Proof . Let f ∈C(R,H) be asymptotically almost periodic and
f(t)=p(t) +ω(t) (5.89) for allt ∈ R, where functionp ∈C(R,H) is almost periodic and limt→+∞|ω(t)| =0.
According to [152, Lemma 4] the equation dx
dt = −x|x|+p(t) (5.90) has a unique almost periodic solutionq∈ C(R,H). Along with (5.89) we consider the equation
dx
dt = −x|x|+p(t) +ω(t), (5.91) whereω(t) =ω(t) for allt≥0 andω(t) =ω(0) ast <0. Denote byϕthe unique bounded onRsolution of (5.91). Letτ ≥ 0. Thenϕ(τ)(t)= ϕ(t +τ) is a unique bounded onR solution of the equation
d y
dt = −y|y|+p
τ(t) +ωτ(t). (5.92)
According toTheorem 5.7.3
Convergence of Some Evolution Equations 169 Note thatω(τ)(t)=ωτ(t) for allt≥0 andωτ(t)=ωτ(0) ast <0 and, consequently,
lim
τ→+∞!!ω
τ!!=0. (5.94)
From (5.93) and (5.94) it follows that limt→+∞|ϕ(t) −q(t)| = 0. Now to finish the
proof of the lemma it is enough to note that the restriction of the functionϕonR+is asymptotically almost periodic solution of (5.86). Corollary 5.41. For any asymptotically almost periodic function f ∈C(R,H) all solutions of (5.86) are asymptotically almost periodic.
Proof . The formulated statement follows fromLemma 5.39and equality (5.88). Theorem 5.7.4. If the mappingf ∈C(R,H) is asymptotically almost periodic, then (5.86) is convergent.
Proof . LetY :=H+(f)= {f(τ)|τ∈R
+}(by bar it is denoted the closure inC(R,H)) and (Y,R+,σ) be a dynamical system of shifts onH+(f). PutX := H×Y and define onXa dynamical system (X,R+,π) by the following rule:π(τ, (x,g))=(ϕ(t,x,g),g(τ)), whereϕ(t,x,g) is a solution of the equation
du
dt = −u|u|+g(t) (5.95) satisfying the initial conditionϕ(0,x,g) = x. Assumeh:= pr2 :X → Y and consider the nonautonomous dynamical system(X,R+,π), (Y,R+,σ),h. Let us show that the constructed nonautonomous dynamical system is convergent.
First of all, let us show that the system (X,R+,π) is compactly dissipative. According toLemma 5.39the system (X,R+,π) is point dissipative, sinceωx =ω(p,q)=H(p,q) for anyx∈Xand, consequently,ΩX =H(p,q) is compact.
LetK⊂Xbe an arbitrary compact set andΣ+K:= {πtx|x∈K,t∈R
+}. Let us show thatΣ+Kis relatively compact. Let{xn} ⊂Σ+K. Then there exist{xn} ⊂K and{tn} ⊆R+ such thatxn=π(xn,tn). Letxn:=(un,gn)∈H×H+(f). SinceKis a compact set, then
the sequences{un}and{gn}can be considered convergent. Assumeu:=limn→+∞unand
g:=limn→+∞gn. By the asymptotical almost periodicity of f we have
lim
n→+∞supt≥0
gn(t)−g(t)=0. (5.96)
Sinceg ∈H+(f), the solutionϕ(t,u,g) of (5.95) is asymptotically almost periodic and, hence, the sequence{ϕ(tn,u,g)}can be considered convergent. Letu := limn→+∞ϕ(tn,
u,g). We will show thatxn→x=(u,g). For this aim we note that
ϕ tn,un,gn −ϕtn,u,g ≤ϕtn,un,gn−ϕtn,u,gn+ϕt,u,gn−ϕ(t,u,g). (5.97)
Putwn(t) := |ϕ(t,u,gn)−ϕ(t,u,g)|andδn:=sup{|gn(t)−g(t)|:t∈R+}. According to [152, page 73] dwn(t) dt ≤ − 1 2w 2 n(t) +δn, (5.98)
and taking into consideration thatwn(0)=0, we obtain
wn(t)≤
2
2δn (5.99)
for allt∈R+. From (5.87), (5.96)–(5.99) it follows that lim n→+∞ϕ tn,un,gn −ϕtn,u,g=0, (5.100)
and, consequently, xn = (ϕ(tn,un,gn),gn) → (u,g) = x. So,Σ+K is relatively compact.
Assume M :=H+(K)=Σ+
K and
J:=Ω(M)= ∩t≥0∪τ≥tπτM. (5.101)
According to [112] the setJis compact and invariant. FromTheorem 5.7.3andLemma 5.39it follows that the unique compact invariant set of the dynamical system (X,R+,π) is the setΩX = H(p,q). So,Ω(M) = J = Ω(X) and, hence, (X,R+,π) is compactly dissipative dynamical system and its Levinson centerJX =ΩX. Now to finish the proof
of the theorem it is sufficient to note that byTheorem 5.7.3JX∩Xycontains at most one
point for anyy∈ωf =JY. The theorem is proved.
(4) LetHbe a Hilbert space. In this point we will consider the equation dx
dt = f(t,x), (5.102)
where f ∈C(R×H,H) satisfies the condition Re%x1−x2,f t,x1 −ft,x2 & ≤ −κx1−x2α (5.103) for allt∈R+andx∈H(κ >0 andα >2). Along with (5.102) we consider the family of equations
d y
dt =g(t,y),
g∈H(f), (5.104)
whereH(f) := {f(τ)|τ∈R}, where f(τ) is the shift of f ontoτ with respect to the variabletand by bar it is denoted the closure inC(R×H,H). Note that along with the function f any functiong ∈H(f) satisfies condition (5.103) with the same constantsκ andα. According to the results of [153, Chapter 2], if the function f ∈ C(R×H,H) satisfies condition (5.103), then for everyu ∈ H andg ∈ H(f) (5.104) has a unique solutionϕ(t,u,g) defined onR+and passing through the pointu∈Hast=0; besides, the mappingϕ : R+×H×H(f) → H is continuous. Put now Y := H(f) and by (Y,R,σ) denote a dynamical system of shifts onH(f). Further, letX :=H×H(f) and π :R+×X → Xbe the mapping defined by the equalityπ(t, (u,g)) =(ϕ(t,u,g),g(t)). Then (X,R+,π) is a semigroup dynamical system. At last, assumeh:=pr2:X→Y. Then (X,R+,π), (Y,R,σ),his a nonautonomous dynamical system generated by (5.102).
Convergence of Some Evolution Equations 171 Definition 5.42. As earlier, (5.102) is called convergent if the nonautonomous dynamical system(X,R+,π), (Y,R,σ),hgenerated by (5.102) is convergent.
InChapter 3we established (seeTheorem 3.8.5andCorollary 3.118) that (5.102) is convergent, if the right-hand side f is asymptotically almost periodic with respect tot and satisfies the condition (5.103) with the parameterα=2. Below we will establish the convergence of (5.102), when f satisfies condition (5.103) with the parameterα > 2. Previously, let us give two auxiliary lemmas.
Lemma 5.43. Letf ∈C(R×H,H) be such that the set{f(τ)|τ∈R}is relatively compact inC(R×H,H) and condition (5.103) is held. Then:
(1) for anyu∈Hthe solutionϕ(t,u,f) of (5.102) is compact onR+(i.e.,ϕ(R+,u,f) is a relatively compact set inH);
(2) for allt∈R+andx1,x2∈H
ϕ t,x1,f −ϕt,x2,f≤x1−x22−α+ (α−2)t 1/(2−α) =x1−x21 +x1−x2α−2(α−2)t 1/(2−α) . (5.105)
Proof . Let us define a functionF∈C(R×H,H) by the following rule
F(t,x) := ⎧ ⎪ ⎨ ⎪ ⎩ f(t,x), for (t,x)∈R+×H, f(0,x), for (t,x)∈R−×H. (5.106)
It is easy to check that the functionFpossesses the next properties: (a) {F(τ)|τ∈R}is relatively compact inC(R×H,H);
(b) Rex1−x2,F(t,x1)−F(t,x2) ≤ −κ|x1−x2|αfor allt∈Randx1,x2∈H. According to [153, Theorem 2.2.3.1] the equation
dx
dt =F(t,x) (5.107)
has a unique compact onRsolutionϕ(t,x0,F) and for every two solutionsϕ(t,x1,F) and ϕ(t,x2,F) there takes place the inequality
ϕ
t,x1,F−ϕt,x2,F≤x1−x22−α+ (2−α)t1/(2−α) (5.108) for allt ∈ R+,x1,x2 ∈ Hand, consequently, limt→+∞|ϕ(t,x1,F)−ϕ(t,x0,F)| =0 for allx ∈H. The last relation imply that all solutions of (5.107) are compact onR+. Now to complete the proof of the lema it is enough to note thatϕ(t,x,f)=ϕ(t,x,F) for all
Lemma 5.44. Letα,κandεbe positive numbers. Then onR+the scalar equation