**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

2_{Departamento 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 equation*u*(*t*) =*Au*(*t*) +*f*(*t*,*u*(*t*)),*t*∈R, when the nonlinearity*f*satisﬁes
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 operator*A*:*D*(*A*)⊂*X*_{→}* _{X}*

_{is such that –}

_{A}_{is a sectorial operator in}the Banach space

*X*

_{and}

_{f}_{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

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-ator*A*, especially the embeddings into known spaces like*Lp*spaces, and the-regularity
properties (see Deﬁnition .) of the nonlinearity*f* 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*Xβ*_{:= (}_{D}_{(}_{A}β_{),}_{ · X}

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

to the operator*A*and by{*eAt*_{}t≥}_{the analytic semigroup generated by}_{A}_{. Furthermore, we}
will suppose thatRe*σ*(–*A*) >*a*> . Consequently, there exists a positive constant*M*such
that*eAt*_{veriﬁes}

*eAt _{x}*

*Xβ*≤*Mt*–*βe*–at*x*X, *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 function*u*:R→*X*_{such that}_{u}_{∈}_{C}_{(}_{R}_{,}* _{X}*+

_{) and veriﬁes}

*u*(*t*) =

*t*

–∞*e*

*A(t–s) _{f}_{s}*

_{,}

_{u}_{(}

_{s}_{)}

_{ds}_{,}

_{t}_{∈}

_{R}

_{.}

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 in*L*_{(}_{}_{), with}_{}_{⊂}_{R}_{a}
bounded smooth domain, the heat equation

*ut*=*Au*+*gh*(*u*), inR×,

*u*= , onR×*∂*, (.)

where*g*:R→Ris a pseudo almost periodic function, the function*h*:R→Rveriﬁes
|*h*(*t*) –*h*(*s*)| ≤*c*( +|*t*|/_{+}_{|}_{s}_{|}/_{)}_{|}_{t}_{–}_{s}_{|}_{, with}_{h}_{()}_{= , and the operator}_{} _{Au}_{=}_{}_{u}_{–}_{au}_{,}_{a}_{> ,}
is considered with domain

*D*(*A*) :=*H*()∩*H*_{}().

Let*L*=with Dirichlet boundary conditions in. Then*L*can be seen as an unbounded
operator in*E*_{}=*L*() with domain*D*(*A*). The scale of the fractional power spaces{*Eα*

}*α*∈R

associated to*L*veriﬁes (see [, ])

*E*_{}*α* →*H**α*(), *α*≥, *E*_{}–*α*=*E*_{}*α*, *α*≥.
Therefore, we get

*Eα*_{} →*Lr*(), for*r*≤

– *α*, ≤*α*<
,

*E*_{}=*L*(),

*Eα*_{}←*Ls*(), for*s*≥

– *α*, –

Let*Lα*be the realization of*L*in*E*_{}*α*. Then we ﬁnd that*Lα*is an isometry from*E*_{}*α*+into*E*_{}*α*

and

*Lα*:*D*(*Lα*) =*Eα*+⊂*E*

*α*

→*E*

*α*

is a sectorial operator. Denote*Xα*

:=*Eα*–,*α*∈R. The fractional power spaces associated
to*L*–:*X*⊂*X*→*X*satisfy

*X*_{}*α* →*Lr*(), for*r*≤

– *α*, ≤*α*<
,

*X*_{}=*L*(),

*Xα*

←*Ls*(), for*s*≥
– *α*,

<*α*≤.

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

*f*(*t*,*ψ*)(*x*) =*g*(*t*)*hψ*(*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*+

*u*–*v _{X}*+

,

for some*c*> . Furthermore*f* is a pseudo almost periodic function (see Deﬁnition .).
For∈(,_{}), it follows from Theorem . that if*g*∞ is small enough then (.) has a
pseudo almost periodic mild solution*u*:R→*X*+_{. This solution is given by}

*u*(*t*,*x*) =

*t*

–∞*e*

*A(t–s) _{g}*

_{(}

_{s}_{)}

_{h}_{u}_{(}

_{s}_{,}

_{x}_{)}

_{ds}_{,}

_{t}_{∈}

_{R}

_{,}

_{x}_{∈}

_{}_{.}

Furthermore,*u*∈PAP(*X*+*θ*_{), for every }_{≤}_{θ}_{<}

.

In Section we will treat in the*Lq*_{setting the general situation,}

*ut*=*Au*+*gh*(*u*), inR×,

*u*= , onR×*∂*,

where*g*:R→Ris a pseudo almost periodic function,*Au*=*u*–*au*,*a*> , and the
func-tion*h*:R→Rveriﬁes

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

**2 Background material**

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

such that for all*α*> there exists*l*(*α*) > such that every interval*I*⊂Rof length*l*(*α*)
contains a number*τ*with the property that

*g*(*t*+*τ*) –*g*(*t*)* _{Y}*<

*α*, ∀

*t*∈R.

Also, let*P*(*Y*) be the set of all bounded continuous functions*φ*:R→*Y*such that

lim

*T→∞*
*T*

*T*

–T

*φ*(*t*)* _{Y}dt*= .

**Deﬁnition .** A continuous function*g*:R→*Y*is called pseudo almost periodic if*g*=

*h*+*φ*, where*h*∈AP(*Y*) and*φ*∈*P*(*Y*).

The functions*h*and*φ* are called, respectively, the almost periodic component and the
ergodic perturbation of the function*g*. As usual, we represent by PAP(*Y*) the set of all
pseudo almost periodic functions*h*: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 function*h*:R×*Y*→*Z*such that

*h*(·,*x*) is an almost periodic function uniformly on bounded sets of*Y*, that is, for all*α*>
and any bounded subset*K*of*Y* there exists*l*(*α*) > such that all interval of length*l*(*α*)
contains a number*τ*with the property that

*h*(*t*+*τ*,*x*) –*h*(*t*,*x*)* _{Z}*≤

*α*,

for all*t*∈Rand*x*∈*K*.

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 by*P*(*Y*,*Z*) the set of all continuous functions*φ*:R×*Y*→*Z*
such that*φ*(·,*x*) :R→*Z*is a bounded function for all*x*∈*Y*and

lim

*T→∞*
*T*

*T*

–T

*φ*(*t*,*x*)* _{Z}dt*= ,

uniformly in*x*∈*Y*.

**Deﬁnition .** A continuous function*g*:R×*Y*→*Z*is called pseudo almost periodic if

*g*=*h*+*φ*, where*g*∈AP(*Y*,*Z*) and*φ*∈*P*(*Y*,*Z*).

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

(*X*_{,}* _{X}*

_{) if there exist}

_{ρ}_{> ,}

_{γ}_{(}

_{}_{) with}

_{ρ}_{≤}

_{γ}_{(}

_{}_{) < , and a positive constant}

_{c}_{, such that}

*g*:*X*+_{→}* _{X}γ*()

_{and}

* _{g}*(

*x*) –

*g*(

*y*)

*()≤*

_{X}γ*c*

+*xρ _{X}*–++

*y*

*ρ*–

*X*+

*x*–*y*X+,

for all*x*,*y*∈*X*+_{.}

The main result of this work basically says that if*f*(*t*,·) is an-regular map relative to the
pair (*X*_{,}* _{X}*

_{) and belongs to}

_{PAP}

_{(}

*+*

_{X}_{,}

*()*

_{X}γ_{) 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 in*X*
without the nonlinearity being deﬁned on*X*.

**3 Abstract results**

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

*t*+*θ*–*βeAtx _{X}*+

*θ*≤

*Me*–at

*x*X

*β*, ≤

*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 functions*f* such that, for*t*∈R,*f*(*t*,·) is an-regular map relative to the
pair (*X*_{,}* _{X}*

_{), satisfying}

*f*(*t*,*x*) –*f*(*t*,*y*)* _{X}γ*()≤

*c*

+*xρ _{X}*–++

*y*

*ρ*–

*X*+

*x*–*y*X+, (.)

*f*(*t*,*x*)* _{X}γ*()≤

*c*

+*xρ _{X}*+

, (.)

for all*x*,*y*∈*X*+_{.}

The following theorem is the main result of this work.

**Theorem .** *Consider f* ∈*F*(,*γ*(),*ρ*,*c*)∩PAP(*X*+_{,}* _{X}γ*()

_{).}

_{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 every}_{}_{≤}_{θ}_{<}_{γ}_{(}_{}_{).}

*Proof* Fix*r*> and let*L*(*r*) =max{*r*–_{+}* _{r}ρ*–

_{, + }

*–*

_{r}ρ_{}}

_{. We claim that the result is true if}

*Mca*–*γ*()*γ*() –*L*(*r*) < .
In fact, let*B*(*r*) be the closed ball

*B*(*r*) :=

*w*∈PAP*X*+:sup

*t∈*R
_{w}_{(}_{t}_{)}

Deﬁne on*B*(*r*) the operator*T*by

(*Tu*)(*t*) =

*t*

–∞*e*

*A(t–s) _{f}_{s}*

_{,}

_{u}_{(}

_{s}_{)}

_{ds}_{.}

The set*B*(*r*) is*T*-invariant. We divide the proof of this claim in two steps.

Step : If*u*∈PAP(*X*+_{) then}_{Tu}_{∈}_{PAP}_{(}* _{X}*+

*θ*

_{), for every }

_{≤}

_{θ}_{<}

_{γ}_{(}

_{}_{). First of all, we show}

that the function

*t*→*ft*,*u*(*t*)

belongs toPAP(*Xγ*()_{). In fact, since}_{f}_{∈}_{PAP}_{(}* _{X}*+

_{,}

*()*

_{X}γ_{) there exist}

_{g}_{∈}

_{AP}

_{(}

*+*

_{X}_{,}

*()*

_{X}γ_{) and}

*φ*∈

*P*(

*X*+,

*Xγ*()) such that

*f*=

*g*+

*φ*. In the same way,

*u*=

*h*+

*ψ*with

*h*∈AP(

*X*+) and

*ψ*∈

*P*(

*X*+). Hence,

*f*·,*u*(·)=*g*·,*h*(·)+*f*·,*u*(·)–*g*·,*h*(·)

=*g*·,*h*(·)+*f*·,*u*(·)–*f*·,*h*(·)+*φ*·,*h*(·).

Since*g*(·,*h*(·))∈AP(*Xγ*()_{), see Lemma ., it remains to show that the function}

*t*→*ft*,*u*(*t*)–*ft*,*h*(*t*)+*φt*,*h*(*t*):=*F*(*t*) +*φt*,*h*(*t*)
belongs to*P*(*Xγ*()). To this end, note that

*T*

*T*

–T

*F*(*t*)* _{X}γ*()

*dt*

≤ *c*
*T*

*T*

–T

*u*(*t*) –*h*(*t*)* _{X}*+

*dt*

+ *c*
*T*

*T*

–T

* _{u}*(

*t*)

*ρ*–+

_{X}*u*(

*t*) –

*h*(

*t*)

*+*

_{X}*dt*

+ *c*
*T*

*T*

–T

*h*(*t*)*ρ _{X}*–+

*u*(

*t*) –

*h*(

*t*)

*+*

_{X}*dt*

≤*c* +sup

*t∈*R

*u*(*t*)*ρ _{X}*–++sup

*t∈*R

*h*(*t*)*ρ _{X}*–+

_{}

*T*

*T*

–T

*ψ*(*t*)* _{X}*+

*dt*.

Thus,*F*∈*P*(*Xγ*()). On the other hand, since*h*(R) is relatively compact in*X*+, for*α*>
we can ﬁnd a ﬁnite number of open balls*Ok*with center*xk*∈*h*(R) and radius less than*α*
such that*h*(R)⊂*m _{k=}Ok*. Clearly, we can writeR=

*m*

*k=Bk*, where*Bk*=*h*–(*Ok*) are open
subsets. Consider

*Ek*:=

*B*, *k*= ,

*Bk*\*k–i=* *Bi*, *k*> ,

then*Ei*∩*Ej*=∅, if*i*=*j*. Let*T*> be such that

*T*

*T*

–T

con-which goes to zero as*t*→*t*+. The case*t*>*t*is similar. Furthermore,
interval of length*l*(*α*) contains a number*τ* such that

*g*(*t*+*τ*) –*g*(*t*)* _{X}γ*()≤

It is not hard to check that

lim

*T→∞φT*(*t*) = .

Next, since*φT*(*s*) is bounded and the function*s*→*s*–+*γ*()–*θe*–asis integrable in [,∞),
using the Lebesgue dominated convergence theorem we have

≤*Mc*

*t*

–∞(*t*–*s*)
–+*γ*()–

*e*–a(t–s)*ds* +*rρ*

≤*Mca*–*γ*()*γ*() – +*rρ*

≤*r*.

Now, consider*u*,*v*∈*B*(*r*). We have

(*Tu*)(*t*) – (*Tv*)(*t*)* _{X}*+

≤*M*

*t*

–∞(*t*–*s*)
–+*γ*()–

*e*–a(t–s)*fs*,*u*(*s*)–*fs*,*v*(*s*)* _{X}γ*()

*ds*≤

*Mc*

*t*

–∞(*t*–*s*)

–+*γ*()–* _{e}*–a(t–s)

_{ +}

_{u}_{(}

_{s}_{)}

*ρ*–

*X*++*v*(*s*)
*ρ*–

*X*+*u*(*s*) –*v*(*s*)* _{X}*+

*ds*

≤*Mc* + *rρ*–

*t*

–∞(*t*–*s*)
–+*γ*()–

*e*–a(t–s)*ds*

sup

*t∈*R

*u*(*t*) –*v*(*t*)* _{X}*+

≤*Mca*–*γ*()*γ*() – + *rρ*–sup

*t∈*R

*u*(*t*) –*v*(*t*)* _{X}*+.

Therefore, the operator*T* :*B*(*r*)→*B*(*r*) is a contraction and by the Banach contraction
principle we ﬁnd that*T* has a unique ﬁxed point.
**Remark .**

• In particular we can obtain an existence theorem in*X*without the nonlinearity being
deﬁned on*X*_{. Notice that we do not assume that}_{f}_{is a well-deﬁned map on}* _{X}*

_{. The}only requirement on

*f*is that it is an-regular map relative to(

*X*

_{,}

**

_{X}_{)}

_{, for some}

_{}_{> }

_{.}• Another important consequence of Theorem . is given when the nonlinear term

*f* ∈*F*(,*γ*(),*c*)∩*Z*, where

*Z*∈AP*X*+_{,}* _{X}γ*()

_{,}

**

_{P}

*X*+_{,}* _{X}γ*()

_{.}

In fact in this situation we can show that the operator*T*:*Z*→*Z*is well deﬁned.
Under the conditions of Theorem . there exists an-regular mild solution*u*of the
problem (.). Furthermore, this solution veriﬁes*u*∈*Y*( +*θ*), for every≤*θ*<*γ*(),
where*Y*(*β*)∈ {AP(*Xβ*_{),}_{P}

(*Xβ*)}.

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

*f* ∈*F*,*γ*(),*ρ*,*c*∩PAP*X*+,*Xγ*()

there exists*λ*> such that for any≤*λ*≤*λ*, and Theorem . applies to the
function*λf*.

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*, and*c*positive constants,
deﬁne

as the family of functions*f* such that, for*t*∈R,*f*(*t*,·) is an-regular map relative to the
pair (*X*_{,}* _{X}*

_{), satisfying}

*f*(*t*,*x*) –*f*(*t*,*y*)* _{X}γ*()≤

*cx*–

*y*X+,

for all*x*,*y*∈*X*+_{.}

**Theorem .** *Consider f*∈*E*(,*γ*(),*ρ*,*c*)∩PAP(*X*+_{,}* _{X}γ*()

_{).}

_{Suppose that}*Mca*–*γ*()*γ*() –< .

*Then the problem*(.)*has a unique pseudo almost periodic-regular mild solution*.
*Fur-thermore*,*this solution veriﬁes u*∈PAP(*X*+*θ*_{),}_{for every}_{}_{≤}_{θ}_{<}_{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) _{f}_{s}*

_{,}

_{u}_{(}

_{s}_{)}

_{ds}_{,}

*t*∈R, is well deﬁned for every ≤*θ*<*γ*(). Furthermore, if*u*,*v*∈PAP(*X*+_{) then}
(*Tu*)(*t*) – (*Tv*)(*t*)* _{X}γ*()≤

*M*

*t*

–∞(*t*–*s*)

–+*γ*()–* _{e}*–a(t–s)

_{f}_{s}_{,}

_{u}_{(}

_{s}_{)}

_{–}

_{f}_{s}_{,}

_{v}_{(}

_{s}_{)}

*Xγ*()

*ds*≤

*Mc*

*t*

–∞(*t*–*s*)
–+*γ*()–

*e*–a(t–s)*u*(*s*) –*v*(*s*)* _{X}*+

*ds*

≤*Mca*–*γ*()*γ*() –sup

*s∈*R

*u*(*s*) –*v*(*s*)* _{X}*+.

Therefore, the operator*T*:PAP(*X*+_{)}_{→}_{PAP}_{(}* _{X}*+

_{) is a [}

*–*

_{Mca}*γ*()

_{}_{(}

_{γ}_{(}

_{}_{)–}

_{}_{)]-contraction}and by the Banach contraction principle

*T*has a unique ﬁxed point. To close this section we remember that in [] the authors introduce a classiﬁcation for a time-independent map

*f*which is-regular, for∈

*I*, relative to the pair (

*X*

_{,}

**

_{X}_{). For}the convenience of the reader we recall this classiﬁcation.

• If*I*= [,]for some> and*γ*() > , we say that*f* is a*subcritical*map relative to
(*X*,*X*).

• If*I*= [,]for some> with*γ*() =*ρ*,∈*I*, and if*f* is not subcritical, then we
say that*f* is a*critical*map relative to(*X*,*X*).

• If*I*= (,]for some> with*γ*() =*ρ*,∈*I*, and*f* is not subcritical or critical,
then we say that*f* is a*double critical*map relative to(*X*,*X*).

• If*I*= [,]for some>> with*γ*() >*ρ*and*f* is not subcritical, critical or
double critical, then we say that*f* is an*ultra*-*subcritical*map relative to(*X*,*X*).
• If*I*= [,]for some>> with*γ*() =*ρ*,∈*I*, and if*f* is not subcritical,

**4 Applications to heat equation**

Let⊂R*N* _{be a bounded smooth domain. In this section we will treat the equation}

*ut*=*Au*+*gh*(*u*), inR×,

*u*= , onR×*∂*, (.)

where*g*∈PAP(R),*Au*=*u*–*au*,*a*> , and the function*h*:R→Rveriﬁes

*h*(*x*) –*h*(*y*)≤*c* +|*x*|*ρ*–_{+}_{|}_{y}_{|}*ρ*–_{|}_{x}_{–}_{y}_{|}_{,} _{ρ}_{> ,} _{(.)}

for some*c*> . 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
the*Lq*_{theory, <}_{q}_{<}_{∞}_{and}_{q}_{=}*N*(*ρ*–)

.

Let*L*=with Dirichlet boundary conditions in. Then*L*can be seen as an unbounded
operator in*E*

*q*=*Lq*() with domain

*D*(*A*) :=*W*,q()∩*W*_{},q().

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

*q*}∈Rassociated to*L*veriﬁes

(see [, ])

*Eα _{q}* →

*H*

_{q}*α*(),

*α*≥, <

*q*<∞,

*E*–*α*

*q* →

*Eα*

*q*

, *α*≥, <*q*<∞,*q*= *q*

*q*– .
Therefore, we get

*Eα _{q}* →

*Lr*(), for

*r*≤

*Nq*

*N*– *qα*, ≤*α*≤

*N*

*q*,
*E** _{q}*=

*Lq*(),

*Eα*

*q*←*Lr*(), for*r*≥

*N*
*N*– *qα*, –

*N*

*q*≤*α*≤.

Let*Lα*be the realization of*L*in*Eqα*. Then we ﬁnd that*Lα*is an isometry from*Eqα*+into*Eqα*
and

*Lα*:*D*(*Lα*) =*E _{q}α*+⊂

*Eα*→

_{q}*Eα*

_{q}is a sectorial operator. Denote*Xα*

*q* :=*Eαq*–,*α*∈R. The fractional power spaces associated
to*L*–:*Xq*⊂*Xq*→*Xq*satisfy

*Xα*

*q* →*Lr*(), for*r*≤

*Nq*

*N*+ *q*– *qα*, ≤*α*<
*q*+*N*

*q* ,
*X _{q}*=

*Lq*(),

*X _{q}α*←

*Lr*(), for

*r*≥

*Nq*

*N*+

*q*–

*qα*,

*q*–*N*

Consider*f*:R×R→Rdeﬁned by

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

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

• If*q*>_{N}N_{–}, then*f* is an-regular map relative to(*X _{q}*,

*X*)for =(

_{q}*q*)≤<

_{N}N_{+q}and

*γ*() =

*ρ*. Therefore

*f*is a critical map.

• If*q*=_{N}N_{–}, then*f* is an-regular map relative to(*X*

*q*,*X**q*)for =(*q*) <<* _{N+q}N* and

*γ*() =

*ρ*. Therefore

*f*is a double critical map.

• If <*q*<* _{N–}N* , then

*f*is an-regular map relative to(

*X*

*q*,*Xq*)for <(*q*) <<* _{N+q}N* ,
with(

*q*) =

_{N}N_{+q}( –

*N*

_{}( –

*)) > , and*

_{q}*γ*() =

*ρ*. Therefore

*f*is an ultra-critical map. A straightforward computation shows that

*f* ∈*F*,*γ*(),*ρ*,*c*∩PAP*X*+,*Xγ*(),

for any∈((*q*),* _{N+q}N* ). If the constants

*M*and

*c*are as in Theorem . then there exists a mild solution

*u*∈PAP(

*X*+

_{) of (.) such that}

_{u}_{∈}

_{PAP}

_{(}

*+*

_{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**

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

Matemática, Universidade Federal de Pernambuco, Recife, PE 50540-740, Brazil.3_{Department of Mathematics, Shanghai}
Jiao Tong University, Shanghai, 200240, People’s Republic of China. 4_{Departamento 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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