Upper bounds for the decay rate in a nonlocal p-Laplacian evolution problem

R E S E A R C H

Open Access

Upper bounds for the decay rate in a nonlocal

p

-Laplacian evolution problem

Carlos Esteve

_{, Julio D Rossi}

_{and Angel San Antolin}

*_{Correspondence:}

[email protected]

1_{Departamento de Análisis}

Matemático, Universidad de Alicante, Ap. correos 99, Alicante, 03080, Spain

Full list of author information is available at the end of the article

Abstract

We obtain upper bounds for the decay rate for solutions to the nonlocal problem

∂

RnJ(x,y)|u(y,t) –u(x,t)|p–2(u(y,t) –u(x,t))dywith an initial condition

u0∈L1(Rn)∩L∞(Rn) and a fixedp> 2. We assume that the kernelJis symmetric,

bounded (and therefore there is no regularizing effect) but with polynomial tails, that is, we assume a lower bounds of the formJ(x,y)≥c1|x–y|–(n+2σ), for|x–y|>c2and

J(x,y)≥c1, for|x–y| ≤c2. We prove thatu(·,t)Lq_(Rn₎≤Ct– n

(p–2)n+2σ(1– 1

q)_forq≥_{1 andt}

large.

MSC: 35K05; 45P05; 35B40

Keywords: nonlocal diffusion; decay rates

1 Introduction

In this paper we deal with nonlocal Cauchy problems of the form

∂tu(x,t) =

RnJ(x,y)

u(y,t) –u(x,t)p–u(y,t) –u(x,t)dy (.)

fort∈R+andx∈Rn withn≥, a fixedp>  and an initial condition u(x, ) =u(x)

satisfyingu∈L_(Rn_)∩L∞_(Rn_{). As regards the kernelJ, we will always assume that it}

is a bounded and symmetric function defined for (x,y)∈Rn_×Rn_{together with the}

in-tegrability conditionJ(·,y)∈L_(Rn_{) for ally∈R}n_{. Under these hypotheses existence and}

uniqueness of a solution follow from a fixed point argument as in [].

Nonlocal problems have been recently widely used to model diffusion processes (see [] and [] for a general nonlocal vector calculus). Problem (.) and its stationary ver-sion have been considered recently in connection with real applications, for example to peridynamics or a recent model for elasticity. We quote for instance [–] and the recent book [].

Our main goal here is to obtain upper bounds for the asymptotic behavior of the solution of (.) ast→+∞. It is expected that the diffusive nature of the equation implies that the solution goes to zero whent→+∞.

To obtain our results the key assumptions are the following lower bounds forJ:

J(x,y)≥c|x–y|–(n+σ) for|x–y|>c and

J(x,y)≥c for|x–y| ≤c

(.)

for certain constantsc,c> , andσ∈(, ). For simplicity we will assumec= .

The main result of this paper reads as follows.

Theorem . Let n≥,q∈[, +∞)andσ∈(, ).Let J be a kernel satisfying(.).Then the solution of(.)associated to an initial condition u∈L_(Rn_)∩L∞_(Rn_{)decays in L}q_(Rn₎

with the upper bound

u(·,t)_Lq_(Rn₎≤Ct

–_(p–)n_n+σ(–q)_{, (.)}

where the constant C depends on u,q,σ,and n.

Let us end the introduction with some comments on the previous bibliography. For the linear case,p= , and for smooth kernelsJwith compact support, it is proven in [] that the solutionuof (.) has the decay estimate

u(·,t)_Lq_(Rn₎≤Ct

–n_(–_q)

for anyq∈[,∞). Note that this decay rate is the same as the one that holds for solutions of the classical Heat equation. In the case of an equation in convolution form, that is, whenJ(x,y) =K(x–y) withKa nonnegative radial function, not necessarily compactly supported, it is proven in [] that the solutions of equations with the form (.) have the decay estimate

u(·,t)_Lq_(Rn₎≤Ct

–_n_σ(–q)_,

provided the function K has a Fourier transform satisfying the expansion Kˆ(ξ) =  –

A|ξ|σ_+o(|ξ|σ_{) forξ∼, whereA>  is a constant. In this case the decay estimate is} anal-ogous to the one for theσ-order fractional heat equation,vt= –(–)σv, withσ∈(, ).

We also note that the convolution form of the equation allows the use of Fourier analysis to obtain this result. However, the use of Fourier analysis is not helpful here due to fact that our operator is not in convolution form. In spite of this difficulty, energy methods can be applied; see [, ]. We also mention the recent reference [], where similar estimates can be found for nonlocal equations with unbounded kernels. We borrow ideas and tech-niques from these references. In particular we use Proposition . of [] (whose proof is included here for completeness). However, we have to point out that in [] and [] only the linear case, that is,p= , was treated, while here we deal with (.) for anyp≥. For examples of kernels with exponential decay bounds we refer to [] and []. Finally, we point out that our results are also valid for unbounded kernels (in the spirit of []) since only lower bounds are assumed in (.).

The case ≤p<  remains open as well as the corresponding estimate for theL∞-norm.

2 Basic facts and preliminaries

First, we need to introduce fractional Sobolev spaces and their seminorms, we refer to [] for details. Forσ∈(, ) andr∈[,∞),Wσ,r_(Rn_{) is the fractional Sobolev space of all}

Lr_(Rn_{) functions with finite fractional seminorm [v]}

σ,r, given by

[v]rσ,r=

Rn

|v(x+z) –v(x)|r

Under these definitions, we have the following fractional Sobolev-type inequality: there exists a constantC>  such that, for eachv∈Wσ,r_(Rn_{) withσr<n, we have}

vr_Ls_(Rn₎≤C[v]r_σ,r, (.)

wheres=nr/(n–σr) (see []).

First, we consider a positive smooth functionψ:Rn_{→Rwith the following properties:}

supp(ψ)⊂B and

Rn

ψ(x)dx= . (.)

With the aid of this function, we split a functionuinto two parts. We will denote the ‘smooth’ part ofuasvand the remaining asw. We let

v(x,t) :=

Rn

ψ(x–z)u(z,t)dz; w(x,t) :=u(x,t) –v(x,t). (.)

Sometimes, for simplicity in the notation and where the context is clear, we will writeu,

v, andwas functions depending only ofx.

As a first property of this decomposition we find that eachLrnorm of the functionsv

andwis controlled by the corresponding norm ofu.

Lemma . Let v and w be given by(.).For each r∈(, +∞),we have

vLr_(Rn₎≤ u_Lr_(Rn₎ and w_Lr_(Rn₎≤ u_Lr_(Rn₎.

Proof We start withv. Denotingr=r/(r– ) the Hölder conjugate of rand using the definition ofv, we have

Rn

v(x)rdx =

Rn _Rn

ψ(x–y)u(y)dy

r

dx

=

Rn _Rn

ψ(x–y)/rψ(x–y)/ru(y)dy

r

dx

≤

Rn _Rn

ψ(x–y)dy

/r

Rn

ψ(x–y)u(y)rdy

/rr

dx

=

Rn u(y)r

Rn

ψ(x–y)dx dy

≤

Rn

u(y)rdy.

The inequality forweasily follows immediately from the triangular inequality inLr.

Now we state a key result to get the desired estimate on the decay rate.

Proposition . Let n≥and let J:Rn_×Rn_→R

+be a kernel satisfying(.),ψ satis-fying(.),β∈[σ, ),and r>max{, β}.Then there exists a constant C> such that,for all u∈Lr(Rn)and v,w defined in(.),we have

[v]r_βr–_,r+wr_Lr_(Rn₎≤C

RnJ(x,y)

u(x) –u(y)rdx dy. (.)

Proof For the estimate concerningw, we have

Rn

w(x)rdx=

Rn

u(x) –v(x)rdx

=

Rn u(x) –

Rn

ψ(x–z)u(z)dz

r

dx

=

Rn _Rn

ψ(x–z)u(x) –u(z)dz

r

dx

=

Rn _Rn

ψ(x–z)/rψ(x–z)/ru(x) –u(z)dz

r

dx.

Applying Hölder’s inequality, we get

Rn

w(x)rdx≤

Rn _Rn

ψ(x–z)dz

r/r

Rn

ψ(x–z)u(x) –u(z)rdz

dx

≤

Rn

Rn

ψ(x–z)u(x) –u(z)rdz dx,

wherer=r/(r– ).

Sinceψ is supported inB, we haveψ(x–z)≤J(x,z) for all|x–z| ≥, and, sinceJ

verifiesJ(x,z)≥cfor|x–z| ≤, there exists a constantCdepending only on|ψ|∞ such

thatψ(x–z)≤CJ(x,z). Then

wr_Lr_(Rn₎≤C

RnJ(x,y)

u(x) –u(y)rdx dy.

Now we deal with the term withv. We split the fractional seminorm as

[v]r_βr–_,r=

|x–y|>

|v(x) –v(y)|r

|x–y|n+β dx dy+

|x–y|≤

|v(x) –v(y)|r

|x–y|n+β dx dy

=:Iext+Iint

and look at these integrals separately. ForIext, using the definition ofvwe have

Iext=

|x–y|>

_Rn

u(x–z) –u(y–z)ψ(z)dz

r

|x–y|–(n+β)dx dy.

Now, we can think of the measureμ(dz) =ψ(z)dzas a probability measure (because of (.)), and since the functiont→ |t|r_{is convex inR, we can apply Jensen’s inequality on}

thedz-integral in right-hand side of the last expression to obtain

Iext≤

|x–y|>

Rn

u(x–z) –u(y–z)rψ(z)dz|x–y|–(n+β)_{dx dy,}

which, after an application of Fubini’s theorem, gives

Iext≤

Rn

ψ(z)

|x–y|>

u(x–z) –u(y–z)r|x–y|–(n+β)dx dy

Then, applying the change x˜=x–z,y˜=y–zin thedx dyintegral and using (.), we conclude

Iext≤

Rn

ψ(z)

|˜x–˜y|>

u(x˜) –u(y˜)r|˜x–y˜|–(n+β)_{dx d˜ y˜}

dz

=

|˜x–y˜|>

u(x˜) –u(y˜)r|˜x–˜y|–(n+β)_{dx d˜ y˜.}

Using this last expression, we obtain from the assumption (.) that

Iext≤C

Rn

J(x,y)u(x) –u(y)rdx dy. (.)

Now we deal withIint. In this case, using the definition ofv, we can write

Iint=

|x–y|<

_Rn

u(z)ψ(x–z) –ψ(y–z)dz

r

|x–y|–(n+β)_{dx dy. (.)}

Note that by using (.), we have for allx,y∈Rn

Rnu(x)

ψ(x–z) –ψ(y–z)dz=u(x) Rn

ψ(x–z)dz–

Rn

ψ(y–z)dz

= ,

and then

Rnu

(z)ψ(x–z) –ψ(y–z)dz=

Rn

u(z) –u(x)ψ(x–z) –ψ(y–z)dz.

Thus, using this equality with (.), we get

Iint=

|x–y|<

_Rn

u(z) –u(x)ψ(x–z) –ψ(y–z)dz

r

|x–y|–(n+β)_{dx dy.}

However, note that if|x–z| ≥ in thedzintegral, since|x–y|<  necessarily|y–z|> . Then, due to the fact thatψis supported in the unit ball, the contribution of the integrand when |x–z| ≥ is null in the dzintegral. Taking this into account, applying Hölder’s inequality into thedz-integral, we have

Iint=

|x–y|<

_|x–z|<u(z) –u(x)ψ(x–z) –ψ(y–z)dz

r

× |x–y|–(n+β)_{dx dy}

≤

|x–y|< |x–z|<

u(z) –u(x)rdz

×

|x–˜z|<

ψ(x–˜z) –ψ(y–˜z)rdz˜

r/r

|x–y|–(n+β)dx dy.

By Fubini’s theorem we can write

Iint=

x∈Rn _|x–z|<

u(z) –u(x)rdz

where

(x) =

|x–y|< |x–˜z|<

ψ(x–z˜) –ψ(y–˜z)rdz˜

r/r

|x–y|–(n+β)dy.

Using the regularity ofψ, we have

(x)≤

|x–y|< |x–z˜|<

Dψr_∞|x–y|rdz˜

r/r

|x–y|–(n+β)_dy

≤ Dψr_∞|B|r/r

|x–y|<

|x–y|r|x–y|–(n+β)_dy,

and sincer> β, we conclude that the last integral is convergent, obtaining

(x)≤Cn,β,rDψr_∞|B|r/r

,

which leads to the following estimate forIint:

Iint≤C

x∈Rn

|x–z|<

u(z) –u(x)rdz dx.

From this, it is easy to get

Iint≤C

|x–z|≤

|u(z) –u(x)|r

( +|x–z|)n+βdz dx,

which, by the use of (.), let us conclude that

Iint≤C

RnJ(x,y)

u(x) –u(y)rdx dy.

This last estimate together with (.) concludes the proof.

3 Proof of Theorem 1.1

As mentioned in the introduction, existence and uniqueness of solutions to problem (.) follows as in []. In fact, the symmetry, boundedness, and integrability assumptions over

Jallow us to perform a fixed point argument to obtain the following result, whose proof is omitted.

Theorem . Let u∈L(Rn)∩L∞(Rn),then there exists a unique solution u∈C([, +∞), L_(Rn_)∩L∞_(Rn_{))of equation(.).This solution satisfiesu(·,t)}

L_(Rn₎≤ u_L_(Rn₎ and u(·,t)L∞(Rn₎≤ u_L∞_(Rn₎for all t≥.

Now, let us introduce the main idea behind the energy methods. To clarify the exposi-tion, let us perform these computations in the local case and next see how we can adapt them to our nonlocal problem with the help of Proposition .. Let us describe briefly how the energy method can be applied to obtain decay estimates for local problems. Let us begin with the simpler case of the estimate for solutions to thep-Lapacian evolution equation inL_{-norm. Letube a solution to}

If we multiply the equation byuand integrate inRn_{, we obtain}

∂t

Rn  u

_{(x,t)dx= –}

Rn

_∇u(x,t)p

dx.

Now we use Sobolev’s inequality

Rn

_∇u(x,t)pdx≥C

Rn

u(x,t)p∗dx

p/p∗

withp∗=pn/(n–p) to obtain

∂t

Rnu

_{(x,t)dx≤–C}

Rn

u(x,t)p∗dx

p/p∗

.

If we use interpolation and thatu(·,t)L_(Rn₎≤C(u) for anyt> , we have

u(·,t)_L_(Rn₎≤u(·,t) α

L_(Rn₎u(·,t)

–α

Lp∗_(Rn₎≤Cu(·,t)

–α

Lp∗_(Rn₎

withαdetermined by

 =α+

 –α

p∗ , that is,α=

 –



p∗

p∗

(p∗– ).

Hence we get

∂t

Rnu

_{(x,t)dx≤–C}

Rnu

_(x,t)dx



–α

from where the decay estimate

u(·,t)_L_(Rn₎≤Ct

–_(_{n(p–)+}n )

, t> ,

follows. To obtain a decay bound foru(·,t)Lq_(Rn₎we can use the same idea multiplying byuq–at the beginning.

Now we are ready to proceed with the proof of our main result.

Proof of Theorem. The symmetry assumption onJallows us to mimic this idea and use an energy approach in order to get Theorem .. Roughly speaking, this assumption allows us to ‘integrate by parts’ (.). Forq=  the proof is finished by Theorem .. Forq>  we multiply the equation byq|u|q–_{uand integrate, obtaining the identity}

∂t

Rn

u(x)qdx= –q 

RnJ(x,y)

u(y) –u(x)p–u(y) –u(x)

×u(y)q–u(y) –u(x)q–u(x)dy dx, (.)

Now we recall the following inequality (whose proof is straightforward): letq>  and

a,b= . Then there exists a constantCdepending only onq, such that

(a–b)|a|q–a–|b|q–b≥C|a–b|q.

Hence, using this inequality with (.), we conclude

∂tuq_Lq_(Rn₎≤–C

RnJ(x,y)

u(y) –u(x)p–+qdy dx=: –CE(u). (.)

Note that we find that theLq_{-norm ofuis decreasing int. At this point we would like to}

use Sobolev’s inequality, which is not available due to the lack of regularizing effect of our nonlocal operator. Instead we will use Proposition ., which involves a good control of the smooth partv(but we have to take care of the rough partw).

Let us fixq≥ (the case  <q<  will be tackled at the end of this proof using interpo-lation). By the definition ofvandwin (.), we have

uq_Lq_(Rn₎≤q–

vq_Lq_(Rn₎+wq_Lq_(Rn₎

. (.)

Now we note thatvbelongs toLpfor allp. Hence, we can interpolate, obtaining

vq_Lq_(Rn₎≤ vq θ

Ls_(Rn₎vq(– θ)

L_(Rn₎,

with

s=n(p+q– )

n– σ ,

whereθ is given by



q= θ

s + ( –θ), that is,θ= s(q– )

q(s– ).

Recallingv_L_(Rn₎≤ u(·,t)_L_(Rn₎≤ u_L_(Rn₎and the Sobolev-type inequality (.), we obtain

vq_Lq_(Rn₎≤C[v]q θ

˜

σ,p+q–, (.)

whereσ˜ = σ(p+q– )–_{and the constantCdepends onu}

,q,σ, andn.

Concerningwwe can also interpolate and obtain

wq_Lq_(Rn₎≤ wq γ

Lp+q–_(Rn₎w

q(–γ)

L_(Rn₎,

withγ given by



q= γ

p+q– + ( –γ), that is,γ =

(p+q– )(q– )

q(p+q– ) .

Note that we are using thatp>  here. Now we use

wL_(Rn₎≤u(·,t)

to get

wq_Lq_(Rn₎≤Cwq γ

Lp+q–_(Rn₎, (.)

withCdepending onu,q,σ, andn.

From (.), (.), and (.) we obtain

uq_Lq_(Rn₎≤C[v]q θ

˜

σ,p+q–+Cw

qγ

Lp+q–_(Rn₎. (.)

Now we use Proposition ., withr=p+q–  andβ=σ, to obtain

vq_Lθs_(Rn₎≤C

E(u) qθ

p+q– _{and w}qγ

Lp+q–_(Rn₎≤C

E(u) qγ

p+q–

and we conclude that

uq_Lq_(Rn₎≤C

E(u) qθ

p+q– _+CE(u) qγ

p+q–_,

that is,

H–uq_Lq_(Rn₎

≤E(u)

withH(z) =Czp+qqθ–_+Cz qγ

p+q–_{. Sinceu}q

Lq_(Rn₎(t)≤ uq_Lq_(Rn₎(recall that theLq-norm of the solution decreases) and _p+q_qθ_– <_p+q_qγ_–, we have

H–uq_Lq_(Rn₎

≥Cuq_Lq_(Rn₎ p+q–

qθ _.

Then, from (.), we obtain

∂tu(·,t) q

Lq_(Rn₎≤–CE(u)≤–CH–

uq_Lq_(Rn₎

≤–Cuq_Lq_(Rn₎ p+q–

qθ _,

from which it follows that

u(·,t)q_Lq_(Rn₎≤Ct

–_p+q––qθ _qθ

,

that is,

u(·,t)q_Lq_(Rn₎≤Ct

–_q((p(_–)q–)_nn_+σ)

,

as we wanted to show.

Now we deal with the case  <q< . We can interpolate, obtaining

u(·,t)q_Lq_(Rn₎≤u(·,t)

qθ

L_(Rn₎u(·,t)

q(–θ)

L_(Rn₎,

with



q= θ

+ ( –θ), that is,θ=   – 

q

As we have already proved that

u(·,t)_L_(Rn₎≤Ct

–_(p–)n_n+σ_

and u(·,t)_L_(Rn₎≤C,

we conclude

u(·,t)_Lq_(Rn₎≤Ct

–_(p–)n_n+σ(–q)

for  <q< .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details

1_{Departamento de Análisis Matemático, Universidad de Alicante, Ap. correos 99, Alicante, 03080, Spain.}2_Departamento

de Matemática, FCEyN UBA, Ciudad Universitaria, Pab 1, Buenos Aires, 1428, Argentina.

Acknowledgements

This work was partially supported by MEC MTM2010-18128 and MTM2011-27998 (Spain).

Received: 11 March 2014 Accepted: 25 April 2014 Published:13 May 2014

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Cite this article as:Esteve et al.:Upper bounds for the decay rate in a nonlocalp-Laplacian evolution problem.