RECOVERING OF LOWER ORDER COEFFICIENTS IN FORWARD- BACKWARD PARABOLIC EQUATIONS

We study the issue of recovering a lower order coefficient depending on spatial variables in a forward-backward parabolic equation of the second order. The overdetermination condition is an analog of the final overdetermination condition. A solution at the initial and final moments of time is given. Equations of this type often appear in mathematical physics, for example, in fluid dynamics, in transport theory, geometry, population dynamics, and some other fields. Conditions on the data are reduced to smoothness assumptions and some inequalities for the norms of the data. So it is possible to say that the obtained results are local in a certain way. Under some condition on the data, we prove that the problem is solvable. Uniqueness of the theorem is also described. The arguments rely on the generalized maximum principle and the solvability of theorems of the periodic direct problem. The results generalize the previous knowledge about the multidimensional case. The used function spaces are the Sobolev spaces.


Introduction
Let G be a bounded. The inverse problems is studied in the cylinder = (0, ) ν are the components of the outer unit normal to Γ . We assume that the coefficients of the operator L and the boundary operator B as well as the corresponding function spaces are real. The definitions of the function spaces involved can be found, for instance, in [1]. The operator L is elliptic, i. e., there exists a constant 0 > 0 δ such that The inverse problems of the form (1)-(3) in the case of positive function ( , ) g x t are studied in many articles (see [2][3][4][5] and the bibliography therein). In our case the function ( , ) g x t can change a sign, i. e., we deal with the forward-backward parabolic equation. Equations of this type often appear in mathematical physics, for example, in fluid dynamics while studying fluid motion with alternating coefficient of viscosity, in transport theory while describing the process of particles motion in some environment. Such equations also occur in geometry, population dynamics, and some other fields. Sufficient number of examples is given in [6]. The boundary value problems for equations of the form (1) are studied in many articles (see, for instance, [7,8]). The inverse problem of finding the right-hand side in (1) is studied in [9,10,12,13]. We generalize here the results of the article [13]. Our conditions on the coefficients are more general (in particular, the function in front of the derivative in time can depend on t ) and moreover, we prove solvability for an arbitrary n ( 3 n ≤ in [13]).
where k t ∂ are generalized derivatives in the Sobolev sense. By 0 W we mean the subspace of W of functions satisfying the homogeneous Dirichlet conditions in S . Define the norm 2 (0, ; ) Next, we present the condition on the data of the problem. We assume that , , (0, ; ( )), ( ,0) = ( , ) ( = 0,1), where /2 p n > for > 2 n and > 1 p for 2 n ≤ ; (i) 2 1 , , x ∈Γ in the case of the Robin boundary conditions. A pair of functions ( , ), ( ) in the case of the Dirichlet boundary conditions, the conditions (2), (3) holds, and where the integral over Γ is absent in the case of the Dirichlet boundary conditions. Consider an auxiliary problem = ( , ) = ( , ), ( , ) , in the case of the Dirichlet boundary condition and the estimate 2 0 (0, ; ') ( ) Proof. We can refer to Theorem 3 in [8], where the corresponding result is stated in an abstract form. We need only to check the conditions of this theorem. In the case of the Dirichlet boundary condition Theorem 1 is reduced to Theorem 3 in [8] after the change of variables = u v + Φ . The corresponding check relies on the embedding theorems and the condition of the theorem.

Main results
In this section we consider the inverse problem in question. To justify the corresponding results below, we employ the generalized maximum principle. So we need to impose some additional conditions on the data. (

Under the conditions (4)-(6), (i)-(viii), there exists a solution
there exists a unique solution to the problem (11), (12) from the class W . This solution satisfies the estimate 2 2 in the case of the Dirichlet boundary condition and the estimate 2 0 ( 0 ,; ' ) () in the case of the Robin boundary conditions. In view of the embedding 1 we have (let, n > 2, for example)

Математика
Bulletin of the South Ural State University Ser. Mathematics. Mechanics. Physics, 2018, vol. 10, no. 4, pp. 23 Using the conditions on the data and (15), we can rewrite (13) in the form where the constants 2 c , 3 c are independent of λ , u. Differentiate the equality (6) with respect t to and take ( ) Take 0 k > and as before assume that R B λ ∈ . First, consider the case of the Robin boundary conditions and assume that 0 ϕ ≠ . Integrating with respect t to and by parts we infer Here we employ the transformations of the type Consider the mapping