INVERSE PROBLEMS FOR EVOLUTION EQUATIONS WITH TIME DEPENDENT OPERATOR-COEFFICIENTS

. In this paper we study an inverse problem with time dependent operator-coeﬃcients. We indicate suﬃcient conditions for the existence and the uniqueness of a solution to this problem. A number of concrete applications to partial diﬀerential equations is described

with α(t) possibly non negative, but such general hypotheses do not guarantee, in particular, that y is an X−valued continuous solution on [0, τ ]. To handle the related inverse problems, more regular solutions seem necessary.
In this paper, we are concerned to give a first treatment to the inverse problem with time dependent coefficients d dt y(t) = A(t)y(t) + f (t)z + h(t), 0 ≤ t ≤ τ, (1.1) It is well known that problems like (1.1)-(1.3) can be handled directly via fixed point argument. Recently in [14], the time independent coefficients case was faced by reducing the problem to a direct problem with perturbed operators A + B. Clearly, one can tempt to extend this strategy to the general case, but the problem is to introduce assumptions guaranteeing that A(t) + B(t) has the required properties which govern a well posed evolution equation.
In order to solve (1.1)-(1.3) according to this strategy, we apply Φ ∈ X * to both sides of equation (1.1) to obtain, by using (1.3), under the additional assumption (1.5) Substituting (1.5) in (1.1), we obtain the following new direct problem to be solved in the variable y(·) Introduce operator B(t) : D(B(t)) = D(A(t)) → X, all is reduced to existence, uniqueness and regularity of a solution y(·) to the problem The essential role of our results is to guarantee that the perturbed operator A(t) + B(t) has a well appropriate behavior. To this point, the results from [9] will play an essential role in the hyperbolic case. To motivate this choice of using perturbation methods, for sake of simplicity, we confine to the parabolic case, cfr. [23]. Then for a given continuous function f : [0, τ ] → C, the strict solution to the problem necessarily has the form where U (t, s), 0 ≤ s ≤ t ≤ τ is the evolution operator generated by A(t). Notice that under suitable regularity assumptions on the data, at least formally, Then all is reduced to find a continuous function f (t) on [0, τ ] such that, assuming g ∈ C 1 ([0, τ ], C), and thus the problem becomes the one to solve the integral equation (1.6) globally. This is not an obvious task, as it can be seen from the monograph [8]. Therefore, it seems that the proposed perturbation argument yields more quickly the desired results.
As previously remarked, solutions to the integro-differential equations of parabolic type and possibly degenerate of the form have been, very recently, discussed in the paper [15], where L(t), K(t, s) are second order differential operators, M (t) is the multiplication operator by m(t, x) ≥ 0 on the ambient space L p (Ω), 0 ≤ t ≤ τ . Theorem 9.1 in [15] on existence and uniqueness of solutions implies only that L(·)y(·) ∈ C ((0, τ ]; L p (Ω)) and thus the related treatment of the inverse problem seems to be not an easy consequence, because more regularity to the solution must be assumed. The contents of the paper are as follows. In Section 2 we recall some results of perturbation that we need. Section 3 contains the main results. In Section 4, some applications of partial differential equations are given to illustrate our abstract results.

Preliminaries. We start this section by recalling some definitions
Of course, similarly, one can introduce stability of the restriction of A(t) to a subspace D of X. The Favard class F of a generator of a C 0 −semigroup in X is the interpolation space (X, D) 1, ∞ . It is known that if X is reflexive, then the Favard class reduces to D.  In particular, see [9], Proposition 6, To solve the problem at all, we need the following essential well known result, precisely, Theorem 5.3, p. 147 in [21], related to the Cauchy problem in a Banach space X affirms that if {A(t)} t∈[0,T ] is a stable family of generators, with a constant domain D, such that ∀ y ∈ D, A(t)y is continuously differentiable, then for all y 0 ∈ D and f ∈ C 1 ([0, T ]; X), the Cauchy problem admits a unique strict solution y(·).
3. Main results. As we noted in the introduction, the inverse problem to find y ∈ C([0, τ ]; D), f ∈ C([0, τ ]; C) such that under the assumption that {A(t)} t∈[0,T ] is a stable family of generators of C 0semigroups with constant domain D, is reduced to the direct problem of finding a strict solution y to the initial value problem Let us assume that for all x ∈ D, A(t)x is continuously differentiable. Take F = D and assume a) in Proposition 2.3. We have Proof. The proof follows easily from Proposition 2.3 and the recalled existence and uniqueness result from Pazy [21].
If D is stable for A(t), too, Theorem 5.2, p. 146 in [21] allows one to establish the following result on space regularity condition Analogous results are obtained for CD-Systems in a separable Banach space. We omit these related results.
In the sequel, we shall devote ourselves to the parabolic case . Suppose that A(t) generates an analytic semigroup in the Banach space X, D(A(t)) ≡ D is independent of t, B(t) ∈ L(D, X) but range(B(t)) ∈ (X, D) θ,∞ .