ABSTRACT SECOND ORDER DIFFERENTIAL EQUATIONS WITH TWO SMALL PARAMETERS AND LIPSCHITZIAN NONLINEARITIES

SECOND ORDER DIFFERENTIAL EQUATIONS WITH TWO SMALL PARAMETERS AND LIPSCHITZIAN NONLINEARITIES In a real Hilbert space H we consider the following singularly perturbed Cauchy problem ε u′′ εδ(t) + δ u ′ εδ(t) + Auεδ(t) + B(uεδ(t)) = f(t), t ∈ (0, T ), uεδ(0) = u0, uεδ(0) = u1, where u0, u1 ∈ H, f : [0, T ] 7→ H, ε, δ are two small parameters, A is a linear self-adjoint operator and B is a nonlinear lipschitzian operator. We study the behavior of solutions uεδ in two di erent cases: ε → 0 and δ ≥ δ0 > 0; ε → 0 and δ → 0, relative to solution to the corresponding unperturbed problem.

(ii) ε → 0 and δ → 0, relative to the solutions to the following unperturbed system: The problem (P εδ ) is the abstract model of singularly perturbed problems of hyperbolicparabolic type in the case (i) and of the hyperbolic-parabolic-elliptic type in the case (ii). Such kind of problems arise in various elds of science and technology, for example, in the mathematical modeling of elasto-plasticity phenomena.
In many works, under various restrictions singularly perturbed Cauchy problems for linear or nonlinear dierential equations of second order of type (P εδ ) were studied. Without pretending to a complete analysis, we will mention the works [2,3,4,5,8,9], in which the reader can nd an extensive bibliography.
In most of the mentioned cases the results were obtained by using the theory of semigroups of linear operators. Dierent to other methods, our approach is based on two key points. The rst one is the relationship between solutions to the Cauchy problem for the abstract linear second order dierential equation and the corresponding problem for the rst order equation. The second key point are a priori estimates of solutions, which are uniform with respect to the small parameter. Moreover, we study the problem (P εδ ) for a larger class of functions, i. e. f ∈ W 1,p (0, T ; H). Also we obtain the convergence rate, as ε → 0, which depends on p.
The organization of this paper is as follows. At the beginning of the next section we present the theorems of existence and uniqueness of solutions to the problems (P εδ ), (P δ ) and some a priori estimates of these solutions. Then we present a relationship between solutions to the problem for the abstract linear second order dierential equation and the corresponding solution to the problem for the rst order equation. In the section 3 we present the main result of the paper. More precisely, we prove the convergence estimates of the dierence of solutions to the problems (P εδ ) and (P δ ) for ε → 0, δ ≥ δ 0 > 0 and also to the problems (P εδ ) and (P 0 ) for ε → 0, δ → 0.
In what follows we will need some notations. Let k ∈ N * , 1 ≤ p ≤ +∞, (a, b) ⊂ (−∞, +∞) and X be a Banach space. By W k,p (a, b; X) denote the Banach space of vectorial distributions u ∈ D (a, b; X), u (j) ∈ L p (a, b; X), j = 0, 1, . . . , k, endowed with the norm If X is a Hilbert space, then W k,2 (a, b; X) is also a Hilbert space with the scalar product The framework of our paper will be determined by the following conditions:

Preliminaries
In this section we remind results about the solvability of problems (P εδ ), (P δ ) and (P 0 ). Since these results do not depend on the positive values of the parameters ε and β, we will put ε = δ = 1. In this case the problem (P εδ ) takes the form: the problem (P δ ) takes the form: and the problem (P 0 ) takes the form: The following theorems were inspired by the work [1] and are completely proved in the work [6].
Theorem 2. Let T > 0. Assume that conditions (HA) and (HB) are fullled. If u 0 ∈ H and f ∈ L 2 (0, T ; H), then there exists a unique function l ∈ C([0, T ]; H), A 1/2 l ∈ L 2 (0, T ; H) such that l satises the equation (2) in the sence of distributions on (0, T ) and the initial condition from (2). This function is called the strong solution to the problem (2).
Theorem 3. Let T > 0 and p > 1. Suppose that conditions (HA) and (HB) are fullled and ω > L. If f ∈ W 1,p (0, T ; H), then the equation Av + B(v) = f has a unique strong solution v ∈ W 1,p (0, T ; H) and For the further consideration we rewrite the problems (P εδ ) and (P δ ) in the form: In what follows we will prove some a priori estimates for solutions to the problems (P µ ) and (P 0 ). Lemma 1. Suppose that q 0 = ω − L > 0 and conditions (HA) and (HB) are fullled. If u 0 ∈ D(A), u 1 ∈ D A 1/2 , F ∈ W 1,1 (0, ∞; H) then for any strong solution U µ to the problem (P µ ) the following estimate then integrating (6) on (0, s), we get Applying Gronwall-Bellman's Lemma to the last inequality, we obtain Under the conditions of Lemma, due to the Theorem 1, we have that the following relations Abstract second order differential equations...

QQ
hold. Taking into account the relations (8), we divide (7) by h and then pass to the limit in the obtained inequality, to get the estimate (5).
To establish the relationship between solutions to the problems (P µ ) and (P 0 ) in the linear case we will dene the kernel of transformation realizing this relationship.
For µ > 0 denote by In the following lemma we present some properties of kernel K(t, τ, µ), used in the proof of the following results.
[7] The function K(t, τ, µ) is solution to the problem ∞)) and possesses the following properties: positive denite operator and F ∈ L ∞ (0, ∞; H). If U µ is a strong solution to the problem (P µ ) with U µ ∈ W 2,∞ (0, ∞; H), AU µ ∈ L ∞ (0, ∞; H), then the function W µ dened by 3 Behaviour of solutions to the problem (P εδ ) In this section we will prove the main result concerning the behavior of the solutions to the problem (P εδ ), in both cases: ε → 0 and δ ≥ δ 0 > 0; ε → 0 and δ → 0, relative to solution to the corresponding unperturbed problem.

Moreover, the estimate (11) implies
Due to the estimates (12) and Lemma 1, we obtain the following estimates holds with M from(9), µ 0 from (5) and Abstract second order differential equations...

QS
By Lemma 3, the function W µ , dened by W µ (s) = ∞ 0 K(s, τ, µ) U µ (τ ) dτ, is a strong solution to the problem W µ (s) + AW µ (s) = F 0 (s, µ), a.e. s > 0, in H, Denote by R(s, µ) = L(s) − W µ (s), where L is the strong solution to the problem (P 0 ) with F instead of F, T = ∞ and W µ is the strong solution to the problem (15). Then, due to Theorem 2, R(·, µ) ∈ W 1 ,∞ loc (0, ∞; H) and R is a strong solution in H to the problem where In what follows we need the following two Lemmas, which will be proved after the proof of the Theorem 4. Lemma 4. Assume the conditions of Theorem 4 are fullled. Then the following estimates: are valid with M from (9), γ from (14) and µ 0 from (5).
Lemma 5. Assume the conditions of Theorem 4 are fullled. Then for the strong solution to the problem (17) the following estimate is true with M from(9), β and γ from(14) and µ 0 from (5).
In what follows we will investigate the behavior of solutions to the problem (P δ ) as δ → 0.