# Asymptotic Periodicity for Strongly Damped Wave Equations

Authors: Claudio Cuevas; Carlos Lizama; Herme Soto
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This work deals with the existence and uniqueness of asymptotically almost-periodic mild solutions for a class of strongly damped semilinear wave equations.


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## Body

## 1. Introduction

LetX be a reflexive Banach space and let A:D(A)⊆X→X be a closed densely defined operator and η>0. Consider the Cauchy problem
(1)utt+2ηA1/2ut+Au=f(t,u,ut),t>0,u(0)=u0∈X1/2,ut(0)=v0∈X,
where X1/2 is the fractional power space associated with A as in [1]. Equations like (1) appear in the literature under the name of strongly damped wave equations. An example of mathematical model represented in form (1) is the wave equation with structural damping (see [2–5]). The strongly damped wave equations has been investigated in several contexts by many authors in the last years, for example, existence [6, 7], global classical solution [6, 8], long-time asymptotic behavior [9–11], attractor [2, 12–16], well-posedness [17], decay estimates [18], blowup [8, 19, 20], controllability [19], bootstrapping, and regularity [21]. Another important aspect of the qualitative study of the solutions of strongly damped wave equations is their asymptotic periodicity. In recent years, the study of periodicity and its various extensions for evolution equations has attracted a great deal of attention of many mathematicians (see [22–32] and references therein).To the best of our knowledge, the study of the existence of asymptotically almost-periodic solutions for strongly damped wave equations of type (1) is a topic not yet considered in the literature.Problem (1) can be written as a first order in time Cauchy problem in Y0:=X1/2×X:
(2)[uv]t+𝒜(1/2)[uv]=F(t,[uv]),t>0,[u(0)v(0)]=[u0v0],
where
(3)𝒜(1/2)=[0-IA2ηA1/2]:D(𝒜(1/2))⊆X1/2×X⟶X1/2×X
is defined by
(4)𝒜(1/2)[φψ]=[-ψAφ+2ηA1/2ψ]for[φψ]∈D(𝒜(1/2))=X1×X1/2,F(t,[uv])=[0f(t,u,v)].Definition 1.
The pair(η,A) is said to be an admissible pair if there exist ψ∈(0,π/2) and M>0 such that
(5)∥(λI-A)-1∥ℒ(X)≤M1+|λ|,
for all λ in the sector Σψ={λ∈ℂ:ψ≤|argλ|≤π}∪{0} and π/2>ψ/2+arg(η+η2-1).
If(η,A) is an admissible pair, by [21, Proposition 2.1] 𝒜(1/2) is a closed operator with 0∈ρ(𝒜(1/2)). Indeed, 𝒜(1/2) has the inverse expressed in the matrix form as
(6)𝒜(1/2)-1=[2ηA-1/2A-1-I0]∈ℒ(X1/2×X).
Moreover, the operator 𝒜(1/2) has compact resolvent whenever A has compact resolvent. By [21, Theorem 2.3] the operator 𝒜(1/2) is sectorial in X1/2×X. The semigroup {e-𝒜(1/2)t:t≥0} generated by -𝒜(1/2) in X1/2×X is exponentially decaying analytic. That is, there are constants K≥1 and C>0 such that
(7)∥e-𝒜(1/2)t∥ℒ(X1/2×X)≤Ke-Ct,t≥0.
Throughout this paper, we always assume that (η,A) is an admissible pair.
This paper has four sections. In the next section, we consider some definitions, technical aspects, and basic properties related to asymptotically almost-periodic functions. In the third section, we obtain general results on the existence of asymptotically almost-periodic (mild) solutions to the problem (1). The main abstract results are Theorems 12, 16, 17, and 19. Finally, in the fourth section we consider several applications. In particular, we consider the following class of partial differential equations:
(8)utt+νb(t)ut+Δ2u-δΔut=μa(t)(h(u)∇·u+g(u)Δu),x∈Ω,t≥0u=Δu=0,x∈∂Ω,t>0,u(0,x)=u0(x),ut(0,x)=v0(x),x∈Ω,
in a bounded smooth domain Ω⊆ℝN and where h and g satisfy certain growth conditions. We prove that if a∈AP(ℝ), b∈C0(ℝ+,ℝ), and |μ|+|ν| is small enough, then the previous problem has an asymptotically almost periodic mild solution. The same type of conclusion is derived for the wide class
(9)utt+Δ2u-δΔut=a(t)|∫Ω‍∇·u(t,x)dx|βΦ0,Δ2u-δΔut=a(t)Δ2u-δΔutlllx∈Ω,t≥0,u=Δu=0,x∈∂Ω,t>0,u(0,x)=u0(x),ut(0,x)=v0(x),x∈Ω,
where Φ0∈Lp(Ω) and p>N/2.


## 2. Preliminaries

In this section, we present some concepts and properties needed to develop the following sections. LetX be a reflexive Banach space. For an interval I⊆ℝ, Cb(I,X) denotes the space formed by the bounded continuous functions from I into X, endowed with the norm of uniform convergence. When X=ℝ, we denote Cb(I) instead of Cb(I,ℝ). The notation C0(ℝ+,X) stands for the subspace of Cb(ℝ+,X) consisting of functions that vanish at infinity. We denote by ℒ(X) the Banach algebra of bounded linear operators defined on X. For r>0, the notation Br(X) stands for the closed ball {x∈X:∥x∥≤r}. For a linear operator A with domain D(A) and range ℛ(A) in X, we represent by σ(A) (resp., ρ(A)) the spectrum (resp., the resolvent set) of A. For λ∈ρ(A), we denote by R(λ,A)=(λI-A)-1 the resolvent operator of A. When A is closed, we denote by [D(A)] the domain of A endowed with the graph norm ∥x∥A=∥x∥+∥Ax∥.Next, we present a brief summary of the main properties of asymptotically almost-periodic functions.Definition 2 (see [33]).
A continuous functionf:ℝ→X is called almost-periodic if for each ε>0 there exists l(ε)>0 such that for every interval of length l(ε) it contains a number τ with the property that ∥f(t+τ)-f(t)∥≤ε for each t∈ℝ.
The previous numberτ is called an ε-translation number for f. We denote by AP(X) the space formed by the almost periodic functions f:ℝ+→X. We note that each almost-periodic function is bounded and uniformly continuous. It is well known that the range ℛ(f)={f(t):t∈ℝ} of an almost periodic function f is relatively compact. The space AP(X) is a Banach space endowed with the norm of uniform convergence.
LetY be a Banach space. We have the following concept of parameter-dependent almost-periodic function.Definition 3.
A continuous functionf:ℝ×Y→X is called almost-periodic in t uniformly for y in compact subsets of Y if for every compact subset K of Y and each ε>0 there exists l(ε)>0 such that every interval I of length l(ε) contains a number τ with the property that ∥f(t+τ,y)-f(t,y)∥≤ε for all t∈ℝ, y∈K.
Henceforth, we abbreviate the terminology by calling almost-periodic fromℝ×Y into X to those functions that are almost-periodic in t uniformly for y in compact subsets of Y, and we denote by AP(ℝ×Y,X) the set formed by the almost-periodic functions from ℝ×Y into X.
The proof of the following result is similar to the proof of [22, Lemma 2.12] and therefore omitted.Lemma 4.
Ifh:ℝ→X1/2×X is an almost-periodic function and
(10)u(t)=∫-∞t‍e-𝒜(1/2)(t-s)h(s)ds,t∈ℝ,
then u is almost-periodic.It is well known that the study of composition of two functions with special properties is important and basic for deep investigations. The following result has been established in [34].Lemma 5.
Letf:ℝ×Y→X be an almost-periodic function and let u:ℝ→Y be an almost-periodic function. Then the function ℝ→X,t→f(t,u(t)) is almost-periodic.We will need the following definition.Definition 6.
A continuous functionf:ℝ+→X is called asymptotically almost-periodic if there exist two functions fap∈AP(X) and φf∈C0(ℝ+,X) such that
(11)f(t)=fap(t)+φf(t),t∈ℝ+.
The function fap is called the almost-periodic part of f. We denote by AAP(X) the space formed by the asymptotically almost-periodic functions f:ℝ+→X. The space AAP(X) is a Banach space endowed with the norm of uniform convergence. Furthermore, AAP(X)=AP(X)⊕C0(ℝ+,X).
In what followsC0(ℝ+×Y,X) denotes the space consisting of continuous functions f:ℝ+×Y→X such that limt→∞f(t,y)=0 uniformly for y in compact subsets of Y.Definition 7.
A continuous functionf:ℝ+×Y→X is called asymptotically almost-periodic if there are two functions g∈AP(ℝ×Y,X) and φ∈C0(ℝ+×Y,X) such that
(12)f(t,y)=g(t,y)+φ(t,y),y∈Y,t∈ℝ+.
We denote by AAP(ℝ+×Y,X) the set consisting of all asymptotically almost-periodic functions from ℝ+×Y into X.
LetI⊆ℝ be an interval. We have the following concept of function uniformly continuous on compacts sets.Definition 8.
A continuous functionf:I×Y→X is called uniformly continuous on compact sets if for all compact set K⊆Y and all ε>0 there is δε,K>0 such that ∥f(t,y1)-f(t,y2)∥≤ε for all t∈I and y1,y2∈K with ∥y1-y2∥≤δε,K.Lemma 9 (see [22]).
Letf:ℝ+×Y→X be an asymptotically almost-periodic and uniformly continuous on compact sets function. Let u:ℝ+→Y be an asymptotically almost-periodic function. Then the function ℝ+→X, t→f(t,u(t)) is asymptotically almost-periodic.The proof of the following result is similar to the proof of [22, Lemma 2.13]. Therefore, we will omit it.Lemma 10.
Ifh:ℝ+→X1/2×X is an asymptotically almost-periodic function and
(13)u(t)=∫0t‍e-𝒜(1/2)(t-s)h(s)ds,t≥0.
Then, u is asymptotically almost-periodic.


## 3. Asymptotically Almost-Periodic Mild Solutions

We recall the following definition that will be essential for us.Definition 11.
Let[u0v0] be in X1/2×X. We say that [u(·)v(·)]:ℝ+→X1/2×X is a mild solution to (2) (or to (1)) if it satisfies the Cauchy integral formula:
(14)[u(t)v(t)]=e-𝒜(1/2)t[u0v0]+∫0t‍e-𝒜(1/2)(t-s)F(s,[u(s)v(s)])ds,[u(t)v(t)]=e-𝒜(1/2)t[u0v0]+∫0t‍e-𝒜(1/2)(t-s)Fllllllllllllllt≥0.Theorem 12.
Letf:ℝ+×X1/2×X→X be an asymptotically almost-periodic function and assume that there exists a locally integrable function Lf:ℝ→ℝ satisfying
(15)∥f(t,u1,v1)-f(t,u2,v2)∥X≤Lf(t)[∥u1-u2∥X1/2+∥v1-v2∥X],
for all [uivi]∈X1/2×X, i=1,2 and each t≥0. If
(16)Ksupt≥0∫0t‍e-C(t-s)Lf(s)ds<1,
where K and C are given in (7). Then, (2) has a unique asymptotically almost-periodic mild solution.Proof.
We define the map𝔉 on the space AAP(X1/2×X) by the expression
(17)𝔉([uv])(t)=e-𝒜(1/2)t[u0v0]𝔉([uv])(t)+∫0t‍e-𝒜(1/2)(t-s)F(s,[u(s)v(s)])ds,t∈ℝ+,
where
(18)[u(·)v(·)]=[uap(·)vap(·)]+[φu(·)φv(·)]
is an asymptotically almost-periodic function; that is,
(19)[uap(·)vap(·)]∈AP(X1/2×X),[φu(·)φv(·)]∈C0(ℝ+,X1/2×X).
Since f∈AAP(ℝ+×X1/2×X,X); then there are two functions Φ∈AP(ℝ+×X1/2×X,X) and Ψ∈C0(ℝ+×X1/2×X,X) so that f=Φ+Ψ.
We set for[ab]∈X1/2×X(20)Φ~(s,[ab])=[0Φ(s,a,b)],Ψ~(s,[ab])=[0Ψ(s,a,b)].
For t∈ℝ+, we have the following decomposition:
(21)𝔉([uv])(t)=∫0t‍e-𝒜(1/2)(t-s)(Φ~(s,[u(s)v(s)])-Φ~(s,[uap(s)vap(s)]))ds+∫-∞t‍e-𝒜(1/2)(t-s)Φ~(s,[uap(s)vap(s)])ds-∫-∞0‍e-𝒜(1/2)(t-s)Φ~(s,[uap(s)vap(s)])ds+e-𝒜(1/2)t[u0v0]+∫0t‍e-𝒜(1/2)(t-s)Ψ~(s,[u(s)v(s)])ds:=𝒢(t)+ℋ(t),
where
(22)𝒢(t)=∫-∞t‍e-𝒜(1/2)(t-s)Φ~(s,[uap(s)vap(s)])ds
and ℋ(t) denotes the remained terms of the previous decomposition.
Next, let us show thatℋ∈C0(ℝ+,X1/2×X). By (7) we have that
(23)t⟶e-𝒜(1/2)t[u0v0]∈C0(ℝ+,X1/2×X).
We observe that
(24)ℬ:={[u(t)v(t)]:t≥0},ℬap:={[uap(t)vap(t)]:t∈ℝ}
are relatively compact in X1/2×X. Since Ψ∈C0(ℝ+×X1/2×X,X), for each ε>0 there exists a constant T>0 such that
(25)∥Ψ~(s,z)∥X1/2×X≤ε
for all t≥2T,z∈ℬ.
We deduce(26)∥∫0t‍e-𝒜(1/2)(t-s)Ψ~(s,[u(s)v(s)])ds∥X1/2×X≤KCe-Ct/2supt∈ℝ+,z∈ℬ∥Ψ~(s,z)∥X1/2×X+KCε.
On the other hand, since Φ∈AP(ℝ+×X1/2×X,X) by [35, Appendix] we get
(27)∥∫-∞0‍e-𝒜(1/2)(t-s)Φ~(s,[uap(s)vap(s)])ds∥X1/2×X≤KCe-Ctsupt∈ℝ,z∈ℬap∥Φ(s,z)∥X,fort∈ℝ+.
Next, we estimate the first term of (21). For ε>0, we choose T>0 big enough so that
(28)∥φu(s)∥X1/2+∥φv(s)∥X≤ε,
for all s≥T, and ∥Ψ~(s,z)∥X1/2×X≤ε for all s≥T and all z∈ℬ∪ℬap. We have the following estimates:
(29)∥∫0t‍e-𝒜(1/2)(t-s)(Φ~(s,[u(s)v(s)])∫0t‍e-𝒜(1/2)(t-s)-Φ~(s,[uap(s)vap(s)]))ds∥X1/2×X≤K∫0t‍e-C(t-s)Lf(s)×(∥u(s)-uap(s)∥X1/2+∥v(s)-vap(s)∥X)ds+K∫0t‍e-C(t-s)∥(s,[uap(s)vap(s)])Ψ~(s,[u(s)v(s)])K∫0t‍e-C(t-s)lll-Ψ~(s,[uap(s)vap(s)])∥X1/2×Xds≤K∫0T‍e-C(t-s)Lf(s)(∥φu(s)∥X1/2+∥φv(s)∥X)ds+K∫Tt‍e-C(t-s)Lf(s)(∥φu(s)∥X1/2+∥φv(s)∥X)ds+(K∫0T‍e-C(t-s)ds)×(supξ∈ℝ+,z∈ℬ∥Ψ~(ξ,z)∥X1/2×Xll+supξ∈ℝ+,z∈ℬap∥Ψ~(ξ,z)∥X1/2×X)+K∫Tt‍e-C(t-s)×(∥∥X1/2×X+∥Ψ~(s,[uap(s)vap(s)])∥∥Ψ~(s,[u(s)v(s)])∥X1/2×X+∥Ψ~(s,[uap(s)vap(s)])∥X1/2×X)ds≤Ke-Ct(∫0T‍eCsLf(s)ds)supξ∈ℝ+∥[φu(ξ)φv(ξ)]∥X1/2×X+εK+KCeCTe-Ct×(supξ∈ℝ+,z∈ℬ∥Ψ~(ξ,z)∥X1/2×Xll+supξ∈ℝ+,z∈ℬap∥Ψ~(ξ,z)∥X1/2×X)+2εKC.
From (23) to (29), we deduce that limt→∞ℋ(t)=0; that is, ℋ∈C0(ℝ+,X1/2×X).
SinceΦ~∈AP(ℝ+×X1/2×X,X1/2×X) and [uapvap]∈AP(X1/2×X), we get from Lemma 5 that
(30)Φ~(·,[uap(·)vap(·)])∈AP(X1/2×X).
Now, by Lemma 4, we obtain that 𝒢∈AP(X1/2×X), and hence 𝔉 is well defined. It suffices to show that the operator 𝔉 has a unique fixed point in AAP(X1/2×X).
For this, we consider that[u1v1],[u2v2]∈AAP(X1/2×X). We can deduce that
(31)∥𝔉[u1v1]-𝔉[u2v2]∥∞=supt≥0∥𝔉([u1v1])(t)-𝔉([u2v2])(t)∥X1/2×X≤Ksupt≥0∫0t‍e-C(t-s)Ksupt≥0lll×∥f(s,u1(s),v1(s))-f(s,u2(s),v2(s))∥Xds≤Ksupt≥0∫0t‍e-C(t-s)Lf(s)Ksupt≥0ll×[∥u1(s)-u2(s)∥X1/2+∥v1(s)-v2(s)∥X]ds≤(Ksupt≥0∫0t‍e-C(t-s)Lf(s)ds)∥[u1v1]-[u2v2]∥∞,
by the contraction principle, 𝔉 has a unique fixed point in AAP(X1/2×X). This completes the proof.Remark 13.
We wish to emphasize that condition (15) is optimal in the sense that the function Lf(·) is locally integrable. This is the largest possible class of Lipschitz constant Lf(·) for which the conclusion of Theorem 12 holds true. However, this condition in Lf(·) makes our analysis much more harder, because to prove Theorem 12 we cannot use the standard composition lemma for asymptotically almost-periodic functions (see Lemma 9). To overcome this difficulty we need to use a suitable decomposition for the natural operator associated with the mild solution (see (14) and (21)). In contrast, we note that in the more restrictive case of Lf(·) to be an integrable bounded function we can use Lemma 9 directly.Corollary 14.
Letf:ℝ+×X1/2×X→X be an asymptotically almost-periodic function that satisfies the Lipschitz condition (15) with Lf(·)≡L. If
(32)KLC<1,
then problem (2) has a unique asymptotically almost-periodic mild solution.Remark 15.
Letf:ℝ+×X1/2×X→X be an asymptotically almost-periodic function that satisfies the Lipschitz condition (15). We can avoid the condition (32) by using the fixed-point iteration method. Indeed, we consider two cases.Case 1.
L
f
(
·
)
≡
L. We consider the following space
(33)AAP0(X1/2×X):={[uv]∈AAP(X1/2×X):u(0)=0,v(0)=0}
endowed with the norm of the uniform convergence. We define the map Γ on the space AAP0(X1/2×X) by
(34)Γ([uv])(t)=∫0t‍e-𝒜(1/2)(t-s)×F(s,[u(s)v(s)]+e-𝒜(1/2)s[u0v0])ds.
Combining Lemmas 9 and 10, we know that Γ is a continuous function from AAP0(X1/2×X) into AAP0(X1/2×X). Moreover, for [uivi]∈AAP0(X1/2×X), i=1,2 we have
(35)∥Γ([u1v1])(t)-Γ([u2v2])(t)∥X1/2×X≤KL∫0t‍e-C(t-s)sup0≤ξ≤s[∥u1(ξ)-u2(ξ)∥X1/2+∥v1(ξ)-v2(ξ)∥X∥u1(ξ)-u2(ξ)∥(ξ)]ds.
With the notation Φ(s)=KLe-Cs and
(36)α(s)=sup0≤ξ≤s[∥u1(ξ)-u2(ξ)∥X1/2+∥v1(ξ)-v2(ξ)∥X],
the previous estimate yields
(37)∥Γ([u1v1])(t)-Γ([u2v2])(t)∥X1/2×X≤(Φ*α)(t),t≥0.
Since Φ*α is a nondecreasing function, proceeding inductively, we can show that
(38)∥Γn([u1v1])(t)-Γn([u2v2])(t)∥X1/2×X≤(Φ*n*α)(t),∥Γn([u1v1])(t)-Γn([u2v2])(t)∥X1/2×X≤lllllllllllt≥0,
where Φ*n denotes the n-fold convolution of Φ with itself.
On the other hand, the map𝒮:Cb(ℝ+)→Cb(ℝ+), given by
(39)(𝒮α)(t)=∫0t‍Φ(t-s)α(s)ds,t≥0,
is a bounded linear map. Moreover, it follows from [36, Theorem2.3.5] that σ(𝒮)={0}, which implies that ∥𝒮n∥=∥Φ*n∥1→0 as n→∞. This shows that Γn is a contraction for n sufficiently large. As a consequence Γ has a fixed point [uv] in AAP0(X1/2×X).
We note that the function(40)[x(t)y(t)]=[u(t)v(t)]+e-𝒜(1/2)t[u0v0]
is an asymptotically almost-periodic mild solution to problem (2).Case 2.
L
f
(
·
) in (15) is an integrable bounded function. By Lemmas 9 and 10 the space AAP(X1/2×X) is invariant under 𝔉 (see (17)). The fixed point iteration method and the following estimate
(41)∥𝔉n[u1v1]-𝔉n[u2v2]∥∞≤(K∥Lf∥1)nn!∥[u1v1]-[u2v2]∥∞
are responsible for the fact that 𝔉 has a unique fixed point in AAP(X1/2×X). This concludes the discussion of Remark 15.
We have the following results of the existence of local type.Theorem 16.
Letf:ℝ+×X1/2×X→X be an asymptotically almost-periodic function that satisfies the Lipschitz condition
(42)∥f(t,u1,v1)-f(t,u2,v2)∥X≤Lf(r)[∥u1-u2∥X1/2+∥v1-v2∥X],
for each t≥0 and [uivi]∈X1/2×X such that ∥ui∥X1/2+∥vi∥X≤r, i=1,2 where Lf:ℝ+→ℝ+ is nondecreasing continuous function such that Lf(0)=0 and f(t,0,0)=0 for all t≥0. Then, there exists ε>0 such that, for each [u0v0]∈X1/2×X satisfying ∥u0∥X1/2+∥v0∥X≤ε, there is a unique asymptotically almost-periodic mild solution of problem (2).Proof.
We chooseλ,r>0 small enough such that λ<1 and K(λ+(1/C)Lf(r))<1. Assume that ∥u0∥X1/2+∥v0∥X≤λr. We consider the space
(43)AAPr:={[uv]∈AAP(X1/2×X):u(0)=u0,v(0)=v0,AAPr:=ll∥u(t)∥X1/2+∥v(t)∥X≤r,t≥0[uv]},
endowed with the norm of the uniform convergence. We consider the map 𝔉 given by (17) on AAPr. By Lemma 9(44)𝔉(AAPr)⊆AAP(X1/2×X).
We next prove that 𝔉(AAPr)⊆AAPr.
In fact, if[uv]∈AAPr, we have
(45)∥𝔉([uv])(t)∥X1/2×X≤Ke-Ct[∥u0∥X1/2+∥v0∥X]+K∫0t‍e-C(t-s)Lf(r)×[∥u(s)∥X1/2+∥v(s)∥X]ds≤K(λ+1CLf(r))r≤r,
which permit us to infer that 𝔉(AAPr)⊆AAPr.
On the other hand, for[uivi]∈X1/2×X,i=1,2, we have that
(46)∥𝔉([u1v1])(t)-𝔉([u2v2])(t)∥X1/2×X≤KLf(r)∫0t‍e-C(t-s)×[∥u1(s)-u2(s)∥X1/2+∥v1(s)-v2(s)∥X]ds≤KCLf(r)∥[u1v1]-[u2v2]∥∞,
which shows that 𝔉 is a contraction from AAPr into itself. Therefore, the assertion holds for ε=λr.Theorem 17.
Letf∈AAP(ℝ+×X1/2×X,X) be a function that satisfies the local Lipschitz condition (42) with Lf(·) a nondecreasing function. Assume that there is a constant r>0 such that
(47)KC(Lf(r+C#)+1r(Lf(C#)C#+supt≥0∥f(t,0,0)∥))<1,
where C#=K[∥u0∥X1/2+∥v0∥X], K, and C are the constant given in (7). Then, there is an asymptotically almost-periodic mild solution of problem (2).Proof.
We define the mapΓ on the closed ball
(48)ℬr={[uv]∈AAP0(X1/2×X):∥u(t)∥X1/2+∥v(t)∥X≤r,AAPr:llt≥0[uv]}
by means of the expression (34).
If[uv]∈ℬr, we have the estimate
(49)∥Γ([uv])(t)∥X1/2×X≤∥Γ([uv])(t)-Γ([00])(t)∥X1/2×X+∥Γ([00])(t)∥X1/2×X≤KLf(r+C#)∫0t‍e-C(t-s)KLf(r+C#)ll×[∥u(s)∥X1/2+∥v(s)∥X]ds+K∫0t‍e-C(t-s)KKklll×∥F(s,e-𝒜(1/2)s[u0v0])∥X1/2×Xds≤KCLf(r+C#)r+KC(Lf(C#)C#+supt≥0∥f(t,0,0)∥)≤KC(1r(Lf(C#)C#+supt≥0∥f(t,0,0)∥))Lf(r+C#)llll+1r(Lf(C#)C#+supt≥0∥f(t,0,0)∥))r≤r.
Moreover, for[uivi]∈ℬr, i=1,2, we have
(50)∥Γ([u1v1])(t)-Γ([u2v2])(t)∥X1/2×X≤KLf(r+C#)∫0t‍e-C(t-s)KLf(r+C#)lll×[∥u1(s)-u2(s)∥X1/2≤KLf(r+C#)mmmm+∥v1(s)-v2(s)∥X]ds≤KLf(r+C#)∥[u1v1]-[u2v2]∥∞.
Using (50), we get that Γ is a contraction on ℬr.In many concrete situations, the operatorA has compact resolvent, which in turn implies that the semigroup e-𝒜(1/2)t generated by the operator -𝒜(1/2) is compact for t>0. To exploit this property of compactness, we need to introduce some preliminaries.LetY be an arbitrary Banach space and let h:[0,∞)→[1,∞) be a nondecreasing continuous function such that h(t)→∞ when t→∞. In next, we denote by Ch(Y) the space
(51)Ch(Y)={u∈C(ℝ+,Y):limt→0u(t)h(t)=0}
endowed with the norm
(52)∥u∥h=supt≥0∥u(t)∥Yh(t).For reference purposes, we state the following property.Lemma 18 (see [26]).
A set𝒦⊆Ch(Y) is relatively compact in Ch(Y) if the following conditions are fulfilled: (C1)
For allb>0, the set 𝒦b={u|[0,b]:u∈𝒦} is relatively compact in C([0,b],Y).(C2)
limt→∞(∥u(t)∥Y/h(t))=0 uniformly for u∈𝒦.To establish our next result we introduce the following condition.(
W
) There is a continuous nondecreasing function Wf:ℝ+→ℝ+ such that
(53)∥f(t,u,v)∥X≤Wf(∥u∥X1/2+∥v∥X),
for all t≥0 and all [uv]∈X1/2×X.We next denote(54)β(L)=supt≥0Kh(t)∫0t‍e-C(t-s)Wf(Lh(s))ds.
We have the following result.Theorem 19.
Assume that the operatorA in (1) has compact resolvent. Suppose, in addition, that the following conditions are fulfilled:(a)
The functionf∈AAP(ℝ+×X1/2×X,X) is uniformly continuous on compact sets and satisfies the condition (W).(b)
For eacha≥0, r>0, and 0≤ξ≤a,
(55)(e-𝒜(1/2)(a+s-ξ)-e-𝒜(1/2)(a-ξ))F(ξ,[uv])⟶0
ass→∞ uniformly for all [uv] so that ∥u∥X1/2+∥v∥X≤r.(c)
For eachL>0,
(56)limt→∞1h(t)∫0t‍e-C(t-s)Wf(Lh(s))ds=0.(d)
For eachε>0, there is δ>0 such that for every [uivi]∈Ch(X1/2×X), i=1,2 with
(57)1h(t)[∥u1(t)-u2(t)∥X1/2+∥v1(t)-v2(t)∥X]≤δ,1hl[∥u1(t)-u2(t)∥X1/2+∥v1(t)-v2(t)∥X]llt≥0
implies that(58)∫0t‍e-C(t-s)∥f(s,u1(s),v1(s))-f(s,u2(s),v2(s))∥Xds≤ε,
for eacht≥0.(e)
liminfξ→∞β(ξ)/ξ<1.
Then, problem (2) has an asymptotically almost-periodic mild solution.Proof.
LetCh0(X1/2×X) be the space consisting of the functions [u(·)v(·)] in Ch(X1/2×X) such that u(0)=0, v(0)=0. It is clear that Ch0(X1/2×X) is a closed subspace of Ch(X1/2×X). We define the operator Γ on Ch0(X1/2×X) by (34). It follows from conditions (a) and (c) that
(59)1h(t)∥Γ([uv])(t)∥X1/2×X≤Kh(t)∫0t‍e-C(t-s)Wf(Lh(s))ds⟶0,1h(t)∥Γ([uv])(t)∥X1/2×Xlllllllt⟶∞,
where
(60)L=∥[uv]∥h+K(∥u0∥X1/2+∥v0∥X).
Thus, we have that Γ:Ch0(X1/2×X)→Ch0(X1/2×X).
We divide the rest of the proof into several steps.(i)
The mapΓ is continuous. For each ε>0 there is δ>0 such that for [uivi]∈Ch0(X1/2×X), i=1,2with(61)∥[u1-u2v1-v2]∥h≤δimplies that(62)1h(t)∥Γ([u1v1])(t)-Γ([u2v2])(t)∥X1/2×X≤1h(t)ε≤ε,which shows the assertion.(ii)
The mapΓ is completely continuous. We take r>0 and we set V=Γ(Br(Ch0(X1/2×X))).Fort≥0, we set
(63)V(t):={[u(t)v(t)]:[uv]∈V}.We first show thatV(t) is a relatively compact set in X1/2×X for each t≥0. It follows from the mean value theorem that V(t)∈tc(𝒦0)¯, where c(𝒦0) denotes the convex hull of
(64)𝒦0={e-𝒜(1/2)(t-s)F(s,[uv]):0≤s≤t,F(s,[uv]):llll∥u∥X1/2+∥v∥X≤ρ(s,[uv])},whereρ=h(t)(r+K(∥u0∥X1/2+∥v0∥X)), and K is a constant given in (7).Since(65)sup{∥F(s,[uv])∥X1/2×X:0≤s≤t,∥u∥X1/2+∥v∥X≤ρ(s,[uv])}≤Wf(ρ),and taking into account thate-𝒜(1/2)t is compact for t>0, we infer that 𝒦0 is a relatively compact set in X1/2×X and consequently V(t) is also relatively compact. Let now b>0 fixed and Vb the set formed by the functions [uv]∈V restricted to the interval [0,b]. We affirm that the set Vb is equicontinuous.In fact, if(66)[uv]=Γ([u~v~]),with[u~v~]∈Br(Ch0(X1/2×X)).Fors≥0, we obtain the following estimate:
(67)∥[u(t+s)-u(t)v(t+s)-v(t)]∥X1/2×X≤K∫tt+s‍e-C(t+s-ξ)Wf(Lh(ξ))dξ+∫0t‍∥[u~(ξ)v~(ξ)](e-𝒜(1/2)ξ[u0v0]e-𝒜(1/2)(t+s-ξ)-e-𝒜(1/2)(t-ξ))llllllllll×F(ξ,[u~(ξ)v~(ξ)]+e-𝒜(1/2)ξ[u0v0])∥X1/2×Xdξ,whereL=r+K(∥u0∥X1/2+∥v0∥X). It is immediate that the first term on the right hand side converges to zero when s→0 and, using condition (b) we obtain the second term on the right-hand side also converges to zero when s→0 and the convergence is independent of the function [u~(·)v~(·)].We now show that(68)limt→∞1h(t)[u(t)v(t)]=0,independent of[u~v~]∈Br(Ch0(X1/2×X)). This assertion is a direct consequence of the following estimate and the condition (c):(69)1h(t)∥[u(t)v(t)]∥X1/2×X≤Kh(t)∫0t‍e-C(t-s)Wf(Lh(s))ds.Combining these assertions with Lemma18, we get that V is a relatively compact set in Ch0(X1/2×X). Since r was chosen arbitrary, this proves that Γ is completely continuous.(iii)
There isr0>0 such that Γ(Br0(Ch0(X1/2×X)))⊆Br0(Ch0(X1/2×X)). In fact, if we assume that the assertion is false; then for all r>0 we can choose [urvr]∈Br0(Ch0(X1/2×X)) such that
(70)1h(t)∥Γ([urvr])(t)∥X1/2×X>rfor allt∈ℝ+. Then
(71)1≤(1+K(∥u0∥X1/2+∥v0∥X)r)×β(r+K(∥u0∥X1/2+∥v0∥X))r+K(∥u0∥X1/2+∥v0∥X),(72)1≤liminfξ→∞β(ξ)ξ,which contradicts condition(e) and establishes the assertion.(iv)
If[uv]∈AAP0(X1/2×X), then the function ℝ+→X1/2×X given by
(73)t⟶[u(t)v(t)]+e-𝒜(1/2)t[u0v0]is inAAP(X1/2×X). Since f∈AAP(ℝ+×X1/2×X,X) is uniformly continuous on compact sets, we have from Lemma 9 that
(74)t⟶F(t,[u(t)v(t)]+e-𝒜(1/2)t[u0v0])∈AAP(X1/2×X).Applying Lemma10, we obtain that Γ(AAP0(X1/2×X))⊆AAP0(X1/2×X). Consequently, combining with (iii) we infer that
(75)Γ(Br0(Ch0(X1/2×X))∩AAP0(X1/2×X)¯h)⊆Br0(Ch0(X1/2×X))∩AAP0(X1/2×X)¯h,where𝒮¯h denotes the closure of 𝒮 in Ch(X1/2×X). Using Schauder fixed point theorem, we deduce that Γ has a fixed point [uv]∈Br0(Ch0(X1/2×X))∩AAP0(X1/2×X)¯h.(v)
Finally, we show that[uv]∈AAP0(X1/2×X). Let ([unvn])n be a sequence in Br0(Ch0(X1/2×X))∩AAP0(X1/2×X) that converges to [uv] for the topology in Ch(X1/2×X). For ε>0, let δ>0 be the constant in condition (d); there is n0∈ℕ so that
(76)1h(t)∥[un(t)-u(t)vn(t)-v(t)]∥X1/2×X≤δ,for allt≥0 and all n≥n0. Therefore, for n≥n0(77)supt≥0∥Γ([unvn])(t)-Γ([uv])(t)∥X1/2×X≤ε.Hence(78)limn→∞∥Γ[unvn]-[uv]∥∞=0.SinceΓ[unvn]∈AAP0(X1/2×X) we get that [uv]∈AAP0(X1/2×X) and completes the proof.Remark 20.
Note that in Theorem19 we do not need to assume that the operator A in (1) has compact resolvent if the following condition holds.(f)
For alla≥0 and r>0 the set {f(t,φ1,φ2):0≤t≤a,||φ1||X1/2+||φ2||X≤r} is relatively compact in X.


## 4. Applications

Suppose thath,g∈C(ℝ,ℝ), a:ℝ→ℝ (b:ℝ+→ℝ) is a bounded continuous function, ν,μ∈ℝ, δ>0 and ρ1,ρ2∈(1,+∞). In a bounded smooth domain Ω⊆ℝN, we consider the following partial differential equation:
(79)utt+νb(t)ut+Δ2u-δΔut=μa(t)(h(u)∇·u+g(u)Δu)utt+νb(t)ut+Δ2u-δΔutΔ2u-δΔutlllllx∈Ω,t≥0,u=Δu=0,x∈∂Ω,t>0,u(0,x)=u0(x),ut(0,x)=v0(x),x∈Ω,
where h and g satisfy the following growth conditions:
(80)|h(s1)-h(s2)|≤Ch|s1-s2|(1+|s1|ρ1-1+|s2|ρ1-1),|h(s1)-h(s2)|≤Ch|h(s1)-h(s2)|≤Chhhhhhhs1,s2∈ℝ,|g(s1)-g(s2)|≤Cg|s1-s2|(1+|s2|ρ2-1+|s1|ρ2-1),|g(s1)-g(s2)|≤Cg|s1-s2|mmmmmms1,s2∈ℝ,
where Ch and Cg are positive constants. Here, we describe the asymptotically almost-periodic behavior of solutions of problem (79) in the Lp-setting. To model this problem in the abstract form (1) we set that η=δ/2, p>N/2, the operator A is defined in Lp(Ω) by Au=ΔD2u (ΔD is the Dirichlet Laplacian in Ω) on the domain
(81)D(ΔD2)={ϕ∈Hp4(Ω):ϕ=Δϕ=0on∂Ω},
where Hp4(Ω)=W4,p(Ω) is the standard Sobolev space (see [37]). With this specification, problem (79) will fall into the abstract formulation (1). Since A1/2=-ΔD, we can choose the angle ψ for the sector
(82)Σψ/2={λ∈ℂ:ψ2≤|argλ|≤πwithψ∈(0,π2)}
as small as needed and therefor  (see [21, Example 4.3]) (η,A) will be an admissible pair for any η>0. From [21, Section 3] we get that
(83)[Lp(Ω)]1/2={ϕ∈Hp2(Ω):ϕ=Δϕ=0on∂Ω}.
We define f:ℝ+×[Lp(Ω)]1/2×Lp(Ω)→Lp(Ω) by
(84)f(t,φ1,φ2)=μa(t)(he(φ1)∇·φ1+ge(φ1)Δφ1)-νb(t)φ2,aaaaaaat∈ℝ+,φ1∈[Lp(Ω)]1/2,φ2∈Lp(Ω),
where θe is the Nemytskii˘ operator associated with θ, and ∇φ represents the divergence of φ. For φ1∈[Lp(Ω)]1/2 and φ2∈Lp(Ω). Using Minkowski's inequality and Sobolev embedding, we have the estimate
(85)∥f(t,φ1,φ2)∥Lp(Ω)≤|μ||a(t)|(∥he(φ1)∥L∞(Ω)+∥ge(φ1)∥L∞(Ω))×∥φ1∥Hp2(Ω)+|ν||b(t)|∥φ2∥Lp(Ω).
Whence f is well defined. We claim that f satisfies (42) with
(86)Lf(r)=C~(|μ|+|ν|)(∥a∥∞+∥b∥∞)×(1+2r+3(rρ1+rρ2)).
Indeed, it is an easy consequence of the following estimates (here C~ will stand for some positive constant independent of φ~1 and φ~2 with φ~1,φ~2∈[Lp(Ω)]1/2). 						(87)∥(he(φ~1)-he(φ~2))∇·φ~1∥Lp(Ω)≤C~∥φ~1-φ~2∥Hp2(Ω)×(1+∥φ~1∥Hp2(Ω)ρ1-1+∥φ~2∥Hp2(Ω)ρ1-1)∥φ~1∥Hp2(Ω),∥he(φ~2)∇·(φ~1-φ~2)∥Lp(Ω)≤C~(1+∥φ~2∥Hp2(Ω)+∥φ~2∥Hp2(Ω)ρ1)∥φ~1-φ~2∥Hp2(Ω),∥ge(φ~1)Δ(φ~1-φ~2)∥Lp(Ω)≤C~(1+∥φ~1∥Hp2(Ω)+∥φ~1∥Hp2(Ω)ρ2)∥φ~2∥Hp2(Ω),∥(ge(φ~1)-ge(φ~2))Δφ~2∥Lp(Ω)≤C~∥φ~1-φ~2∥Hp2(Ω)×(1+∥φ~1∥Hp2(Ω)ρ2-1+∥φ~2∥Hp2(Ω)ρ2-1)∥φ~2∥Hp2(Ω).
Let K be a compact subset of [Lp(Ω)]1/2×Lp(Ω). For φ1∈[Lp(Ω)]1/2 and φ2∈Lp(Ω), we set that
(88)fap(t,φ1,φ2)=μa(t)(he(φ1)∇·φ1+ge(φ1)Δφ1),Φf(t,φ1,φ2)=νb(t)φ2,MK=sup{∥[φ1φ2]∥[Lp(Ω)]1/2×Lp(Ω):[φ1φ2]∈K}.
We have the following estimates:
(89)∥Φf(t,φ1,φ2)∥Lp(Ω)≤|ν||b(t)|MK,(90)∥fap(t+τ,φ1,φ2)-fap(t,φ1,φ2)∥Lp(Ω)≤|μ||a(t+τ)-a(t)|×(∥he(φ1)∥L∞(Ω)+∥ge(φ1)∥L∞(Ω))∥φ1∥Hp2(Ω)≤C~|μ||a(t+τ)-a(t)|×(1+∥φ1∥Hp2(Ω)+∥φ1∥Hp2(Ω)ρ1+∥φ1∥Hp2(Ω)ρ2)∥φ1∥Hp2(Ω)≤C~(K)|a(t+τ)-a(t)|,
where C~(K) is a constant depending on K.Ifb∈C0(ℝ+,ℝ) from (89), we get
(91)Φf∈C0(ℝ+×[Lp(Ω)]1/2×Lp(Ω),Lp(Ω)),
and (90) implies that
(92)fap∈AP(ℝ×[Lp(Ω)]1/2×Lp(Ω),Lp(Ω)).
Hence,
(93)f∈AAP(ℝ+×[Lp(Ω)]1/2×Lp(Ω),Lp(Ω)).
Applying Theorem 17, we have the following result.Proposition 21.
Under the previous conditions, ifa∈AP(ℝ), b∈C0(ℝ+,ℝ), and |μ|+|ν| is small enough, then problem (79) has an asymptotically almost-periodic mild solution.Example 22.
We consider the following partial differential equation(94)utt+νt+1ut+Δu2-δΔut=μ(cost+cos2t)|u|ρ-1(u∇·u+uΔu),=μ(cost+cos2t)lllllx∈Ω,t≥0,u=Δu=0,x∈∂Ω,t>0,u(0,x)=u0(x),ut(0,x)=v0(x),x∈Ω.
If |μ|+|ν| is small enough by Proposition 21, problem (94) has an asymptotically almost-periodic mild solution. In fact, take h(u)=g(u)=u|u|ρ-1, a(t)=cost+cos2t, and b(t)=1/(t+1).
The next application is a modification of problem (79). Let k be a fixed nonnegative integer and let c:ℝ→ℝ+ and b:ℝ+→ℝ be two bounded continuous functions. Suppose that f∈AAP(ℝ+×ℝN×ℝ×ℝ,ℝ) satisfies the following growth condition:
(95)|f(t,x,u,w)-f(t,x,u~,w~)|≤c(t)[|w-w~|+|u-u~|(1+|u|k+|u~|k)],
for each t≥0, x∈ℝN, u,u~,w,w~∈ℝ. We consider the following partial differential equation
(96)utt+b(t)ut+Δ2u-δΔut=f(t,x,u,∇·u),utt+b(t)ut+Δ2u-δΔut=x∈Ω,t≥0,u=Δu=0,x∈∂Ω,t>0,u(0,x)=u0(x),ut(0,x)=v0(x),x∈Ω.
We model (96) in the abstract form (1) in a similar way to problem (79). That is we set η=δ/2, p>N (hence Hp2(Ω)↪C1(Ω) which will be useful to get some a priori estimates) and we consider the operator A as defined in (81) and (82). Suppose that
(97)supt≥0(∫Ω‍|f(t,x,0,0)|pdx)1/p<∞.
We define F~:ℝ+×[Lp(Ω)]1/2×Lp(Ω)→Lp(Ω) by
(98)F~(t,φ1,φ2)(x)=f(t,x,φ1(x),∇·φ1(x))-b(t)φ2(x).
We observe that F~ is well defined. In fact, we get the following estimates:
(99)(∫Ω‍|f(t,x,φ1(x),∇·φ1(x))|pdx)1/p≤c(t)((∫Ω‍|∇·φ1(x)|pdx)1/p+(∫Ω‍|φ1(x)|p(1+|φ1(x)|k)pdx)1/p+(∫Ω‍|f(t,x,0,0)|pdx)1/p)≤∥c∥∞(+(∫Ω‍|f(t,x,0,0)|pdx)1/p∥∇·φ1∥Lp(Ω)+(1+∥φ1∥L∞(Ω)k)∥φ1∥Lp(Ω)+(∫Ω‍|f(t,x,0,0)|pdx)1/p).
On the other hand, for φ1,φ~1∈[Lp(Ω)]1/2 and φ2,φ~2∈Lp(Ω) we have the following estimate:
(100)∥F~(t,φ1,φ2)-F~(t,φ~1,φ~2)∥Lp(Ω)≤4max{∥c∥∞,∥b∥∞}(1+∥φ1∥Hp2(Ω)k+∥φ~1∥Hp2(Ω)k)×(∥φ1-φ~1∥Hp2(Ω)+∥φ2-φ~2∥Lp(Ω)),
which means that F~ satisfies (42) with LF~(r)=4max{∥c∥∞,∥b∥∞}(1+2rk).
Sincef∈AAP(ℝ+×ℝN×ℝ×ℝ,ℝ), there are two functions
(101)fap∈AP(ℝ×ℝN×ℝ×ℝ,ℝ),φf∈C0(ℝ+×ℝN×ℝ×ℝ,ℝ),
so that f=fap+φf. Let K be a compact subset of [Lp(Ω)]1/2×Lp(Ω). For φ1∈[Lp(Ω)]1/2 and φ2∈Lp(Ω), we set
(102)F~ap(t,φ1,φ2)(x)=fap(t,x,φ1(x),∇·φ1(x)),Φ~F~(t,φ1,φ2)(x)=φf(t,x,φ1(x),∇·φ1(x))-b(t)φ2(x).
If (φ1,φ2)∈K, then
(103)∥Φ~F~(t,φ1,φ2)∥Lp(Ω)≤(∫Ω‍|φf(t,x,φ1(x),∇·φ1(x))|pdx)1/p+|b(t)|MK,
where MK is given by (88).
We note that using Sobolev embedding we have that there is a positive constantc~>0 depending only on K so that
(104)supx∈Ω|φ1(x)|≤c~,supx∈Ω|∇·φ1(x)|≤c~.
Taking into account the fact that Y=Ω¯×Bc~(ℝ2) is a compact subset of ℝN+2, (4.29) and (103). It follows that limt→∞Φ~F~(t,φ1,φ2)=0 uniformly for (φ1,φ2)∈K. Hence,
(105)Φ~F~∈C0(ℝ+×[Lp(Ω)]1/2×Lp(Ω),Lp(Ω)).
As an immediate consequence of Definition 3 and using the fact that Ω is bounded we have that
(106)F~ap∈AP(ℝ×[Lp(Ω)]1/2×Lp(Ω),Lp(Ω)).
Since F~=F~ap+Φ~F~, we conclude that
(107)F~∈AAP(ℝ+×[Lp(Ω)]1/2×Lp(Ω),Lp(Ω)).Proposition 23.
Under the previous conditions, ifmax{∥c∥∞,∥b∥∞} is small enough, then problem (96) has an asymptotically almost-periodic mild solution.Proof.
We argue as follows. Let us chooser>0 such that K/rCsupt≥0(∫Ω‍|f(t,x,0,0)|pdx)1/p is small enough, where K and C are constants given in (7). Then, condition (47) is fulfilled. Now applying Theorem 17 we conclude the proof.Leta:ℝ+→ℝ be a bounded continuous function and let β be in (0,1) and δ>0. In a bounded smooth domain Ω⊆ℝN, we consider the following partial differential equation:
(108)utt+Δ2u-δΔut=a(t)|∫Ω‍∇·u(t,x)dx|βΦ0,utt+Δ2u-δΔut=a(t)=a(t)x∈Ω,t≥0,u=Δu=0,x∈∂Ω,t>0,u(0,x)=u0(x),ut(0,x)=v0(x),x∈Ω,
where Φ0∈Lp(Ω), p>N/2. We model (108) in the abstract form (1); that is, we set η=δ/2 and we consider the operator A as in (81) and (82). We define f:ℝ+×[Lp(Ω)]1/2×Lp(Ω)→Lp(Ω) by
(109)f(t,φ1,φ2)(x)=a(t)|∫Ω‍∇·φ1(ξ)dξ|βΦ0(x).
To establish our next result, we assume the following conditions.(H1)
Leta(·)=aap(·)+φa(·) be in AAP(ℝ+), with aap(·)∈AP(ℝ) and φa(·)∈C0(ℝ+).(H2)
There is a continuous nondecreasing functionh:[0,∞)→[1,∞) such that h(t)→∞ as t→∞ and
(110)limt→∞1h(t)∫0t‍e-C(t-s)h(s)βds=0,supt≥0∫0t‍e-C(t-s)|a(t)|h(s)βds<∞.Proposition 24.
Under the previous conditions, problem (108) has an asymptotically almost-periodic mild solution.Proof.
Letq>0 be given so that (1/q)+(1/p)=1. We can infer the following estimate:
(111)∥f(t,φ1,φ2)∥Lp(Ω)≤∥a∥∞∥Φ0∥Lp(Ω)|Ω|β/q∥φ1∥Hp2(Ω)β.
Hence, we can define Wf in (W) by Wf(ξ)=||a||∞||Φ0||Lp(Ω)|Ω|β/qξβ. Let K be a compact subset of [Lp(Ω)]1/2×Lp(Ω) and let MK be as in (88). We set
(112)fap(t,φ1,φ2)(x)=aap(t)|∫Ω‍∇·φ1(ξ)dξ|βΦ0(x),Φf#(t,φ1,φ2)(x)=φa(t)|∫Ω‍∇·φ1(ξ)dξ|βΦ0(x),[φ1φ2],[ψ1ψ2]∈K.
We have the following estimates:(113)∥fap(t+τ,φ1,φ2)-fap(t,φ1,φ2)∥Lp(Ω)≤|Ω|β/qMKβ∥Φ0∥Lp(Ω)|aap(t+τ)-aap(t)|,(114)∥Φf#(t,φ1,φ2)∥Lp(Ω)≤|Ω|β/qMKβ∥Φ0∥Lp(Ω)|φa(t)|,(115)∥f(t,φ1,φ2)-f(t,ψ1,ψ2)∥Lp(Ω)≤|Ω|β/q∥a∥∞∥Φ0∥Lp(Ω)∥φ1-ψ1∥Hp2(Ω)β.
It follows from (113) and (114) that f∈AAP(ℝ+×[Lp(Ω)]1/2×Lp(Ω),Lp(Ω)). In addition, from the estimate (115), we obtain that f is uniformly continuous on compact sets.
From(H2), for [uivi]∈Ch([Lp(Ω)]1/2×Lp(Ω)),i=1,2, we can infer that
(116)1h(t)∫0t‍e-C(t-s)Wf(Lh(s))ds=∥a∥∞∥Φ0∥Lp(Ω)|Ω|β/qLβ(1h(t)∫0t‍e-C(t-s)h(s)βds)⟶0,e-C(t-s)Wf(Lh(s))dse-C(t-s)Wf(Lh(s))dslllllllllllllt⟶∞,(117)∫0t‍e-C(t-s)∥f(s,u1(s),v1(s))-f(s,u2(s),v2(s))∥Lp(Ω)ds≤|Ω|β/q∥Φ0∥Lp(Ω)∫0t‍e-C(t-s)|a(s)|∥u1(s)-u2(s)∥Hp2(Ω)βds=|Ω|β/q∥Φ0∥Lp(Ω)∫0t‍e-C(t-s)|a(s)|h(s)β×(∥u1(s)-u2(s)∥Hp2(Ω)h(s))βds≤|Ω|β/q∥Φ0∥Lp(Ω)supt≥0(∫0t‍e-C(t-s)|a(t)|h(s)βds)×[supt≥0(1h(t)[∥u1(t)-u2(t)∥Hp2(Ω)|Ω|β/q∥Φ0∥Lp(Ω)22+∥v1(t)-v2(t)∥Lp(Ω)]1h(t))]β.
Therefore, conditions (c) and (d) of Theorem 19 are satisfied. A straightforward computation shows that (e) holds. We can prove that the set {f(t,φ1,φ2):0≤t≤a,||φ1||Hp2(Ω)+||φ2||Lp(Ω)≤r} is relatively compact in Lp(Ω). In fact, first, we denote by u~ the zero extension of u outside Ω; that is,
(118)u~(x)={u(x)ifx∈Ω,0ifx∈ℝN∖Ω.
For every number ε>0, there exists a number δ>0 and a subset G⊂⊂Ω such that for every ϱ∈ℝN with |ϱ|<δ(119)∫Ω‍|Φ~0(x+ϱ)-Φ~0(x)|pdx<(εrβ∥a∥∞|Ω|β/q)p,∫Ω∖G‍|Φ0(x)|pdx<(εrβ∥a∥∞|Ω|β/q)p.
Hence,
(120)∫Ω‍|f~(t,φ1,φ2)(x+ϱ)-f~(t,φ1,φ2)(x)|pdx≤[rβ∥a∥∞|Ω|β/q]p×∫Ω‍|Φ~0(x+ρ)-Φ~0(x)|pdx≤ε,∫Ω∖G‍|f(t,φ1,φ2)(x)|pdx≤[rβ∥a∥∞|Ω|β/q]p×∫Ω∖G‍|Φ0(x)|pdx≤ε.
From [38, Theorem 2.21] the desired assertion follows. Note finally that using Remark 20 we have that (108) has an asymptotically almost periodic mild solution. This ends the proof.



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*Source: 308616-2013-09-11.xml*