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R E S E A R C H

Open Access

Periodicity and ergodicity for abstract

evolution equations with critical

nonlinearities

Bruno de Andrade

1

, Claudio Cuevas

2*

, Jin Liang

3

and Herme Soto

4

*Correspondence:

cch@dmat.ufpe.br

2Departamento de Matemática,

Universidade Federal de Pernambuco, Recife, PE 50540-740, Brazil Full list of author information is available at the end of the article

Abstract

We study the existence of pseudo almost periodic mild solutions for the abstract evolution equationu(t) =Au(t) +f(t,u(t)),t∈R, when the nonlinearityfsatisfies certain critical conditions. We apply our abstract results to the heat equation. MSC: Primary 35B15; secondary 35K05; 58J35

Keywords: pseudo almost periodic solutions; critical nonlinearities; growth conditions

1 Introduction

We observe that real systems usually exhibit internal variations or are submitted to ex-ternal perturbations. In many situations we can assume that these variations are approx-imately periodic in a broad sense. In the literature several concepts have been studied to represent the idea of approximately periodic function (see []). Most of the works deal with asymptotically periodic functions and almost periodic functions (e.g., [, ]). As a natural generalization of the classical almost periodicity in the sense of Bochner, the concept of pseudo almost periodicity as well as its various extensions have been studied extensively by many researchers in the last  years (cf.,e.g., [–]). Pseudo almost periodic func-tions have many applicafunc-tions in several problems, for example in the theory of functional differential equations, integral equations, and partial differential equations. The concept of pseudo almost periodicity, which is the central issue in this work, was introduced by Zhang [–] in the early s. Our purpose in this work is to analyze the existence of pseudo almost periodic mild solutions of the abstract problem

u(t) =Au(t) +ft,u(t), t∈R, (.) where the linear operatorA:D(A)⊂XXis such that –Ais a sectorial operator in the Banach spaceXandf is a continuous function. Many authors have considered ab-stract linear evolution equations in Banach spaces (see [–] and references therein). In particular several results on the existence of pseudo almost periodic solutions to dif-ferential equations of the type (.) have been established. Stimulated by these works, in this paper, we will investigate further the corresponding problem where the nonlinearity of the nonlinear term of (.) has critical growth. Note that there is much interest in the

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study of nonlinearities when they are assumed to be a critical cause and intrinsic difficult. We need to understand the scale of fractional powers spaces associated to the linear oper-atorA, especially the embeddings into known spaces likeLpspaces, and the-regularity properties (see Definition .) of the nonlinearityf in this scale of spaces. It makes the analysis much more technical and harder. Problems with this kind of nonlinearity appear frequently in the heat equation theory. However, to the best of our knowledge, there are few results for these problems.

We will denote by:= (D(Aβ), · X

β),β≥, the fractional power spaces associated

to the operatorAand by{eAt}t≥the analytic semigroup generated byA. Furthermore, we will suppose thatReσ(–A) >a> . Consequently, there exists a positive constantMsuch thateAtverifies

eAtx

Mtβe–atxX, t> ,β≥. (.) The main scope of this work is to provide sufficient condition to ensure the existence of a pseudo almost periodic-regular mild solution of (.). By an-regular mild solution of (.) we understand a continuous functionu:R→Xsuch thatuC(R,X+) and verifies

u(t) =

t

–∞e

A(t–s)fs,u(s)ds, tR.

The notion of-regular mild solution for abstract semilinear evolution equations was in-troduced by Arrieta and Carvalho in [] (see also []) where a detailed discussion was presented considering the importance of these types of solutions.

A typical example that we will treat in this work is to consider inL(), withRa bounded smooth domain, the heat equation

ut=Au+gh(u), inR×,

u= , onR×, (.)

whereg:R→Ris a pseudo almost periodic function, the functionh:R→Rverifies |h(t) –h(s)| ≤c( +|t|/+|s|/)|ts|, withh()= , and the operator Au=uau,a> , is considered with domain

D(A) :=H()∩H().

LetL=with Dirichlet boundary conditions in. ThenLcan be seen as an unbounded operator inE=L() with domainD(A). The scale of the fractional power spaces{

}α∈R

associated toLverifies (see [, ])

EαHα(), α≥, Eα=Eα, α≥. Therefore, we get

Lr(), forr≤ 

 – α, ≤α<  ,

E=L(),

Ls(), fors≥ 

 – α, – 

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Letbe the realization ofLinEα. Then we find thatis an isometry fromEα+intoEα

and

:D() =+⊂E

α

→E

α

is a sectorial operator. Denote

 :=–,α∈R. The fractional power spaces associated toL–:X⊂X→Xsatisfy

XαLr(), forr≤ 

 – α, ≤α<  ,

X=L(),

 ←Ls(), fors≥   – α,

<α≤.

To rewrite (.) in the abstract form (.) consider the function

f(t,ψ)(x) =g(t)(x).

It follows from Section , for t∈R fixed, that the function f(t,·) is an ultra-critical -regular map relative to the pair (X,X), for every∈(,). In fact, we see, for each

t∈R, that

f(t,·) :X+X

 

is well defined and satisfies

f(t,u) –f(t,v) X

 

cg∞ +u/X+

 +v / X+

uvX+

 ,

for somec> . Furthermoref is a pseudo almost periodic function (see Definition .). For∈(,), it follows from Theorem . that ifg∞ is small enough then (.) has a pseudo almost periodic mild solutionu:R→X+. This solution is given by

u(t,x) =

t

–∞e

A(t–s)g(s)hu(s,x)ds, tR,x.

Furthermore,u∈PAP(X+θ), for every θ<

.

In Section  we will treat in theLqsetting the general situation,

ut=Au+gh(u), inR×,

u= , onR×,

whereg:R→Ris a pseudo almost periodic function,Au=uau,a> , and the func-tionh:R→Rverifies

h(x) –h(y)≤c +|x|ρ–+|y|ρ–|xy|, ρ> ,

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2 Background material

Consider Banach spaces,YandZ, with norms · Yand · Z, respectively. LetAP(Y) be the set of all almost periodic functions, that is, the set of all continuous functionsg:R→Y

such that for allα>  there existsl(α) >  such that every intervalI⊂Rof lengthl(α) contains a numberτwith the property that

g(t+τ) –g(t)Y<α, ∀t∈R.

Also, letP(Y) be the set of all bounded continuous functionsφ:R→Ysuch that

lim

T→∞  T

T

–T

φ(t)Ydt= .

Definition . A continuous functiong:R→Yis called pseudo almost periodic ifg=

h+φ, whereh∈AP(Y) andφP(Y).

The functionshandφ are called, respectively, the almost periodic component and the ergodic perturbation of the functiong. As usual, we represent by PAP(Y) the set of all pseudo almost periodic functionsh:R→Y. We remark that this set provided with the uniform convergence norm · ∞is a Banach space. In fact, we find that (PAP(Y), · ∞) is a closed subset of the Banach space of bounded continuous functions.

In this work we will need a special notion of pseudo almost periodic functions. To this end, we represent byAP(Y,Z) the set of all continuous functionh:R×YZsuch that

h(·,x) is an almost periodic function uniformly on bounded sets ofY, that is, for allα>  and any bounded subsetKofY there existsl(α) >  such that all interval of lengthl(α) contains a numberτwith the property that

h(t+τ,x) –h(t,x)Zα,

for allt∈RandxK.

The following lemma will be very useful in this work.

Lemma .([]) If f ∈AP(Y;Z)and h∈AP(Y),then the function f(·,h(·))∈AP(Z). Likewise, we represent byP(Y,Z) the set of all continuous functionsφ:R×YZ such thatφ(·,x) :R→Zis a bounded function for allxYand

lim

T→∞  T

T

–T

φ(t,x)Zdt= ,

uniformly inxY.

Definition . A continuous functiong:R×YZis called pseudo almost periodic if

g=h+φ, whereg∈AP(Y,Z) andφP(Y,Z).

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Definition . For>  we say that a map g is an-regular map relative to the pair

(X,X) if there existρ> ,γ() withργ() < , and a positive constantc, such that

g:X+Xγ()and

g(x) –g(y)Xγ()≤c

 +X–++y ρ– X+

xyX+,

for allx,yX+.

The main result of this work basically says that iff(t,·) is an-regular map relative to the pair (X,X) and belongs toPAP(X+,Xγ()) then we will have existence of a pseudo almost periodic-regular mild solution to (.). Furthermore this solution belongs toPAP(X+θ)

for all ≤θ <γ(). It is important to remark that we obtain an existence result inX without the nonlinearity being defined onX.

3 Abstract results

We start this section observing that from (.) follows the estimate

t+θβeAtxX+θMe–atxXβ, ≤β≤ +θ≤.

This estimate will be very useful for our purpose. We recommend to the reader the book [] for more details and information on sectorial operators and fractional power spaces. In this work we consider the following class of nonlinearities: with , γ(),ρ, and c

positive constants, define

F,ρ,γ(),c

as the family of functionsf such that, fort∈R,f(t,·) is an-regular map relative to the pair (X,X), satisfying

f(t,x) –f(t,y)Xγ()≤c

 +X–++y ρ– X+

xyX+, (.)

f(t,x)Xγ()≤c

 +X+

, (.)

for allx,yX+.

The following theorem is the main result of this work.

Theorem . Consider fF(,γ(),ρ,c)∩PAP(X+,Xγ()).If the constant c is small

enough then the problem(.)has a pseudo almost periodic-regular mild solution. Fur-thermore,this solution verifies u∈PAP(X+θ),for everyθ<γ().

Proof Fixr>  and letL(r) =max{r–+rρ–,  + rρ–}. We claim that the result is true if

Mcaγ()γ() –L(r) < . In fact, letB(r) be the closed ball

B(r) :=

w∈PAPX+:sup

t∈R w(t)

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Define onB(r) the operatorTby

(Tu)(t) =

t

–∞e

A(t–s)fs,u(s)ds.

The setB(r) isT-invariant. We divide the proof of this claim in two steps.

Step : Ifu∈PAP(X+) thenTuPAP(X+θ), for every θ<γ(). First of all, we show

that the function

tft,u(t)

belongs toPAP(()). In fact, sincef PAP(X+,Xγ()) there existgAP(X+,Xγ()) and φP(X+,()) such thatf =g+φ. In the same way,u=h+ψwithh∈AP(X+) and ψP(X+). Hence,

f·,u(·)=g·,h(·)+f·,u(·)–g·,h(·)

=g·,h(·)+f·,u(·)–f·,h(·)+φ·,h(·).

Sinceg(·,h(·))∈AP(()), see Lemma ., it remains to show that the function

tft,u(t)–ft,h(t)+φt,h(t):=F(t) +φt,h(t) belongs toP(()). To this end, note that

 T

T

–T

F(t)Xγ()dt

cT

T

–T

u(t) –h(t)X+dt

+ cT

T

–T

u(t)ρX–+u(t) –h(t)X+dt

+ cT

T

–T

h(t)ρX–+u(t) –h(t)X+dt

c +sup

t∈R

u(t)ρX–++sup

t∈R

h(t)ρX–+

T

T

–T

ψ(t)X+dt.

Thus,FP(()). On the other hand, sinceh(R) is relatively compact inX+, forα>  we can find a finite number of open ballsOkwith centerxkh(R) and radius less thanα such thath(R)⊂mk=Ok. Clearly, we can writeR=

m

k=Bk, whereBk=h–(Ok) are open subsets. Consider

Ek:=

B, k= ,

Bk\k–i= Bi, k> ,

thenEiEj=∅, ifi=j. LetT>  be such that 

T

T

–T

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con-which goes to zero ast→t+. The caset>tis similar. Furthermore, interval of lengthl(α) contains a numberτ such that

g(t+τ) –g(t)Xγ()≤

It is not hard to check that

lim

T→∞φT(t) = .

Next, sinceφT(s) is bounded and the functionss–+γ()–θe–asis integrable in [,∞), using the Lebesgue dominated convergence theorem we have

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Mc

t

–∞(ts) –+γ()–

e–a(t–s)ds +

Mcaγ()γ() – +

r.

Now, consideru,vB(r). We have

(Tu)(t) – (Tv)(t)X+

M

t

–∞(ts) –+γ()–

e–a(t–s)fs,u(s)–fs,v(s)Xγ()dsMc

t

–∞(ts)

–+γ()–e–a(t–s) +u(s)ρ–

X++v(s) ρ–

X+u(s) –v(s)X+ds

Mc + –

t

–∞(ts) –+γ()–

e–a(t–s)ds

sup

t∈R

u(t) –v(t)X+

Mcaγ()γ() – + –sup

t∈R

u(t) –v(t)X+.

Therefore, the operatorT :B(r)→B(r) is a contraction and by the Banach contraction principle we find thatT has a unique fixed point. Remark .

• In particular we can obtain an existence theorem inXwithout the nonlinearity being defined onX. Notice that we do not assume thatf is a well-defined map onX. The only requirement onf is that it is an-regular map relative to(X,X), for some> . • Another important consequence of Theorem . is given when the nonlinear term

fF(,γ(),c)∩Z, where

Z∈APX+,Xγ(),P

X+,Xγ().

In fact in this situation we can show that the operatorT:ZZis well defined. Under the conditions of Theorem . there exists an-regular mild solutionuof the problem (.). Furthermore, this solution verifiesuY( +θ), for every≤θ<γ(), whereY(β)∈ {AP(),P

()}.

• An important example of our result will be the case that for a function

fF,γ(),ρ,c∩PAPX+,()

there existsλ> such that for any≤λλ, and Theorem . applies to the functionλf.

For the sake of completeness, we include a result on globally Lipschitz maps. To this end, we consider the following class of nonlinearities: with,γ(),ρ, andcpositive constants, define

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as the family of functionsf such that, fort∈R,f(t,·) is an-regular map relative to the pair (X,X), satisfying

f(t,x) –f(t,y)Xγ()≤cxyX+,

for allx,yX+.

Theorem . Consider fE(,γ(),ρ,c)∩PAP(X+,Xγ()).Suppose that

Mcaγ()γ() –< .

Then the problem(.)has a unique pseudo almost periodic-regular mild solution. Fur-thermore,this solution verifies u∈PAP(X+θ),for everyθ<γ().

Proof In fact, a similar procedure to Step  in Theorem . shows that the map T :

PAP(X+θ)PAP(X+θ) defined by

(Tu)(t) =

t

–∞e

A(t–s)fs,u(s)ds,

t∈R, is well defined for every ≤θ<γ(). Furthermore, ifu,v∈PAP(X+) then (Tu)(t) – (Tv)(t)Xγ()≤M

t

–∞(ts)

–+γ()–e–a(t–s)fs,u(s)fs,v(s) ()dsMc

t

–∞(ts) –+γ()–

e–a(t–s)u(s) –v(s)X+ds

Mcaγ()γ() –sup

s∈R

u(s) –v(s)X+.

Therefore, the operatorT:PAP(X+)PAP(X+) is a [Mcaγ()(γ()–)]-contraction and by the Banach contraction principleT has a unique fixed point. To close this section we remember that in [] the authors introduce a classification for a time-independent mapf which is-regular, forI, relative to the pair (X,X). For the convenience of the reader we recall this classification.

• IfI= [,]for some> andγ() > , we say thatf is asubcriticalmap relative to (X,X).

• IfI= [,]for some> withγ() =ρ,I, and iff is not subcritical, then we say thatf is acriticalmap relative to(X,X).

• IfI= (,]for some> withγ() =ρ,I, andf is not subcritical or critical, then we say thatf is adouble criticalmap relative to(X,X).

• IfI= [,]for some>> withγ() >ρandf is not subcritical, critical or double critical, then we say thatf is anultra-subcriticalmap relative to(X,X). • IfI= [,]for some>> withγ() =ρ,I, and iff is not subcritical,

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4 Applications to heat equation

Let⊂RN be a bounded smooth domain. In this section we will treat the equation

ut=Au+gh(u), inR×,

u= , onR×, (.)

whereg∈PAP(R),Au=uau,a> , and the functionh:R→Rverifies

h(x) –h(y)≤c +|x|ρ–+|y|ρ–|xy|, ρ> , (.)

for somec> . It is well known that there are many works on equations like (.). In par-ticular, we suggest to the reader the references [–, ]. We are specially interested in theLqtheory,  <q<andq=N(ρ–)

 .

LetL=with Dirichlet boundary conditions in. ThenLcan be seen as an unbounded operator inE

q=Lq() with domain

D(A) :=W,q()∩W,q().

It is well known that the scale of fractional power spaces{

q}∈Rassociated toLverifies

(see [, ])

qHqα(), α≥,  <q<∞,

Eα

q

q

, α≥,  <q<∞,q= q

q– . Therefore, we get

qLr(), forrNq

N– , ≤α

N

q, Eq=Lq(),

qLr(), forr

N N– , –

N

qα≤.

Letbe the realization ofLinEqα. Then we find thatis an isometry fromEqα+intoEqα and

:D() =Eqα+⊂qq

is a sectorial operator. Denote

q :=Eαq–,α∈R. The fractional power spaces associated toL–:Xq⊂Xq→Xqsatisfy

qLr(), forr

Nq

N+ q– , ≤α< q+N

q , Xq=Lq(),

XqαLr(), forrNq N+ q– ,

qN

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Considerf:R×R→Rdefined by

f(t,x) =g(t)h(x).

We can to classify this function in the following way (see [, Lemma ]):

• Ifq>NN–, thenf is an-regular map relative to(Xq,Xq)for =(q)≤<NN+q and γ() =ρ. Thereforef is a critical map.

• Ifq=NN–, thenf is an-regular map relative to(X

q,Xq)for =(q) <<N+qN and γ() =ρ. Thereforef is a double critical map.

• If <q<N–N , thenf is an-regular map relative to(X

q,Xq)for <(q) <<N+qN , with(q) =NN+q( –N( –q)) > , andγ() =ρ. Thereforef is an ultra-critical map. A straightforward computation shows that

fF,γ(),ρ,c∩PAPX+,(),

for any∈((q),N+qN ). If the constantsMandcare as in Theorem . then there exists a mild solutionu∈PAP(X+) of (.) such thatuPAP(X+θ), for every θ<γ().

5 Conclusions

We have studied the existence of pseudo almost periodic mild solutions for abstract evolu-tion equaevolu-tion when the nonlinearity satisfies certain critical condievolu-tions. Investigaevolu-tions in such directions are technically more complicated. The difficulty when we have the critical growth relies on the fact that the nonlinear terms are ‘too large’ and cannot be controlled a priori by the linear terms. Therefore, for this case, the standard approach to semilin-ear equations,e.g., [, ], fails (see also []). To achieve our results we use the scale of fractional power spaces and-regularity theory developed by Arrieta and Carvalho in [] (this paper was very inspiring for us). Note that problems with this kind of nonlinear-ity appear frequently in the heat equation theory. We have applied our abstract results to such equations. Our results are new and contribute to the development of the asymptotic periodicity theory of evolution equations.

Competing interests

The authors declare that they have no competing interest.

Authors’ contributions

Each of the authors contributed to each part of this study equally and approved the final version of this manuscript.

Author details

1Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão, SE 49100-000, Brazil.2Departamento de

Matemática, Universidade Federal de Pernambuco, Recife, PE 50540-740, Brazil.3Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China. 4Departamento de Matemática y Estadística, Universidad de La Frontera, Casilla 54D, Temuco, Chile.

Acknowledgements

B de Andrade is partially supported by CNPQ/Brazil under Grant 100994/2011-3. C Cuevas is partially supported by CNPQ/Brazil under Grant 478053/2013-4. H Soto is partially supported by DIUFRO (Universidad de La Frontera) under Grant DI15-0050.

Received: 7 October 2014 Accepted: 25 December 2014

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References

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