Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2008, Article ID 405092, 13 pages doi:10.1155/2008/405092
Research Article Existence and Uniqueness of Pseudo Almost Automorphic Mild Solutions to Some Classes of Partial Hyperbolic Evolution Equations Zhanrong Hu1, 2 and Zhen Jin1, 2 1 2
Department of Mathematics, North University of China, Taiyuan 030051, China School of Mechatronic Engineering, North University of China, Taiyuan 030051, China
Correspondence should be addressed to Zhanrong Hu,
[email protected] Received 1 June 2008; Revised 4 July 2008; Accepted 10 September 2008 Recommended by Leonid Shaikhet We will establish an existence and uniqueness theorem of pseudo almost automorphic mild solutions to the following partial hyperbolic evolution equation d/dtut ft, But Aut gt, Cut, t ∈ R, under some assumptions. To illustrate our abstract result, a concrete example is given. Copyright q 2008 Z. Hu and Z. Jin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction Let X, · be a Banach space, and let Xα , ·α for 0 < α < 1 be an arbitrary abstract intermediate Banach space between DA domain of the operator A and X. In this paper, we deal with sufficient conditions for the existence and uniqueness of pseudo almost automorphic mild solutions to equation d ut ft, But Aut gt, Cut, dt
t ∈ R,
1.1
where A is a sectorial linear operator on a Banach space X and σA ∩ iR ∅, B, C are bounded linear operators on Xα , f : R × X → Xβ , g : R × X → X are jointly continuous functions. This turns out to be a nontrivial problem due to the complexity and importance of pseudo almost automorphic functions e.g., see 1–8 and the references therein. Upon making some additional assumptions, it will be shown that 1.1 admits a unique Xα -valued pseudo almost automorphic mild solution. Applications include the study
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of pseudo almost automorphic mild solutions to some nonlinear heat equation: ut t, x uxx t, x aut, x gt, qxut, x, ut, 0 ut, π 0,
t ∈ R, x ∈ 0, π,
t ∈ R,
1.2
where a ∈ R is a constant, and g, q are continuous functions. As a natural and important generalization of almost automorphy as well as pseudo almost periodicity, pseudo almost automorphy has recently been investigated for more details, see 1–3. Moreover, Liang et al. in 2 established a composition theorem about pseudo almost automorphic functions; Xiao et al. in 3 obtained sufficient conditions for the existence and uniqueness of pseudo almost automorphic mild solutions to semilinear differential equations in Banach spaces. In particular, they proved in 3 that the space PAAR, X, ·∞ of pseudo almost automorphic functions, endowed with sup norm, is a Banach space, thus pushed the door open to study pseudo almost automorphic mild solutions to various differential equations. The method used here was firstly developed in 9, in which Diagana established an existence and uniqueness theorem of pseudo almost periodic mild solutions to 1.1 under some similar assumptions. Combining bounded invariance theorem of pseudo almost automorphic functions established in 6 with composition theorem of pseudo almost automorphic functions and completeness of the space PAAR, X, ·∞ , this√paper extends the results in 9, since such function as gt, u u sin1/2 cos t cos 2t 2 2 maxk∈Z {e−t±k } cos u, t ∈ R, u ∈ X is not pseudo almost periodic, while they are pseudo almost automorphic. It should be pointed out that one can also make use of 1.1 to study several types of evolution equations including partial functional differential equations, integro-differential equations, and reaction diffusion equations. 2. Preliminaries In this section, we collect some preliminary facts from 9–11 that will be used later. Throughout this paper, N, Z, R, and C stand for the sets of positive integer, integer, real and complex numbers; X, ·, Y, ·Y stand for Banach spaces. If A is a linear operator on X, then ρA, σA, DA, kerA, RA stand for the resolvent set, spectrum, domain, kernel, and range of A. The space BX, Y denotes the Banach space of all bounded linear operators from X into Y equipped with natural norm ABX,Y supx∈X, x / 0 AxY /x. If Y X, it is simply denoted by BX with ABX supx∈X, x / 0 Ax/x. 2.1. Sectorial linear operators and analytic semigroups Definition 2.1. A linear operator A : DA ⊂ X → X not necessarily densely defined is said to be sectorial if the following hold. There exist constants ω ∈ R, θ ∈ π/2, π, and M > 0 such that ρA ⊃ Sθ,ω : {λ ∈ C : λ / ω, | argλ − ω| < θ}, M Rλ, A ≤ , λ ∈ Sθ,ω , |λ − ω| where Rλ, A λI − A−1 for each λ ∈ ρA.
2.1
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Remark 2.2. If A is sectorial, then it generates an analytic semigroup T tt≥0 , which maps 0, ∞ into BX such that there exist M0 , M1 > 0 with T t ≤ M0 eωt ,
t > 0,
tA − ωIT t ≤ M1 eωt ,
t > 0.
2.2
Definition 2.3. A semigroup T tt≥0 is said to be hyperbolic, if there exist a projection P and constants M, δ > 0 such that each T t commutes with P , kerP is invariant with respect to T t, T t : RQ → RQ is invertible and T tP x ≤ Me−δt x
for t ≥ 0,
T tQx ≤ Meδt x for t ≤ 0,
2.3
where Q : I − P and T t : T −t−1 for t ≤ 0. Recall that if a semigroup T tt≥0 is analytic, then T tt≥0 is hyperbolic if and only if σA ∩ iR ∅ see, e.g., 11, Definition 1.14 and Proposition 1.15, page 305. 2.2. Intermediate Banach spaces Definition 2.4. Let α ∈ 0, 1. A Banach space Xα , ·α is said to be an intermediate space between DA and X, if DA ⊂ Xα ⊂ X and there is a constant C > 0 such that xα ≤ Cx1−α xαA ,
x ∈ DA,
2.4
where ·A is the graph norm of A. Concrete examples of Xα include DAα for α ∈ 0, 1, the domains of the fractional powers of A, the real interpolation spaces DA α, ∞, α ∈ 0, 1, defined as follows: DA α, ∞ :
x ∈ X : xα sup t1−α A − ωIe−ωt T tx < ∞ , 0 0, δ > 0, Mα > 0, and γ > 0 such that T tQxα ≤ Cαeδt x for t ≤ 0, T tP xα ≤ Mαt−α e−γt x
for t > 0.
2.6
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Lemma 2.6 see 9. Let 0 < α < β < 1. For the hyperbolic analytic semigroup T tt≥0 , there exist constants c > 0, δ > 0, and γ > 0 such that AT tQxα ≤ ceδt xβ
for t ≤ 0,
AT tP xα ≤ ctβ−α−1 e−γt xβ
2.7
for t > 0.
3. Pseudo almost automorphic functions In this section, we recall some recent results on almost automorphic functions and pseudo almost automorphic functions. Let CR, X denote the collection of continuous functions from R into X. Let BCR, X denote the Banach space of all X-valued bounded continuous functions equipped with the sup norm u∞ : supt∈R ut for each u ∈ BCR, X. Similarly, CR × X, X denotes the collection of continuous functions from R × X into X, BCR × X, X denotes the collection of all bounded continuous functions f : R × X → X. Definition 3.1 see 8. A function f ∈ CR, X is said to be almost automorphic if for every sequence of real numbers σn n∈N , there exists a subsequence sn n∈N such that lim
lim ft sn − sm ft
m → ∞ n → ∞
for each t ∈ R.
3.1
This limit means that gt lim ft sn
3.2
ft lim gt − sn
3.3
n → ∞
is well defined for each t ∈ R, and n → ∞
for each t ∈ R. The collection of all such functions will be denoted by AAR, X. Theorem 3.2 see 7. Assume f, g : R → X are almost automorphic, and λ is any scalar. Then the following holds true: : f−t are almost automorphic; 1 f g, λf, fτ t : ft τ and ft 2 the range Rf of f is precompact, so f is bounded; 3 if {fn } is a sequence of almost automorphic functions and fn → f uniformly on R, then f is almost automorphic. Theorem 3.3 see 7. If one equips AAR, X with the sup norm u∞ supt∈R ut, then AAR, X turns out to be a Banach space. One sets 1 r AA0 R, X : f ∈ BCR, X : lim fsds 0 . r → ∞ 2r −r
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Definition 3.4 see 2. A function f ∈ BCR, X is said to be pseudo almost automorphic if it can be decomposed as f g φ, where g ∈ AAR, X and φ ∈ AA0 R, X. The collection of such functions will be denoted by PAAR, X. Theorem 3.5 see 3. If one equips PAAR, X with the sup norm u∞ supt∈R ut, then PAAR, X turns out to be a Banach space. Definition 3.6 see 2. A function f ∈ CR × X, X is said to be almost automorphic if f ∈ CR × X, X is almost automorphic in t ∈ R uniformly for all x ∈ K, where K is any bounded subset of X. That is to say, for every sequence of real numbers σn n∈N there exists a subsequence sn n∈N such that gt, x lim ft sn , x n → ∞
3.5
is well defined in t ∈ R, for all x ∈ K, and ft, x lim gt − sn , x n → ∞
3.6
for each t ∈ R and x ∈ K. Denote by AAR × X, X the collection of all such functions. One also defines AA0 R × X, X as the collection of functions f ∈ BCR × X, X such that 1 lim r → ∞ 2r
r −r
ft, xdt 0
3.7
uniformly for any bounded subset of X. Definition 3.7 see 2. A function f ∈ BCR × X, X is said to be pseudo almost automorphic if it can be decomposed as f g φ, where g ∈ AAR × X, X and φ ∈ AA0 R × X, X. The collection of such functions will be denoted by PAAR × X, X. Theorem 3.8 see 2. Assume f g φ ∈ PAAR × X, X with gt, x ∈ AAR × X, X, φt, x ∈ AA0 R × X, X, satisfying the following conditions: 1 gt, x is uniformly continuous in any bounded subset K ⊂ X uniformly for t ∈ R; 2 ft, x is uniformly continuous in any bounded subset K ⊂ X uniformly for t ∈ R. If x· ∈ PAAR, X, then f·, x· ∈ PAAR, X. Remark 3.9. If Xα for α ∈ 0, 1 is an intermediate space between DA and X, BCR, Xα , AAR, Xα , PAAR, Xα are equipped with the α-sup norm: u∞,α supt∈R utα , then they all constitute Banach spaces in view of Definition 2.4, Theorems 3.3 and 3.5. Theorem 3.10 see 6. Let u ∈ PAAR, Y, B ∈ BY, X. If vt : But for each t ∈ R, then v ∈ PAAR, X.
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4. Pseudo almost automorphic mild solutions In this section, combining Theorem 3.10 with composition theorem of pseudo almost automorphic functions Theorem 3.8 and completeness of the space PAAR, X, ·∞ Theorem 3.5, we will establish the existence and uniqueness theorem of pseudo almost automorphic mild solutions to 1.1 under the following assumptions. H1 For 0 < α < β < 1, Xβ → Xα → X are continuously embedded and there exist k0 > 0, kα > 0 such that u ≤ k0 uβ , uα ≤ kαuβ for each u ∈ Xβ . H2 A is a sectorial linear operator on a Banach space X and σA ∩ iR ∅. H3 f M ϕ ∈ PAAR × X, Xβ with M ∈ AAR × X, Xβ , ϕ ∈ AA0 R × X, Xβ and g N ψ ∈ PAAR × X, X with N ∈ AAR × X, X, ψ ∈ AA0 R × X, X. H4 The functions f, g are uniformly Lipschitz with respect to the second argument in the sense that: there exists L > 0 such that ft, x − ft, yβ ≤ Lx − y, gt, x − gt, y ≤ Lx − y,
4.1
for all t ∈ R and x, y ∈ X. H5 M, N are uniformly continuous in any bounded subset K ⊂ X uniformly for t ∈ R. H6 The operators B, C ∈ BXα , X and ω maxBBXα ,X , CBXα ,X . Definition 4.1 see 9. A function u ∈ BCR, Xα is said to be a mild solution to 1.1 if s → AT t − sP fs, Bus is integrable on −∞, t, s → AT t − sQfs, Bus is integrable on t, ∞ and
ut −ft, But −
t −∞
t −∞
AT t − sP fs, Busds
T t − sP gs, Cusds −
∞
∞
AT t − sQfs, Busds
t
4.2
T t − sQgs, Cusds
t
for each t ∈ R. Lemma 4.2. Let assumptions (H1)–(H6) hold. Consider the nonlinear operator Λ1 defined by
Λ1 ut
t −∞
AT t − sP fs, Busds −
Then, Λ1 maps PAAR, Xα into itself.
∞ t
AT t − sQfs, Busds.
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Proof. From H1, H3, H4, H5, we deduce that f M ϕ ∈ PAAR × X, X with M ∈ AAR × X, X, ϕ ∈ AA0 R × X, X, satisfying the following conditions. 1 Mt, x is uniformly continuous in any bounded subset K ⊂ X uniformly for t ∈ R. 2 ft, x is uniformly Lipschitz with respect to the second argument in the sense that there exists k0 L > 0 such that ft, x − ft, y ≤ k0 Lx − y,
4.4
for all t ∈ R and x, y ∈ X. Let u ∈ PAAR, Xα . Since B ∈ BXα , X, it follows from Theorem 3.10 that Bu· ∈ PAAR, X. Setting h· f·, Bu· and applying Theorem 3.8, we get that h ∈ PAAR, X. Using H1 and Theorem 3.10, we obtain that h ∈ PAAR, Xβ . Now, write h ξ η where ξ ∈ AAR, Xβ and η ∈ AA0 R, Xβ , then
Λ1 ut
t −∞
AT t − sP ξsds
t ∞
t −∞
AT t − sQξsds
AT t − sP ηsds
t ∞
4.5 AT t − sQηsds.
Set
Ξt Πt
t −∞
t
−∞
AT t − sP ξsds AT t − sP ηsds
t ∞
AT t − sQξsds,
t
∞
4.6 AT t − sQηsds.
Next, we show that Ξ ∈ AAR, Xα and Π ∈ AA0 R, Xα . Now, to prove that Ξ ∈ AAR, Xα . Let us take a sequence σn n∈N and show that there exists a subsequence sn n∈N such that lim
lim Ξt sn − sm − Ξtα 0 for each t ∈ R.
m → ∞ n → ∞
4.7
Since ξ ∈ AAR, Xβ , there exists a subsequence sn n∈N such that lim
lim ξt sn − sm − ξtβ 0
m → ∞ n → ∞
for each t ∈ R.
4.8
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On the other hand, we have
Ξt sn − sm − Ξt
tsn −sm −∞
AT t sn − sm − sP ξsds −
tsn −sm ∞
∞
∞
0
−
∞
−∞
−∞
AT t sn − sm − sQξsds −
AT sP ξt sn − sm − sds −
0
t
0
AT sQξt sn − sm − sds −
AT t − sP ξsds
t
∞
AT t − sQξsds
AT sP ξt − sds
0 −∞
AT sQξt − sds
4.9
AT sP ξt sn − sm − s − ξt − sds
0
−
0 −∞
AT sQξt sn − sm − s − ξt − sds.
Passing to the norm ·α , 0 < α < 1, remembering the triangle inequalities, 2.7, then we obtain that Ξt sn − sm − Ξtα ≤
∞
AT sP ξt sn − sm − s − ξt − sα ds
0
≤c
0 −∞
∞
AT sQξt sn − sm − s − ξt − sα ds 4.10 β−α−1 −γs
s
e
ξt sn − sm − s − ξt − sβ ds
0
c
0 −∞
eδs ξt sn − sm − s − ξt − sβ ds.
Thus, 4.8 and Lebesgue dominated convergence theorem lead to 4.7, therefore, to Ξ ∈ AAR, Xα . To finish the proof, we will prove that Π ∈ AA0 R, Xα . It is obvious that Π ∈ BCR, Xα , the left task is to show that 1 r → ∞ 2r lim
r −r
Πtα dt 0.
4.11
Πtα dt ≤ I J,
4.12
Using 2.7, we have 1 0 ≤ lim r → ∞ 2r
r −r
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where t c r dt t − sβ−α−1 e−γt−s ηsβ ds I : lim r → ∞ 2r −r −∞ ∞ c r lim dt σ β−α−1 e−γσ ηt − σβ dσ r → ∞ 2r −r 0 ∞
r 1 lim c σ β−α−1 e−γσ ηt − σβ dt dσ, r →∞ 0 2r −r r ∞ c J : lim dt eδt−s ηsβ ds r → ∞ 2r −r t r 0 c dt eδσ ηt − σβ dσ lim r → ∞ 2r −r −∞ 0
r δσ 1 e ηt − σβ dt dσ. lim c r → ∞ −∞ 2r −r
4.13
Then, by the Lebesgue dominated convergence theorem and the fact that η ∈ AA0 R, Xβ , one has I J 0. Hence, Π ∈ AA0 R, Xα and we end the proof. Lemma 4.3. Let assumptions (H2)–(H6) hold. Define the nonlinear operator Λ2 by Λ2 ut
t −∞
T t − sP gs, Cusds −
∞
T t − sQgs, Cusds.
4.14
t
Then, Λ2 maps PAAR, Xα into itself. Proof. The proof is similar to that of Lemma 4.2, so we omit it. Theorem 4.4. Under the assumptions (H1)–(H6), partial hyperbolic evolution equation 1.1 admits a unique pseudo almost automorphic mild solution if
c Cα cΓβ − α MαΓ1 − α < 1. Θ Lω kα δ δ γ 1−α γ β−α
4.15
Proof. Firstly, define the nonlinear operator Λ on BCR, Xα by Λut −ft, But −
t −∞
t −∞
AT t − sP fs, Busds
T t − sP gs, Cusds −
∞ t
−ft, But − Λ1 ut Λ2 ut.
∞
AT t − sQfs, Busds
t
T t − sQgs, Cusds
4.16
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Let u ∈ PAAR, Xα , then f·, Bu· ∈ PAAR, Xβ ⊂ PAAR, Xα , as proved in Lemma 4.2. Together with Lemmas 4.2 and 4.3, it follows that the operator Λ maps PAAR, Xα into itself. Secondly, we will show that Λ admits a unique fixed point in PAAR, Xα . Let v, w ∈ PAAR, Xα , then the triangle inequality reads Λvt − Λwtα ≤ ft, Bvt − ft, Bwtα Λ1 vt − Λ1 wtα Λ2 vt − Λ2 wtα .
4.17
By H1, H5, and H6, we obtain ft, Bvt − ft, Bwtα ≤ kαLBvt − Bwt ≤ kαLωv − w∞,α .
4.18
By H2, Lemma 2.6, H5, and H6, we obtain
Λ1 vt − Λ1 wtα ≤
t −∞
AT t − sP fs, Bvs − fs, Bwsα ds
∞
AT t − sQfs, Bvs − fs, Bwsα ds
t
≤ cLωv − w∞,α
t −∞
cLωv − w∞,α ≤ cLωv − w∞,α
t − sβ−α−1 e−γt−s ds
∞
4.19
eδt−s ds
t
Γβ − α 1 . δ γ β−α
Similarly, by H2, Lemma 2.5, H5, and H6, we obtain
Λ2 vt − Λ2 wtα ≤
t −∞
T t − sP gs, Cvs − gs, Cwsα ds
∞
T t − sQgs, Cvs − gs, Cwsα ds
t
≤ MαLωv − w∞,α
t
CαLωv − w∞,α ≤ Lωv − w∞,α
−∞
t − s−α e−γt−s ds
∞ t
eδt−s ds
MαΓ1 − α Cα . δ γ 1−α
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Combining the above inequality together, we obtain Λv − Λw∞,α ≤ Θv − w∞,α , where
c Cα cΓβ − α MαΓ1 − α Θ Lω kα . δ δ γ 1−α γ β−α
4.21
Clearly, if Θ < 1, then the operator Λ becomes a strict contraction on PAAR, Xα . Remembering that PAAR, Xα equipped with the α-sup norm: u∞,α supt∈R utα is a Banach space by Remark 3.9, the classical Banach fixed-point theorem leads to the desired conclusion.
5. Application Example 5.1. Take X : C0, π equipped with the sup norm. Define the operator A by Aϕξ : ϕ
ξ aϕξ,
∀ξ ∈ 0, π, ϕ ∈ DA,
5.1
where DA : {ϕ ∈ C2 0, π, ϕ0 ϕπ 0} ⊂ C0, π and a ∈ R is a constant. Clearly, A is sectorial, and hence is the generator of an analytic semigroup. If a / n2 , then A generates a hyperbolic analytic semigroup T tt≥0 on X. Let
Xα DA α, ∞
⎧ ⎪ ⎨C2α 0, π, ⎪ ⎩C2α 0, π, ν
if 0 < α
