EXISTENCE OF ASYMPTOTICALLY ALMOST AUTOMORPHIC SOLUTIONS FOR A THIRD ORDER DIFFERENTIAL EQUATION

This paper deals with results on existence of asymptotically almost automorphic solutions for a third order in time abstract differential equation which model, on one side, high intensity ultrasound in acoustic wave propagation, while on the other side, vibrations of flexible structures possessing internal material damping. We established the asymptotically almost automorphy of the output solution subject to the asymptotically almost automorphy of the input distur- bance.


Introduction
It is well known, that the dynamics of linear vibrations of elastic structures are mathematically governed by the wave equation.However, the dynamics of elastic vibrations of flexible structures are actually nonlinear in practice.In 1998, Bose and Gorain [6] studied a more realistic model of vibrations of elastic structure in which the stress is not simply proportional to the strain.As a result, they shown that the dynamics of vibrations of elastic structures are governed by the following third order differential equation (1.1) αu ′′′ (t) + u ′′ (t) − β∆u(t) − γ∆u ′ (t) = 0, t ≥ 0 with suitable boundary and initial conditions, and where α, β, γ are positive constants.This third order in time equation displays, even in the linear version, a variety of dynamical behaviors for their solutions that depend on the physical parameters in the equation.These range from non-existence and instability to exponential stability (in time) [21].Concerning qualitative properties, Bose and Gorain studied boundary stabilization and obtained the explicit exponential energy decay rate for the solution of (1.1) subject to mixed boundary conditions (see [6,7,16,17] and references therein).Motivated by these works, abstract linear equations of the form where A is a closed linear operator acting in a Banach space X and f is a Xvalued function has been treated in recent papers [9,14,15,11].We emphasize that the abstract Cauchy problem associated with (1.2) is in general ill-posed, see e.g.[30].We mention that models related to (1.2) have been recently also considered in [21], where (1.1) is called Moore-Gibson-Thompson equation, and the nonlinear version is referred to as the Jordan-Moore-Gibson-Thompson-Westervelt equation.
In [21] equation (1.1) arise as a model in acoustics, more precisely in high intensity ultrasound.The results in [21] for (1.2) assumed that A is a selfadjoint operator defined on a Hilbert space H and rewrite the equation as a first order abstract system on the phase space D(A 1/2 ) × D(A 1/2 ) × H. However it is well known that in order to analyze higher order equations in an abstract setting, a direct approach leads in some situations to better results than those obtained by a reduction to a first-order equation, see e.g.[8] and [13].
Our purpose in this paper is analyze and to prove the existence of asymptotically almost automorphic mild solutions for an abstract semilinear equation of the form (1.3) αu ′′′ (t) + u ′′ (t) − βAu(t) − γAu ′ (t) = f (t, u(t), u ′ (t), u ′′ (t)), α, β, γ ∈ R + , with appropriate initial conditions.The motivation for incorporating f as an input disturbance in the governing differential equation arises from the fact that very small amount of these, are always present in real materials as long as the system vibrates.Hence, is also reasonable the study of existence of asymptotically almost automorphic solutions when f (t, x) is asymptotically almost automorphic in t; that is, asymptotically almost automorphic stability of the system.
A surprising fact is that in order to get asymptotic behavior, some initial conditions should be forced to be zero.This leads to an unexpected property that is not present in the study of the same qualitative property for the Cauchy problem of order less than 3, see [1].
To achieve our goal we use a mixed method, combining tools of certain strongly continuous families in operator theory, introduced in [11], and fixed point theory.
This paper is organized as follows: The preliminary Section 2 collects results essentially contained in [23] and standard literature of almost automorphic and asymptotically almost automorhic functions (see [18], [19]).In particular we establish a result of composition for asymptotically almost automorphic functions (see Lemma 2.7) which is very important in our investigations.In Section 3 we first recall from [11] sufficient conditions for existence of solutions for equation (1.3).In fact, Proposition 3.1 gives a complete description of the solutions in terms of (α, β, γ)-regularized families.It corresponds to an extension of the standard variation of parameters formula.Then, we study conditions for existence and uniqueness of asymptotically almost automorphic solutions.We have two situations: In the linear case, we can ensure conditions for existence of asymptotically almost automorphic solution (see Theorem 3.3).For the semilinear case, we establish sufficient conditions for existence of asymptotically almost automorphic mild solutions (see EJQTDE, 2012 No. 53, p. 2 Theorem 3.3, Theorem 3.5, Theorem 3.9 and Theorem 3.11).In an special case, we are also able to prove existence of mild solution with nonlocal conditions (Theorem 4.12).Finally, we show that our abstract results apply to equation (1.3) in case of A = ∆, the Laplacian.

Preliminaries
Let α, β, γ ∈ R, α = 0 be given.In what follows we denote In order to give a consistent definition of mild solution for equation (1.3) based on an operator theoretical approach, we introduce the following definition (see [20] for a recent discussion about the concept of mild solutions for nonlinear equations and [26] for the approach that we will use in this paper).Definition 2.1.Let A be a closed and linear operator with domain D(A) defined on a Banach space X.We call A the generator of an (α, β, γ)-regularized family {R(t)} t≥0 ⊂ B(X) if the following conditions are satisfied: (R1) R(t) is strongly continuous on R + and R(0) = 0; (R2) R(t)D(A) ⊂ D(A) and AR(t)x = R(t)Ax for all x ∈ D(A), t ≥ 0; (R3) The following equation holds: for all x ∈ D(A), t ≥ 0. In this case, R(t) is called the (α, β, γ)-regularized family generated by A.
We recall the following result which provide a wide class of generators of (α, β, γ)regularized families.
Theorem 2.4 ( [14]).Let −B be a positive selfadjoint operator on a Hilbert space H such that αβ ≤ γ .Then B is the generator of a bounded (α, β, γ)-regularized family on H.
Let us recall the notion of almost automorphic and asymptotically almost automorphic which shall come into play later on.Definition 2.5.A continuous function f : R → X is said to be almost automorphic if for every sequence of real numbers (s ′ n ) n∈N there exists a subsequence is well defined for each t ∈ R, and If the convergence above is uniform in t ∈ R, then f is almost periodic in the classical Bochner's sense.
EJQTDE, 2012 No. 53, p. 4 Almost automorphic, as a generalization of the classical concept of an almost periodic function, was introduced in the literature by S. Bochner and recently studied by several authors, including [4,5,12] among others.A complete description of their properties and further applications to evolution equations can be found in the monographs [18] and [19] by G. M. N'Guérékata.
We remark that the set of all almost automorphic functions, denoted by AA(X), endowed with the sup norm is a Banach space.We define the set AA(R × X; X) which consists of all functions f : R × X → X such that f (•, x) ∈ AA(X) uniformly for each x ∈ K, where K is any bounded subset of X.
Let C 0 (R + , X) be the subspace of BC(R + , X) such that lim t→∞ x(t) = 0 and C 0 (R + × Y, X) denotes the space of all continuous functions h : R + × Y → X such that lim t→∞ h(t, x) = 0 uniformly for x in any compact subset of Y .

Definition 2.6. A continuous function
We observe that AAA(X) is a Banach space with the sup norm.The next lemma will be very useful for our results.Lemma 2.7.[27] Let X and Y be Banach spaces.Suppose that f ∈ AAA(R×Y ; X) and g are uniformly continuous on any bounded subset K ⊂ Y , uniformly for t ≥ 0, . We say f is n times differentiable asymptotically almost automorphic and we denote AAA n (X) the set of functions n times differentiable asymptotically almost automorphic.
The set AAA n (X) is a Banach space with norm For more details see [28], pages 1316-1317.

Asymptotically almost automorphic solutions
Let α, β, γ ∈ (0, ∞).Consider the linear equation The following result gives a complete description of the solutions for equation (3.1) in terms of (α, β, γ)-regularized families.It corresponds to an extension of the standard variation of parameters formula for the second order Cauchy problem.
is a solution of (3.1).

The following assumption was introduced in [11]:
(ED) There are constants M > 0 and ω > 0 such that We say in short that R(t) and R ′ (t) are exponentially stable.We introduce the following condition: (ED) * There are constants M > 0 and ω > 0 such that The following result on regularity of the convolution under asymptotically almost automorphic functions is one of the keys to obtain our results.Lemma 3.2.Let R(t) be an exponentially stable (α, β, γ)-regularized family on X with generator A. If f ∈ AAA(X) then the function We note that G(t) ∈ AA(X) by [2, Lemma 3.1], now we claim that H(t) → 0 as t → ∞.In fact, for each ǫ > 0 there exists a T > 0 such that h(s) ≤ ǫ for all s > T .Then for all t > 2T we deduce Therefore, lim t→∞ H(t) = 0, that is, H ∈ C 0 (R + , X).This completes the proof.
We begin our results on existence of asymptotically almost automorphic functions for the linear equation, with the following theorem.Theorem 3.3.Let R(t) be an (α, β, γ)-regularized family on X with generator A that satisfies assumption ) and y, z ∈ D(A 2 ).From Proposition 3.1 we have that the solution for Eq.(3.1) is given by From Lemma 3.2 we have that g(t) = t 0 R(t − s)f (s)ds belongs to AA(X).On the other hand, if t → ∞ we have that αR ′ (t)y ≤ α y Me −ωt → 0, R(t)y ≤ y Me −ωt → 0 and αR(t)z ≤ α z Me −ωt → 0. Therefore, u ∈ AAA(X).
From now we study the semilinear version of Eq. (3.1).We consider first the initial value problem where α, β, γ ∈ (0, ∞), A is the generator of a (α, β, γ)-regularized family R(t) and f : R + × X → X is a suitable function.Definition 3.4.[11] Let R(t) be an (α, β, γ)-generalized family on X with generator A. A continuous function u : R + → X satisfying the integral equation where y, z ∈ X is called a mild solution to the equation (3.3).
EJQTDE, 2012 No. 53, p. 7 We study conditions to existence and uniqueness of a mild solution for equation (3.3) when the function f is Lipschitz continuous.Theorem 3.5.Let R(t) be an (α, β, γ)-regularized family on X with generator A that satisfies assumption (ED).Let f ∈ AAA(R + × X, X) and suppose that there exists an integrable bounded function L : R + → R + such that Then equation (3.3) has a unique asymptotically almost automorphic mild solution.
In general, we get the following estimate < 1 for n sufficiently large, by the fixed point iteration method Λ has a unique fixed point u ∈ AAA(X).This completes the proof.
The appropriate concept of mild solution reads now as follows.
A continuous function u : R + → X satisfying the integral equation where z ∈ D(A) is called a mild solution to the equation (3.4).
for all t ∈ R. By the other hand, we have EJQTDE, 2012 No. 53, p. 9 Since α 1 , α 2 , α 1 , α 2 are bounded functions, there exist a bounded set K ⊂ X, such that In similar way, we prove lim ) .Now we will prove the second part.Note that Then for all ε > 0, there exists T > 0, such that Since g is uniformly continuous on any bounded subset K ⊂ X, uniformly for t ≥ 0, we have for all ε ′ > 0, there exists δ > 0, such that x − x ′ < δ, y − y ′ < δ, and x, x ′ , y, y ′ ∈ K, where K is any bounded subset of X then In particular, we take ε = δ, then Furthemore x(•), α 1 (•), y(•), α 2 (•) are bounded functions, then exists a bounded set K ⊂ X, such that x(t), α 1 (t), y(t), α 2 (t) ∈ K, for all t ∈ R.Then, we have Following the same lines of the proof of Lemma 3.7, we get the following result for the space AAA 2 (R × X × X × X, X).Lemma 3.8.Let X and Y be Banach spaces.Suppose that f ∈ AAA(R × X × X × X; X) and g is uniformly continuous on any bounded subset K ⊂ X, uniformly for t ≥ 0, where The following is our second main result in this paper.Theorem 3.9.Let R(t) be an (α, β, γ)-regularized family on X with generator A that satisfies assumption (ED).Let f ∈ AAA(R + × X × X, X) and suppose there exists constants L 1 , L 2 such that max{L 1 , L 2 } < 2w M and Then equation (3.4) has a unique differentiable asymptotically almost automorphic mild solution.
Proof.We define the operator Λ on the space AAA 1 (X) by , then Λu(t) ∈ AAA(X).Furthermore, using R(0) = 0 and z ∈ D(A), we have By Lemma 3.7 where L = max{L 1 , L 2 }.In similar way, we have This proves that Λ is a contraction, so by the Banach fixed point theorem there exists a unique u ∈ AAA 1 (X) such that Λu = u, proving the theorem.
We have a similar result using the condition (ED * ).
Similarly we have the following result for the space AAA 2 (R + × X × X × X, X).
Theorem 3.11.Let R(t) be an (α, β, γ)-regularized family on X with generator A that satisfies assumption Then equation (3.4) has a unique twice differentiable asymptotically almost automorphic mild solution.
Proof.The proof is similar to the proof of Theorem 3.9, using the inequality

Existence of mild solutions with nonlocal conditions
In this section, we use the Hausdorff measure of noncompactness and a fixed point argument to prove the existence of a mild solution for an special case of equation (3.4) with a nonlocal initial condition.More precisely, we consider (4.7) where A is the generator of a (α, β, γ)-regularized family R(t) and f : I × X → X, g : C([0, 1]; X) → X are suitable functions.
In order to give our main result, we consider the following assertions (H1) A generates a norm continuous (for t > 0) (α, β, γ)-regularized family R(t).
We note that this measure of noncompactness satisfies interesting regularity properties.For more information, we refer to [3].We are now in position to establish the following result.Proof.