∂u
∂t = ∆u+F(t, x, ut(ã, x)) (3.76)
and ∂u
∂t = ∆u+F(t, x, ut(ã, x)) + ¯f(t, x) (3.77) inR×Ω together with the boundary condition
∂u
∂n+κ(x)u= 0 on R×∂Ω, (3.78)
where Ω is a bounded domain in RJ with smooth boundary ∂Ω (e.g.,∂Ω∈C2+α for someα∈ (0,1)),and ∆ and ∂/∂nrespectively denote the Laplacian operator inRJ and the exterior normal derivative at∂Ω,and moreoverκ∈C1+α(∂Ω) with κ(x)≥0 on ∂Ω.As the spaceX, we take
X=C( ¯Ω),
where ¯Ω is Ω∪∂Ω and C( ¯Ω) is the Banach space of all real-valued continuous functions on ¯Ω with the supremum norm, and we define the operator A by the closed extension of the operator ∆ with domain D(∆) = {ξ ∈ C2( ¯Ω) : ∂ξ/∂n+ κξ= 0 on ∂Ω}.By [212, Theorem 2], Agenerates an analytic compact semigroup {T(t)}t≥0 onX. In fact,{T(t)}t≥0 is represented as
[T(t)φ](x) = Z
Ω
U(t;x, y)φ(y)dy, x∈Ω,¯ (3.79) whereU(t;x, y) is the fundamental solution of the heat equation with the boundary condition (3.78) (cf. [68], [116], [108]).
OnF and ¯f we impose the following conditions:
(H1’) F:R×Ω×C¯ g0(R)7→Ris continuous in (t, x, φ)∈R×Ω×C¯ g0(R) and linear in φ; heregis a (fixed) continuous nonincreasing function such that g(0) = 1 andg(s)→ ∞ass→ −∞;
(H2’) f¯:R×Ω¯ 7→Ris continous in (t, x)∈R×Ω;¯
(H3’) F(t, x, φ) is almost periodic intuniformly for (x, φ)∈Ω×C¯ g0(R),and ¯f(t, x) is almost periodic int uniformly forx∈Ω,¯ whereg is the function in (H1’).
Set
B=Cg0(C( ¯Ω)), and defineF :R× B 7→Xandf :R7→Xby
F(t, φ)(x) =F(t, x, φ(ã, x)), (t, x, φ)∈RìΩ¯ì B,
and
f(t)(x) = ¯f(t, x), (t, x)∈R×Ω,¯
whereφ(ã, x) is an element inCg0(R) defined by φ(θ, x) =φ(θ)(x) forθ∈R−, and g is the function in (H1’). Observe that if the setW is compact inB, then the set {φ(ã, x) : φ∈ W, x ∈Ω}¯ is compact in Cg0(R). From this fact, we can easily see that (Hi’) implies (Hi) fori= 1,2,3.Then (3.77) together with (3.78) is represented as the (abstract) equation (3.61). From (3.79) we can see that the (mild) solution u(ã, σ, φ, F+f) of (5.7) through (σ, φ) satisfies the relation
u(t, x) = Z
Ω
U(t−σ;x, y)φ(y)dy (3.80)
+ Z t
σ
Z
Ω
U(t−s;x, y){F(s, y, u(s+ã, y)) + ¯f(s, y)}dyds for (t, x)∈[σ,∞)×Ω,¯ whereu(t, x) =u(t, σ, φ, F+f)(x).Henceforth, the function usatisfying (3.80) will be called a (mild) solution of (3.77) and (3.78).
In what follows, in order to ensure theBC-TS of the null solution of (3.77) and (3.78), we consider the following condition;
(H4’) there exist positive constantsaandb, a < b,such that sup
t, x
|F(t, x, ξ) +bξ(0)| ≤a|ξ|BC, ξ∈BC(R−;R).
It is easy to check that ifF(t, x, ξ) =−bξ(0) +R0
−∞k(t, s, x)ξ(s)dswith supt, xR0
−∞|k(t, s, x)|ds≤a < b, then (H4’) is satisfied.
Lemma 3.9 Assume (H1’), (H3’) and(H4’). Then the null solution of (3.77) and (3.79)isBC-TS.
Proof. Letε >0 andσ∈Rbe given. To prove Lemma 3.9, it suffices to certify thatφ∈BC(R−;X) with|φ|BC < εandh∈BC([σ,∞);X) with supt≥σ|h(t)|X<
(b−a)εimply|u(t, σ, φ, h)|X< εfort≥σ.If this is not the case, then there exists a τ > σ such that|u(τ, σ, φ, h)|X =ε and |u(t, σ, φ, h)|X < εfor t < τ.Setq(t) = F(t, vt) +bv(t) +h(t) fort∈[σ, τ], wherev(t) :=u(t, σ, φ, F+f+h). The function q: [σ, τ]7→X is continuous, and it satisfies |q(t)|X < bεon [σ, τ] by (H4’). Take a sequence{Qn} ⊂ C1([σ, τ]×Ω;¯ R) with the properties thatQ := sup{|Qn(t, x)| : t ∈ [σ, τ], x ∈ Ω, n¯ = 1,2, . . .} < bε and that supσ≤t≤τ|qn(t)−q(t)|X → 0 as n→ ∞, whereqn(t)(x) =Qn(t, x), x∈Ω. Let¯ Vn(t, x) be the (classical) solution of linear partial differential equation
∂u
∂t = ∆u−bu+Qn(t, x) in (σ, τ]×Ω
which satisfies the boundary condition (3.78) on (σ, τ]×∂Ω and the initial condition Vn(σ, x) = φ(0)(x), x ∈ Ω. Set¯ vn(t)(x) = Vn(t, x) for x ∈ Ω. Then¯ vn(t) =
T(t−σ)φ(0) +Rt
σT(t−s){−bvn(s) +qn(s)}ds for t ∈ [σ, τ] (cf. [116, Theorems 2.9.1 and 2.9.2]). Applying Gronwall’s lemma, one can see thatvn(t) tends toT(t− σ)φ(0) +Rt
σT(t−s){−bv(s) +q(s)}ds =v(t) uniformly on [σ, τ] as n → ∞. Set k = max(|φ(0)|X, Q/b), and take a positive integer n0 so that k+ 1/n0 < ε. We claim that
|Vn(t, x)|< k+ 1/n, (t, x)∈[σ, τ]×Ω,¯ (3.81) for alln≥n0.If this claim is not true, then there exists somen≥n0 and (t0, x0)∈ (σ, τ]×Ω such that¯ |Vn(t0, x0)|=k+1/n(sayVn(t0, x0) =k+1/n) and|Vn(t, x)|<
k+ 1/non [σ, t0)×Ω.¯ SetW(t, x) =Vn(t, x)−k−1/nfor (t, x)∈[σ, τ]×Ω.¯ Then W(t0, x0) = 0 andW(t, x)<0 for all (t, x)∈[σ, t0)×Ω.¯ Moreover, we get
∂W
∂t (t, x) = ∂Vn
∂t (t, x)
= ∆W(t, x)−b(W(t, x) +k+ 1/n) +Qn(t, x),
and hence ∆W(t, x) −(∂/∂t)W(t, x) − bW(t, x) = b(k + 1/n) − Qn(t, x) ≥ Q− Qn(t, x) ≥ 0 on (σ, t0]×Ω. If x0 ∈ Ω, then W(t, x) ≡ 0 on [σ, t0]×Ω¯ by the strong maximum principle (e.g., [183, Theorems 3.3.5, 3.3.6 and 3.3.7]), which is a contradiction because of Vn(σ, x) = φ(0)(x) < k+ 1/n. We thus ob- tain x0 ∈ ∂Ω and W(t, x) < 0 on [σ, t0]×Ω, and hence ∂W/∂n > 0 at (t0, x0) by the strong maximum principle, again. However, this is impossible because of (∂W/∂n)(t0, x0) = (∂Vn/∂n)(t0, x0) = −κ(x0)Vn(t0, x0) ≤ 0. Consequently, the claim (3.81) must hold true. Letting n→ ∞in (3.81), we get |v(t)|X≤k < ε for allt∈[σ, τ], which is a contradiction to|v(τ)|X=ε.This completes the proof.
Combining Theorem 3.19 with Lemma 3.9, we can immediately obtain the fol- lowing result.
Proposition 3.6 Assume(H1’)-(H4’). Then there exists an almost periodic (mild) solution of(3.77)and(3.78).
Under the additional assumption on F(t, x, ξ) and ¯f(t, x), we can deduce the existence of an almost periodic classical solution of (3.77) and (3.78).
Proposition 3.7 In addition to(H1’)-(H4’), assume thatF(t, x, ξ)andf¯(t, x)are locally H¨older continuous in (t, x) ∈ R×Ω¯ uniformly for ξ ∈ Cg0(R) in bounded sets. Then there exists an almost periodic classical solution of(3.77) and(3.78).
Proof. Combining Theorem 3.19 with Lemma 3.9, we see that (3.77) has an almost periodic solution (say,p(t)), which is continuously differentiable and satisfies (3.78) on R. Set P(t, x) = p(t)(x) for (t, x) ∈ R×Ω.¯ Clearly P(t, x) is almost periodic intuniformly forx∈Ω.¯ It remains to prove thatP is a classical solution of (3.77) and (3.78). SetQ(t, x)(=Q(t)(x)) =F(t, x, P(t+ã, x)) + ¯f(t, x) for (t, x)∈ R×Ω.¯ The functionQ:R×Ω¯ 7→Ris continuous in (t, x),and moreover it is locally H¨older continuous intuniformly forx∈Ω as seen in the proof of Theorem 3.20. We¯ assert thatQ(t, x) is H¨older continuous inx∈Ω uniformly for¯ tin bounded sets. If
the assertion is true, then Q(t, x) is H¨older continuous on [s, r]×Ω for any¯ r > s;
consequentlyP(t, x) =R
ΩU(t−s;x, y)P(s, y)dy+Rt s
R
ΩU(t−τ;x, y)Q(τ, y)dydτ is a classical solution of (3.77) and (3.78), as required (cf. [116, Theorem 2.7.1]). Now, in order to prove the assertion we first show that the functionx∈Ω¯ 7→P(t+ã, x)∈ Cg0(R) is Lipschitz continuous, uniformly fort ∈R. By the same reason as in the proof of Theorem 3.20, we can assume that the semigroup{T(t)}t≥0satisfies (3.62).
Take a constantαwith 1/2< α <1. Recall that (−A)α, the fractional power of−A, is a closed linear operator. It is known (e.g., [90, Theorem 1.6.1]) that the Banach space D((−A)α), which is the space of the definition domain of (−A)α equipped with the graph norm, is continuously imbedded to the spaceC1( ¯Ω).Ift > σ,then
|(−A)αp(t)|X = |(−A)α(T(t−σ)p(σ) + Z t
σ
T(t−s)Q(s)ds)|X
≤ cα(t−σ)−αe−λ(t−σ)|p(σ)|X+ +
Z t σ
cα(t−s)−αe−λ(t−s)dsãsup
s∈R
|Q(s)|X. Passing to the limit asσ→ −∞,we get supt∈R|(−A)αp(t)|X<∞; hence supt∈R|P(t,ã)|C1( ¯Ω)(=:C)<∞.Then
sup{|x−y|−1|P(t+θ, x)−P(t+θ, y)|:t∈R, θ≤0, x6=y} ≤C, and consequently
sup{|x−y|−1|P(t+ã, x)−P(t+ã, y)|C0
g(R):t∈R, x6=y} ≤CJ
by (3.25). Thus the functionx∈Ω¯ 7→P(t+ã, x)∈Cg0(R) is Lipschitz continuous, uniformly fort∈R. Therefore, in virtue of the H¨older continuity ofF(t, x, ξ) and f¯(t, x), we can see that Q(t, x) is H¨older continuous in x ∈ Ω uniformly for¯ t in bounded set, as required.