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Abstract
We investigate the uniqueness of a solution for an inverse problem of determining a spacewise-dependent source in some non-linear parabolic and hyperbolic problems. The material coefficients appearing in the governing equations may depend on both space and time. The aim is to identify a spacewise-dependent source from the usual initial and boundary conditions and the final-time over-determination. We formulate conditions on coefficients which guarantee the uniqueness of a solution of the inverse problem.
AMS Subject Classification:
Introduction
Transient problems of parabolic and hyperbolic types have been intensively studied in the last decades. The theory for direct problems is nicely elaborated in many mathematical papers and books. One of the classical inverse problems (IPs) is the source determination. This can depend both on space and on time. This general setting is up to now too hard to attack. Therefore, most of the papers in the literature deal with linear partial differential equations (PDEs) and the unknown source term depends either on space or on time. The aim of this paper is to address uniqueness of a solution to the IP of the space-dependent source determination from a final-time measurement. We consider two classes of non-linear model problems: parabolic and hyperbolic.
The first part of this paper deals with a semilinear parabolic problem with applications in reactive contaminant transport in the saturated zone, cf. [Citation1, Chap. 15]. The governing PDE is nothing else than the continuity equation. Adopting the Darcy law for diffusion in a saturated zone one can get a linear parabolic PDE for the contaminant concentration d, which depends on the time variable . Reaction transformation can be formally captured via augmenting the source by a generalized mass loss rate
, which depends on the sorbed concentration s. The usual linear first-order form is
with some given constants
and
. This can be formally resolved as
We can see that the right-hand side of the governing PDE for d will depend on the time integral of
. Considering a more general relation between contaminant and its sorption, one can even get a time integral of a non-linear function of concentration.
We study a problem of source identification from given final over-determination for the non-linear parabolic equation with Dirichlet boundary conditions. To formulate this situation mathematically, we assume that we have a non-homogeneous and non-isotropic body, denoted by , occupying a bounded domain in
, where
. The unknown functions
and
obey
(1)
(1) where the final time
. Here,
is a symmetric linear elliptic operator of the second-order with coefficients depending on both space and time. The precise form of
will be specified later depending on the situation under consideration. The data functions
as well as the coefficients appearing in the operator
, are given. We show that the final data
(2)
(2) together with appropriate conditions on data uniquely determine the solution.
The second part of this paper is devoted to source identification from given final over-determination for the wave equation with a non-linear damping subject to the Dirichlet boundary conditions. We consider the following model problem with unknown functions and
(the other data functions appearing in the problem setting are known)
(3)
(3) We show that the final over-determination () together with suitable conditions on data uniquely determines the solution.
Parabolic problem
This IP for a linear parabolic PDE has mainly been considered in a few theoretical papers, for example, [Citation2–Citation9]. Recently, see for example [Citation10], an iterative procedure was proposed and analysed for finding the source given from () in the case of time-independent operators . Very nice and simple proof technique for uniqueness has been developed in [Citation11]. Our goal is to apply this technique to the non-linear problems () and ().
We shall work in a variational framework; therefore, we introduce some standard notations, first. We assume that has a Lipschitz boundary
. The space
consists of square integrable functions on
with the usual norm
and scalar product
. The space
denotes the standard Sobolev space on
, i.e. the space of functions with generalized derivatives in
. Due to the smoothness of the boundary of
, the trace of functions in
to the boundary is well defined and
consists of functions with
.
The first theorem deals with a steady-state differential operator .
Theorem 2.1
Consider a linear differential operatorwith bounded (discontinuous) coefficient obeying
and
Let
and
. Then, there exists at most one spacewise-dependent source
such that () together with the condition () hold.
Proof
The paper [Citation8] assumed that , so
. Therefore,
was strictly accretive and
was dissipative. Following the Lumer–Phillips theorem cf. [Citation12, Section 1.4], we see that
generates a contractive semigroup, which was the crucial point in the proof of uniqueness in [Citation8].
Please note that we do not assume that . If
is bounded, then the operator
obeys the Garding inequality:
Let us have two solutions
and
to () and (). Set
and
. Then, we see that
and
. For
we get the following weak formulation
(4)
(4) We have to show that
and
. First, we show that
and later that
. We have to get rid of
at the first stage. The main idea is very simple, namely
Thus, we set
into () and integrate in time over
to get
The coefficients
and
are time-independent! Simple calculations give
because of
and
.
Analogously, we may writeIntegration by parts yields
Collecting all the results above, we arrive at
Due to the fact that
is monotonically increasing, we deduce that
and
is a constant in time. Therefore,
Putting this information into (), we obtain
from which we conclude that
.
Next theorem generalizes Theorem 2.1 to a transient differential operator .
Theorem 2.2
Consider a linear differential operatorwith
and
(5)
(5) All the coefficients appearing in the operator
are assumed to be bounded for any
. Moreover, assume that
Let
and
. Then, there exists at most one spacewise-dependent source
such that () together with the condition () hold.
Proof
The proof follows the same line as in Theorem 2.1. The main difference is in handling of the time-dependent coefficients. Assume that we have two solutions and
to () and (). We denote
and
and recall that
and
. We have to show that
and
.
For , we get (). Setting
and integrating in time over
to see that
Monotonicity of
which is found in the proof of Theorem 2.1 gives
Integrating by parts and using
and
, we have
and
Collecting the results above, we deduce that
and
Now, returning back to (), we conclude that
.
The matrix obeys (), hence it is positive definite with respect to both space and time-dependent entries. Thus, there exists a unique positive-definite square root
. We have to be careful because generally
This equality holds true for e.g. diagonal matrices. More generally,
iff the matrix commutes with its derivative which is equivalent to
for all
and
, see [Citation13, Citation14].
If we want to express conditions of Theorem 2.2 for scalar coefficients, we will have () and both coefficients A(x, t) and c(x, t) must decrease in time.
Hyperbolic problem
The IP with the final over-determination for parabolic problems admits at most one solution, but the situation for hyperbolic equations can be quite different. One can also find some non-uniqueness examples, cf. [Citation5, Section 7.2] In this section, we address the uniqueness results for the IPs () and (). We will show that the damping term helps to establish the uniqueness results for hyperbolic case. We will use a similar proof technique as in the parabolic event. The first theorem deals with a steady-state differential operator L.
Theorem 3.1
Consider a linear differential operatorwith bounded (discontinuous) coefficients obeying
and
Let
and
. Then, there exists at most one spacewise-dependent source
such that () together with the condition () holds.
Proof
Suppose that we have two solutions and
to () and (). Set
and
. Please note that
and
. We have to show that
and
.
For , we get
(6)
(6) We put
and integrate in time over
to get
(7)
(7) The last two terms on the left-hand side vanish due to the same reasoning as in Theorem 2.1. Therefore, we may write
Here, we can see why we do need the damping term. Without it, we would have that
, which gives no guarantee that
.
Employing the fact that the function g is strictly monotonically increasing, we get that , i.e. u is constant in time. Therefore,
Returning back to (), we conclude that
.
Next theorem addresses the uniqueness result for a transient operator L in a hyperbolic problem.
Theorem 3.2
Consider a linear differential operatorwith
and
All the coefficients appearing in the operator L are assumed to be bounded for any (x,t). Moreover, assume that
Let
and
. Then, there exists at most one spacewise-dependent source
such that () together with the condition () hold.
Proof
Let us have two solutions and
to () and (). Put
and
. Please note that
and
. We will show that
and
.
The relation () holds true. The last two terms on the left-hand side of () are non-negative, due to the same reasoning as in Theorem 2.2. Thus, we see thatAccording to the strict monotonicity of the function g, we obtain that
, i.e. u is constant in time. Looking at
we get
, Inspecting the relation (), we conclude that
.
Conclusions
In this work, we have studied the uniqueness of two inverse source problems: parabolic () and (); and hyperbolic () and (). We have showed that under reasonable physical conditions on the data functions, the final over-determination is sufficient for the uniqueness of a space dependent source.
Acknowledgments
This work was supported by the BOF/GOA-project No. 01G006B7 of Ghent University.
Notes
The symbol denotes the transpose of
.
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