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Abstract and Applied Analysis
Volume 2015 (2015), Article ID 393624, 42 pages
First-Order Regular and Degenerate Identification Differential Problems
1Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
2Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy
3Hirai Sanso 12-13, Takarazuka 665-0817, Japan
Received 19 August 2013; Accepted 11 February 2014
Academic Editor: Baoxiang Wang
Copyright © 2015 A. Favini et al. 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.
We are concerned with both regular and degenerate first-order identification problems related to systems of differential equations of weakly parabolic type in Banach spaces. Several applications to partial differential equations and systems will be given in a subsequent paper to show the fullness of our abstract results.
The basic aim of this paper consists in extending the results in  and in solving some identification problems in a product space , where is a Banach space , endowed with a norm , related to systems of two possibly degenerate first-order differential equations in time. More precisely, we will consider the differential problem: determine a pair of functions and , , such that In particular, the regular choice corresponds to the case where The starting point for this paper is provided by  (cf. also ) where the identification degenerate problem is studied in the Banach space under assumptions of weak parabolicity (cf. ) on the linear closed linear operators and .
Existence of solutions to evolution equations with matrix-valued operator coefficients has been considered very recently by Engel  where no degeneration is involved.
Abstract systems of parabolic equations of relevant importance in applications are described in the monograph by Yagi .
Here we study both the regular and the degenerate cases using the basic results in  and the different methods introduced therein. This will allow us to handle, in particular, systems of elliptic-parabolic equations and systems of degenerate parabolic equations in different functional spaces.
We indicate now the plan of the paper.
In Section 2, we will consider an identification problem for first-order regular systems of differential equations.
In Section 3, we will deal with identification problems for first-order in time systems of PDE’s, treating in particular some nonstandard boundary conditions.
Section 4 contains the main contributions to the degenerate case. The first two results—Theorems 14 and 18—can be easily extended to matrix-valued operators of the form where operators and map from to and , respectively, while operators and map from to and , respectively, and being a suitable Banach space.
Different levels of degeneration for operator matrices not entering the time derivative will be considered and corresponding conditions for the solvability of the related identification problems will be given.
Section 6 contains a number of applications to systems of PDE’s enlightening the concrete applications and the strict conditions to be satisfied by the single equations to guarantee both existence, uniqueness, and the regularity of solutions.
The Appendix section describes a general approach to inverse problems using a perturbation theory of generators. In such a way, the inverse problem under consideration is reduced to a direct problem with a new generator.
A number of specific examples illustrate the extension and the strength of this method.
2. Identification Problems for First-Order Regular Differential Systems in Banach Spaces
Let be a complex Banach space with norm and let The space is endowed with the product norm.
Let , , , be four closed linear operators satisfying the following properties, where denotes the resolvent set of a linear closed operator :(H1) and ;(H2) for all ;(H3) for all ;(H4) for all ;(H5) for all ;(H6), , , ;(H7), .
We consider here the problem consisting in recovering the unknown scalar function in the following differential system in : subject to the initial conditionsand to the additional information First we consider the following resolvent system in , where and : that we rewrite in the vector form: Note that according to our assumption (H3) we get Observe now that if Then for all the linear operator admits an inverse in satisfying the estimate Hence the solution to system (9) is given by Therefore, for all , we deduce the estimates Since , we conclude that the operator matrix satisfies the following bound in the product space : where
Remark 1. If operators and are bounded, then ; thus, is trivially verified.
Remark 2. If , then . Indeed, for any it follows Moreover, the closed graph theorem implies . This and the previous estimate imply The relation , along with H9, implies . Consequently, the set relations and cannot occur simultaneously.
Theorem 3. Let and be the generators of two differentiable (not necessarily densely defined) semigroups of parabolic type in the complex Banach space X satisfying (H1)–(H3). Let and be two linear closed operators satisfying properties (H4)–(H6) and let , , be two linear functionals with properties (H7). Let be a triplet of real numbers such that (cf. (19)) Let the data satisfy the properties Then the identification problem admits a unique global solution .
Remark 4. If for some and , then , . Thus implies . In this case, owing to (19), we have , so that we need to require .
In most applications, , so that this relation reduces to .
3. Identification Problems for Regular First-Order in Time Systems of PDE’s
In this section, we will deal with some identification problems related to systems of PDE’s.
Problem 5. Let be a bounded domain in with a -boundary .
We want to recover the scalar function in the initial-boundary value problem: under the additional information Here are two second-order linear elliptic differential operators with real-valued -coefficients on such that being a positive constant.
Assume for the time being that and are multiplication operators defined, respectively, by two -functions and . In this case we have .
Concerning the linear boundary differential operators defined by assume that , , , and , are real-valued functions and vector fields on such that . and are real -tangential operators on , and standing for the conormal derivatives associated with the matrices and , respectively; that is, denoting the unit outward normal vector to at .
Assume further (cf. page 515 in ) that the vector field (resp., ) does not vanish on (resp., ) and the function (resp., ) has zeros of even order not greater than (resp., ) along the integral curve of (resp., ) passing through (resp., ) at . In other words, the so-called -condition holds with and , respectively.
It is shown on page 516 in  that the operator and defined by satisfy in the resolvent estimates for all complex in a sector , with and suitable large.
We are in the subelliptic case: Choose now and make the following assumptions: Then our identification problem admits a unique global solution .
We consider now the more general case where , , and is the multiplication operator by .
From Remark 4 we get , so that reduces to , that is, to . Since we are compelled to require that either of or must coincide with .
Note that, if , then the corresponding boundary value problem is elliptic and this holds if and only if for all (cf. , on page 515). In this case so that .
Choose now and make the same assumptions as in the previous case, except for . The related condition has to be changed to the following: Then the given identification problem admits a unique global solution .
A corresponding result holds when .
Problem 6. We note that in Problem 5 the domain of the operator-matrix is a product of domains. However, Corollary 31 allows to handle also decoupled domains. For this purpose, we will consider a problem related to a reaction diffusion model describing a man-environment epidemic system investigated in . Such a model consists in a parabolic equation coupled with an ordinary differential equation via a boundary feedback operator (cf. also ).
In order to obtain stability results the authors linearize the model and arrive at the following evolution system, where and stand, respectively, for the concentration of the infection agent and the density of the infective population at time and point : where is a bounded domain in with a smooth boundary , is the Laplacian, , , are nonnegative functions, and denoted the outward normal derivative on .
We define , and denote by the multiplication operator induced by the function . Moreover, we introduce the operator matrix
It can be proved (cf. [7, page 126]) that generates an analytic semigroup on .
Consider then the identification problem consisting in finding a triplet , being a scalar function satisfying the direct Problem (38) as well as the additional condition: where , , , are fixed points in . As a consequence of [2, Theorem 3.2] we get
Notice that the interpolation spaces are well characterized both in an abstract form and for many boundary conditions (cf. [4, 9]), but in this concrete case it seems to be difficult to translate them as on page 321 in . Therefore, one can use the more restrictive assumptions that can be easily checked.
Problem 8. We solve here an identification problem in Hölder spaces.
Let be a bounded domain in with a smooth boundary . Then , , and denote the Banach space of all functions in whose derivatives of order are all Hölder continuous with exponent . Such a space will be endowed with the natural norm .
We introduce now some notation and assumption. As usual, denotes a multi-index and we associate with it the monomial differential operator , . Finally, for any (fixed) and let the functions , , satisfy, for some positive constant , the relations: Introduce now the linear operator in defined by A remarkable result by von Wahl (cf.  on page 241) establishes the resolvent bound for a large , where .
For our application is not restrictive to assume that such a condition is satisfied for all . In this case the resolvent estimate holds with .
Let now be another linear differential operator of order , with properties similar to those of , defined by where .
Likewise as above, the spectral estimate holds with .
Notice that and imply and .
Let now and be linear differential operators, with smooth coefficients (e.g., in ), defined by where and .
In view of Satz II on page 239 in  we have where We introduce now the operator matrix We can now apply Theorem 3 with , and Under such hypotheses the identification problem and being two fixed elements in , admits a unique solution provided that the following conditions are satisfied: (H1), , , ;(H2), , ;(H3), , .Notice that, if and , then and condition is satisfied.
Problem 9. Here we solve an identification problem related to the Ornstein-Uhlenbeck operator in , . For this purpose, we refer to the monograph . Such an operator is the prototype of an elliptic operator with unbounded coefficients and is defined on smooth functions by where and denote, respectively, the gradient and the Hessian matrix of , while and are constant matrices, being strictly positive definite and the spectrum of being contained in the left complex half-plane .
It is well known that the realization of in the weighted space , where generates an analytic semigroup with domain , where and is endowed with the norm , where .
Moreover, the spectrum of is the discrete set denoting the (distinct) eigenvalues of .
We note also that (cf. [11, Theorem ]) Whence, since the operator is continuously invertible, we easily deduce the estimates Likewise we get the estimates Finally, recall that the dual space to can be identified with , .
Consider now the linear differential operator defined by where , are uniformly continuous and bounded functions in .
The previous estimates yield Consider then the linear operator where are uniformly continuous and bounded functions in .
Reasoning as above, we conclude that satisfies the estimate Consider now the identification problem: where , , , , and Theorem 3 applies with , . Here , , , so that . Assume further that possess the additional properties: Then the identification Problems (66)–(321) admit a unique strict global solution with , .
We note that a corresponding result still holds if either of the lower order operators is replaced with .
Since is an unbounded domain of cone type, the real interpolation spaces , are well characterized. Exactly we have (cf. [9, Theorem ]): Assume now that our data possess the following properties: Therefore, we can apply Theorem 3 or Proposition 5.1 in  and deduce the same conclusion as above.
We can analogously deal with the case when either of the lower order operators is replaced with .
Problem 10. In  the following remarkable result is proved. Let be a Hilbert space, let be a densely defined, strictly positive self-adjoint operator, and let be a constant complex-valued matrix. Then the operator , defined by generates a strongly continuous analytical semigroup in if and only if (cf. [12, page 311]) the eigenvalues of the matrix satisfy (i) if is complex and (ii) if is real.
This result allows to deal with a lot of very important problems related to evolution PDE’s. We confine ourselves to describe an identification problem related to one of such models.
Suppose that generates an analytical semigroup in . Then owing to [1, Theorem 2.1, page 45] (as improved in a following paper for Al Horani and Favini) the identification problem admits a unique global solution , provided that Let us apply this result to the following thermoelastic PDE problem with simplified “hinged” homogeneous boundary conditions and Dirichlet thermal boundary conditions (cf. [12, page 317]), where the reference bounded domain has a smooth boundary : determine a triplet of functions and such that where and denotes the duality pairing between and its dual space.
Introduce now the positive self-adjoint operator defined by This definition implies The previous differential equations read now Introduce then the Banach space , the new variable , the vector , and the operator defined by Then system can be rewritten as the single equation: In [12, page 318] it is shown that operator generates an analytic semigroup on if and only if the operator generates a strongly continuous analytic semigroup in the space .
Indeed, in this case we have , where the constant matrix is defined by Consider then the eigenvalue equation and observe that (i) since and , the matrix admits a real negative eigenvalue ; (ii) all the real eigenvalues of are strictly negative; (iii) all the nonreal eigenvalues have negative real parts , since the sum of is . Consequently, all the eigenvalues of have negative real parts so that generates an analytic semigroup in (resp., ).
Let us consider first our identification problem in . Since we must require Moreover, we assume