ

4m(1−¹_p)−^|γ_4m^|+θ t /2 0

s^{−(1−4mθ )}ds

C|a| t

2 ₋ⁿ

4m(1−_p¹)−^|_4m^γ|

u₀⁽ⁿ₁^{+4m−θ )/n}

and t t /2

a· ∇^θ

D^γK(t−s)

1ψu(s)

pdsC|a| t t /2

(t−s)^−|γ|+θ4m u(s)^{(n+4m−θ )/n}

p(n+4m−θ )/nds

C|a|u0⁽ⁿ₁^{+4m−θ )/n} t t /2

(t−s)^−|γ4m|+θs^{−4mp1 (p(n+4m−θ )−n)}ds

C|a|u0⁽ⁿ₁^{+4m−θ )/n} t

2 ₋ ¹

4mp(p(n+4m−θ )−n) t 2

1−^|γ_4m^|+θ

=C|a| t

2

_−4mⁿ(1−¹_p)−^|4m^γ|

u0⁽ⁿ₁^{+4m−θ )/n}. Eventually, combining the last estimates to (11) yields (7) in the case|γ| +θ2m−1.

To deal with the case|γ| +θ=2m, we rewrite (10) as D^γu(t )=D^γK(t )∗u0+

t 0

a· ∇^θ−1

D^γK(t−s)

∗ ∇ψ u(s)

ds

=D^γK(t )∗u₀+ t 0

a· ∇^θ−1

D^γK(t−s)

∗ψ u(s)

∇u(s)ds. (12)

The same decomposition as in the first case yields D^γu(t )

pD^γK(t )

pu01+ t /2 0

a· ∇^θ−1

D^γK(t−s)

pψ u(s)

∇u(s)

1ds

+ t t /2

a· ∇^θ

D^γK(t−s)

1ψ u(s)

∇u(s)

pds.

Using (8), (7) (for the case|γ| =1), (11) (since|γ| +θ−1=2m−1) and Lemma 2.1 implies t /2

0

a· ∇^θ−1

D^γK(t−s)

pψ u(s)

∇u(s)

1ds

t /2 0

a· ∇^θ−1

D^γK(t−s)

pψu(s)

1∇u(s)

∞ds

t /2 0

a· ∇^θ−1

D^γK(t−s)

pu(s)^{(4m−θ )/n}

(4m−θ )/n∇u(s)

∞ds

C|a|u₀^(4m₁ ^{−θ )/n} t /2 0

(t−s)^{−4mn(1−1p)−|}

γ|+θ−1

4m s^{−(1−n+θ4m)}s^−n+14m ds

C|a|u₀^(4m₁ ^{−θ )/n} t

2 ₋ⁿ

4m(1−_p¹)−^|γ|+_4m^θ−1 t /2 0

s^{−(1−θ4m−1)}ds

C|a| t

2 ₋ⁿ

4m(1−_p¹)−^|_4m^γ|

u₀^(4m₁ ^{−θ )/n}

and t t /2

a· ∇^θ−1

D^γK(t−s)

1ψ

u(s)

∇u(s)

pds

t t /2

a· ∇^θ−1

D^γK(t−s)

1ψu(s)

p∇u(s)

∞ds

t t /2

a· ∇^θ−1

D^γK(t−s)

1u(s)^{(4m−θ )/n}

p(4m−θ )/n∇u(s)

∞ds

C|a|u0^(4m₁ ^{−θ )/n} t t /2

(t−s)^{−|γ|+4mθ−1}s^{−4mp1 (p(4m−θ )−n)}s^−n4m+1ds

C|a|u0^(4m₁ ^{−θ )/n} t

2

_−4mp¹ (p(4m−θ )−n)−ⁿ4m⁺¹ t 2

1−^|γ|+4m^θ−1

=C|a| t

2

_−4mⁿ(1−p¹)−4m^|γ|

u₀^(4m₁ ^{−θ )/n}.

Therefore, (7) is verified in the case|γ| +θ=2m,θ >1 and this ends the proof of Theorem 2.1. 2

Remarks 2.1.1. It is worth mentioning that (7) is verified form1 in the case|γ|+θ2m−1 (m=1 is a particular case of Lemma 2.1 sinceθ1,|γ| +θ1 and then|γ| =0). The choicem2 is adopted in order to guarantee the estimates on the gradient∇u(t )pused in the proof of the second case|γ| +θ=2m.

2. The condition (6) (resp. (8)) implies that for allδ >0 there existsC_δ>0 such that ψ (t )Cδ|t|^{(n+4m−θ )/n}

(resp.|ψ(t )|Cδ|t|^{(4m−θ )/n}) for allt∈ {s∈R| |s|δ}.

3. It seems possible that under additional assumptions (like (6) and (8)) on the successive derivatives ofψ, we can establish (7) for|γ|2m−1. This remark also applies to (14) in the forthcoming theorem.

The property (7) suggests to study the problem of the large time behavior oft^{4mn(1−1p)+|γ|4m}D_x^γu(t, x)inL^p-norm.

Indeed, in the following result we show that, fortlarge, the higher derivativesD^γ_xu(t )of the solution behave like the corresponding derivativesD_x^γK(t )of the heat kernel. More precisely,

Theorem 2.2.Assume thatm2and

|tlim|→0

ψ (t )

|t|^1+4mn−θ =0. (13)

Then, for allu0∈L¹(Rⁿ)such that

Rⁿu0(x)dx=M, the solutionuof(1)satisfies for allp∈ [1,∞],

tlim→∞t^{4mn(1−p1)+|γ|4m}D^γxu(x, t )−MD^γxKt(x)

p=0 (14)

for all multi-indexγ∈Nⁿsuch that|γ| +θ2m−1. If in addition

|tlim|→0

ψ(t )

|t|^4mn−θ =0, (15)

then the property(14)remains valid when|γ| +θ=2mandθ >1.

Proof. Suppose that|γ| +θ2m−1 and write D^γu(t )−MD^γK(t+1)=

D^γu(t+1)−MD^γK(t )

−M

D^γK(t+1)−D^γK(t ) . It follows from (10) that

D^γu(t+1)−MD^γK(t+1)=A1(t )−MA2(t )+A3(t ), where

A1(t ):=D^γK(t )∗u(1)−MD^γK(t ), A2(t ):=D^γK(t+1)−D^γK(t ), A3(t ):=

t 0

a· ∇^θ

D^γK(t−s)

∗ψ

u(s+1) ds.

Since

Rⁿu(x,1)dx=

Rⁿu0(x)dx=M, it then follows by (4) that

tlim→∞t^{4mn(1−1p)+|}

γ|

4mA1(t )

p=0. (16)

On the other hand, Lemma 1.1 yieldsA2(t )pt^{−4mn(1−1p)−|}

γ| 4m−1

and then

tlim→∞t^{4mn(1−1p)+|γ|4m}A2(t )

p=0. (17)

Note that (16) and (17) are verified for allγ∈Nⁿsuch that|γ|2m−1 and allp∈ [1,∞].

Taking into account these estimates, it remains to show that

tlim→∞t^{4mn(1−1p)+|}

γ|

4mA3(t )

p=0 (18)

for allp∈ [1,∞]and allγ∈Nⁿsuch that|γ| +θ2m−1.

We haveA3(t )pB1+B2where B1:=

t /2 0

a· ∇^θ

D^γK(t−s)

pψ

u(s+1)

1ds

B2:=

t t /2

a· ∇^θ

D^γK(t−s)

1ψ

u(s+1)

pds.

If we setξ₀(s):=ψ (s)/|s|^{(n+4m−θ )/n}, then ψu(s)

pξ0

u(s)_∞u(s)^{(n+4m−θ )/n}

p(n+4m−θ )/n. (19)

Therefore, using successively (11), (19) and Lemma 2.1 involves estimates onB1andB2as follows B1 C|a|

t /2 0

(t−s)^{−4mn(1−p1)−|γ|+θ4m} ξ0

u(s+1)_∞u(s+1)^{(n+4m−θ )/n}

(n+4m−θ )/nds

C|a| t /2 0

(t−s)^{−4mn(1−p1)−|}

γ|+θ

4m ξ₀

u(s+1)

∞(s+1)^{−(1−4mθ)}ds

C|a| t

2 ₋ⁿ

4m(1−_p¹)−^|γ_4m^|+θt /2 0

ξ₀

u(s+1)_∞(s+1)^{−(1−4mθ )}ds :=C|a|

t 2

₋ⁿ

4m(1−_p¹)−^|γ_4m^|+θ

X(t ) (i)