This article was downloaded by: [Portland State University]On: 15 October 2014, At: 13:54Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Inverse Problems in Science andEngineeringPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gipe20

Some uniqueness theorems for inversespacewise dependent source problemsin nonlinear PDEsM. Slodičkaa

a Faculty of Engineering and Architecture, Department ofMathematical Analysis, Ghent University, Ghent, Belgium.Published online: 05 Aug 2013.

To cite this article: M. Slodička (2014) Some uniqueness theorems for inverse spacewise dependentsource problems in nonlinear PDEs, Inverse Problems in Science and Engineering, 22:1, 2-9, DOI:10.1080/17415977.2013.823418

To link to this article: http://dx.doi.org/10.1080/17415977.2013.823418

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Inverse Problems in Science and Engineering, 2014Vol. 22, No. 1, 2–9, http://dx.doi.org/10.1080/17415977.2013.823418

Some uniqueness theorems for inverse spacewise dependent sourceproblems in nonlinear PDEs

M. Slodicka∗

Faculty of Engineering and Architecture, Department of Mathematical Analysis, Ghent University,Ghent, Belgium

(Received 5 July 2013; final version received 5 July 2013)

We investigate the uniqueness of a solution for an inverse problem of determininga spacewise-dependent source in some non-linear parabolic and hyperbolic prob-lems. The material coefficients appearing in the governing equations may dependon both space and time. The aim is to identify a spacewise-dependent source fromthe usual initial and boundary conditions and the final-time over-determination.We formulate conditions on coefficients which guarantee the uniqueness of asolution of the inverse problem.

Keywords: semilinear parabolic problem; wave equation with non-lineardamping; inverse problem; uniqueness

AMS Subject Classification: 35R30

1. Introduction

Transient problems of parabolic and hyperbolic types have been intensively studied in thelast decades. The theory for direct problems is nicely elaborated in many mathematicalpapers 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 toattack. Therefore, most of the papers in the literature deal with linear partial differentialequations (PDEs) and the unknown source term depends either on space or on time. Theaim of this paper is to address uniqueness of a solution to the IP of the space-dependentsource determination from a final-time measurement. We consider two classes of non-linearmodel problems: parabolic and hyperbolic.

The first part of this paper deals with a semilinear parabolic problem with applicationsin reactive contaminant transport in the saturated zone, cf. [1, Chap. 15]. The governingPDE is nothing else than the continuity equation. Adopting the Darcy law for diffusion ina saturated zone one can get a linear parabolic PDE for the contaminant concentration d,which depends on the time variable t ∈ [0, T ]. Reaction transformation can be formallycaptured via augmenting the source by a generalized mass loss rate ∂t s, which depends onthe sorbed concentration s. The usual linear first-order form is

∂t s = Kr (Kdd − s)

∗Email: marian.slodicka@ugent.be

© 2013 Taylor & Francis

Dow

nloa

ded

by [

Port

land

Sta

te U

nive

rsity

] at

13:

54 1

5 O

ctob

er 2

014

Inverse Problems in Science and Engineering 3

with some given constants Kr and Kd . This can be formally resolved as

s(t) = e−Kr t s(0)+ Kr Kd

∫ t

0e−Kr (t−ξ)d(ξ)ξ.

We can see that the right-hand side of the governing PDE for d will depend on the timeintegral of d . 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 forthe non-linear parabolic equation with Dirichlet boundary conditions. To formulate thissituation mathematically, we assume that we have a non-homogeneous and non-isotropicbody, denoted by �, occupying a bounded domain in R

n , where n ≥ 1. The unknownfunctions u and f obey⎧⎪⎪⎨

⎪⎪⎩ut (x, t)+ Lu(x, t) = f (x)+ h(x, t)+

∫ t

0g(u(x, s))ds in �× (0, T ),

u(x, t) = α(x, t) on ∂�× (0, T ),u(x, 0) = u0(x) for x ∈ �,

(1)

where the final time T > 0. Here, L is a symmetric linear elliptic operator of the second-order with coefficients depending on both space and time. The precise form of L will be spec-ified later depending on the situation under consideration. The data functions h, g, α and u0as well as the coefficients appearing in the operator L , are given. We show that the finaldata

u(x, T ) = ψT (x). (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 Dirichletboundary conditions. We consider the following model problem with unknown functions uand f (the other data functions appearing in the problem setting are known)⎧⎪⎪⎨

⎪⎪⎩

utt (x, t)+ g(ut (x, t))+ Lu(x, t) = f (x)+ h(x, t) in �× (0, T ),u(x, t) = α(x, t) on ∂�× (0, T ),u(x, 0) = u0(x) for x ∈ �,ut (x, 0) = v(x) for x ∈ �.

(3)

We show that the final over-determination (2) together with suitable conditions on datauniquely determines the solution.

2. Parabolic problem

This IP for a linear parabolic PDE has mainly been considered in a few theoretical papers,for example, [2–9]. Recently, see for example [10], an iterative procedure was proposed andanalysed for finding the source given from (2) in the case of time-independent operators L .Very nice and simple proof technique for uniqueness has been developed in [11]. Our goalis to apply this technique to the non-linear problems (1) and (2).

We shall work in a variational framework; therefore, we introduce some standardnotations, first. We assume that � has a Lipschitz boundary ∂�. The space L2(�) consistsof square integrable functions on � with the usual norm ‖ · ‖ and scalar product ( · , · ).

Dow

nloa

ded

by [

Port

land

Sta

te U

nive

rsity

] at

13:

54 1

5 O

ctob

er 2

014

4 M. Slodicka

The space H1(�) denotes the standard Sobolev space on�, i.e. the space of functions withgeneralized derivatives in L2(�). Due to the smoothness of the boundary of�, the trace offunctions in H1(�) to the boundary is well defined and H 1

0 (�) consists of functions withu|∂� = 0.

The first theorem deals with a steady-state differential operator L .

Theorem 2.1 Consider a linear differential operator

Lu(x, t) = ∇ · (−A(x)∇u(x, t))+ c(x)u(x, t),

with bounded (discontinuous) coefficient obeying A(x) = Atr (x)1 and

ξ tr · A(x)ξ =n∑

i, j=1

ai, j (x)ξiξ j ≥ C |ξ |2 ∀ξ ∈ Rn .

Let u0, ψT ∈ L2(�) and g′ ≥ 0. Then, there exists at most one spacewise-dependentsource f ∈ L2(�) such that (1) together with the condition (2) hold.

Proof The paper [8] assumed that c ≥ 0, so (Lu, u) ≥ C0‖u‖2H1(�)

. Therefore, Lwas strictly accretive and −L was dissipative. Following the Lumer–Phillips theorem cf.[12, Section 1.4], we see that L generates a contractive semigroup, which was the crucialpoint in the proof of uniqueness in [8].

Please note that we do not assume that c ≥ 0. If c is bounded, then the operator L obeysthe Garding inequality:

(Lu, u) =∫�

Lu u ≥ C0‖∇u‖2 − C‖u‖2, ∀u ∈ H10 (�).

Let us have two solutions 〈u1, f1〉 and 〈u2, f2〉 to (1) and (2). Set u = u1 − u2 andf = f1 − f2. Then, we see that u(x, 0) = 0 and u(x, T ) = 0. For ϕ ∈ H 1

0 (�) we get thefollowing weak formulation

(ut , ϕ)+ (A∇u,∇ϕ)+ (cu, ϕ) = ( f, ϕ)+(∫ t

0[g(u1(s))− g(u2(s))]ds, ϕ

). (4)

We have to show that u = 0 and f = 0. First, we show that u = 0 and later that f = 0. Wehave to get rid of f at the first stage. The main idea is very simple, namely

∫ T

0f (x)ut (x, t)dt = f (x)u(x, T )− f (x)u(x, 0) = 0.

Thus, we set ϕ = ut (x, t) into (4) and integrate in time over (0, T ) to get

∫ T

0‖ut‖2 dt +

∫ T

0(A∇u,∇ut ) dt +

∫ T

0(cu, ut )dt

=∫ T

0

(∫ t

0[g(u1)− g(u2)], ut (t)

)dt.

Dow

nloa

ded

by [

Port

land

Sta

te U

nive

rsity

] at

13:

54 1

5 O

ctob

er 2

014

Inverse Problems in Science and Engineering 5

The coefficients A = Atr and c are time-independent! Simple calculations give∫ T

0(A∇u,∇ut )dt =

∫ T

0

(A

12 A

12 ∇u,∇ut

)dt =

∫ T

0

(A

12 ∇u, A

12 ∇ut

)dt

=∫ T

0

(A

12 ∇u, ∂t

(A

12 ∇u

))dt = 1

2

∫ T

0∂t

∥∥∥A12 ∇u

∥∥∥2dt = 0

because of u(x, 0) = 0 and u(x, T ) = 0.Analogously, we may write

∫ T

0(cu, ut )dt = 1

2

∫ T

0

(c, ∂t u

2)

dt = 0.

Integration by parts yields∫ T

0

(∫ t

0[g(u1)− g(u2)], ut (t)

)dt =

(∫ t

0[g(u1)− g(u2)], u(t)

)∣∣∣∣T

0

−∫ T

0(g(u1)− g(u2), u1 − u2)dt

= −∫ T

0(g(u1)− g(u2), u1 − u2)dt.

Collecting all the results above, we arrive at∫ T

0‖ut‖2dt +

∫ T

0(g(u1)− g(u2), u1 − u2)dt = 0.

Due to the fact that g is monotonically increasing, we deduce that∫ T

0‖ut‖2dt = 0 =⇒ ut = 0 a.e. in �× [0, T ]

and u is a constant in time. Therefore,

u(x, 0) = 0 =⇒ u(x, t) = 0.

Putting this information into (4), we obtain

( f, ϕ) = 0 ∀ϕ ∈ H10 (�),

from which we conclude that f = 0. �

Next theorem generalizes Theorem 2.1 to a transient differential operator L .

Theorem 2.2 Consider a linear differential operator

Lu(x, t) = ∇ · (−A(x, t)∇u(x, t))+ c(x, t)u(x, t)

with Atr = A and

ξ tr · Aξ =n∑

i, j=1

ai, j (x, t)ξiξ j ≥ C |ξ |2, ∀ξ ∈ Rn (5)

Dow

nloa

ded

by [

Port

land

Sta

te U

nive

rsity

] at

13:

54 1

5 O

ctob

er 2

014

6 M. Slodicka

All the coefficients appearing in the operator L are assumed to be bounded for any (x, t).Moreover, assume that

ξ tr · ∂t

(A

12

)A

12 ξ ≤ 0, ∀ξ ∈ R

n and ∂t c(·, t) ≤ 0, ∀t ∈ [0, T ].

Let u0, ψT ∈ L2(�) and g′ ≥ 0. Then, there exists at most one spacewise-dependent sourcef ∈ L2(�) such that (1) together with the condition (2) hold.

Proof The proof follows the same line as in Theorem 2.1. The main difference is inhandling of the time-dependent coefficients. Assume that we have two solutions 〈u1, f1〉and 〈u2, f2〉 to (1) and (2). We denote u = u1 − u2 and f = f1 − f2 and recall thatu(x, 0) = 0 and u(x, T ) = 0. We have to show that u = 0 and f = 0.

For ϕ ∈ H10 (�), we get (4). Setting ϕ = ut (t) and integrating in time over (0, T ) to see

that ∫ T

0‖ut‖2dt +

∫ T

0(A∇u,∇ut )dt +

∫ T

0(cu, ut )dt

=∫ T

0

(∫ t

0[g(u1(s))− g(u2(s))] ds, ut (t)

)dt.

Monotonicity of g which is found in the proof of Theorem 2.1 gives∫ T

0

(∫ t

0[g(u1(s))− g(u2(s))] ds, ut (t)

)dt ≤ 0.

Integrating by parts and using u(x, 0) = 0 and u(x, T ) = 0, we have∫ T

0(cu, ut )dt = 1

2

∫ T

0

(c, ∂t u

2)

dt = −1

2

∫ T

0

(∂t c, u2

)dt ≥ 0

and∫ T

0(A∇u,∇ut )dt =

∫ T

0

(A

12 A

12 ∇u,∇ut

)dt

=∫ T

0

(A

12 ∇u, A

12 ∇ut

)dt

=∫ T

0

(A

12 ∇u, ∂t

(A

12 ∇u

))dt −

∫ T

0

(A

12 ∇u, ∂t

(A

12

)∇u

)dt

= 1

2

∫ T

0∂t

∥∥∥A12 ∇u

∥∥∥2dt −

∫ T

0

(A

12 ∇u, ∂t

(A

12

)∇u

)dt

= −∫ T

0

(∂t

(A

12

)A

12 ∇u,∇u

)dt

≥ 0.

Collecting the results above, we deduce that∫ T

0‖ut‖2dt = 0 =⇒ ut = 0

andu(x, 0) = 0 =⇒ u(x, t) = 0.

Now, returning back to (4), we conclude that f = 0. �

Dow

nloa

ded

by [

Port

land

Sta

te U

nive

rsity

] at

13:

54 1

5 O

ctob

er 2

014

Inverse Problems in Science and Engineering 7

The matrix A obeys (5), hence it is positive definite with respect to both space and time-dependent entries. Thus, there exists a unique positive-definite square root A

12 . We have to

be careful because generally

∂t A = ∂t

(A

12 A

12

)= ∂t

(A

12

)A

12 + A

12 ∂t

(A

12

) = 2∂t

(A

12

)A

12 .

This equality holds true for e.g. diagonal matrices. More generally, ∂t A = 2∂t

(A

12

)A

12 iff

the matrix commutes with its derivative which is equivalent to A(x, t)A(x, s) = A(x, s)A(x, t) for all t, s and x , see [13,14].

If we want to express conditions of Theorem 2.2 for scalar coefficients, we willhave (5) and both coefficients A(x, t) and c(x, t) must decrease in time.

3. 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 somenon-uniqueness examples, cf. [5, Section 7.2] In this section, we address the uniquenessresults for the IPs (3) and (2). We will show that the damping term g(ut ) helps to establishthe uniqueness results for hyperbolic case. We will use a similar proof technique as in theparabolic event. The first theorem deals with a steady-state differential operator L.

Theorem 3.1 Consider a linear differential operator

Lu(x, t) = ∇ · (−A(x)∇u(x, t))+ c(x)u(x, t),

with bounded (discontinuous) coefficients obeying A(x) = Atr (x) and

ξ tr · Aξ =n∑

i, j=1

ai, jξiξ j ≥ C |ξ |2 ∀ξ ∈ Rn .

Let u0, ψT ∈ L2(�) and g′ > 0. Then, there exists at most one spacewise-dependentsource f ∈ L2(�) such that (3) together with the condition (2) holds.

Proof Suppose that we have two solutions 〈u1, f1〉 and 〈u2, f2〉 to (3) and (2). Setu = u1 − u2 and f = f1 − f2. Please note that u(x, 0) = 0, ut (x, 0) = 0 and u(x, T ) = 0.We have to show that u = 0 and f = 0.

For ϕ ∈ H10 (�), we get

(utt , ϕ)+ (g(∂t u1)− g(∂t u2), ϕ)+ (A∇u,∇ϕ)+ (cu, ϕ) = ( f, ϕ) . (6)

We put ϕ = ut (x, t) and integrate in time over (0, T ) to get

1

2‖ut (T )‖2 +

∫ T

0(g(∂t u1)− g(∂t u2), ut )dt +

∫ T

0(A∇u,∇ut )dt +

∫ T

0(cu, ut )dt = 0.

(7)

Dow

nloa

ded

by [

Port

land

Sta

te U

nive

rsity

] at

13:

54 1

5 O

ctob

er 2

014

8 M. Slodicka

The last two terms on the left-hand side vanish due to the same reasoning as in Theorem2.1. Therefore, we may write

1

2‖ut (T )‖2 +

∫ T

0(g(∂t u1)− g(∂t u2), ∂t u1 − ∂t u2)dt = 0.

Here, we can see why we do need the damping term. Without it, we would have that‖ut (T )‖ = 0, which gives no guarantee that u = 0.

Employing the fact that the function g is strictly monotonically increasing, we get thatut = 0, i.e. u is constant in time. Therefore,

u(x, 0) = 0 =⇒ u(x, t) = 0.

Returning back to (6), we conclude that f = 0. �

Next theorem addresses the uniqueness result for a transient operator L in a hyperbolicproblem.

Theorem 3.2 Consider a linear differential operator

Lu(x, t) = ∇ · (−A(x, t)∇u(x, t))+ c(x, t)u(x, t)

with Atr = A and

ξ tr · Aξ =n∑

i, j=1

ai, j (x, t)ξiξ j ≥ C |ξ |2

All the coefficients appearing in the operator L are assumed to be bounded for any (x,t).Moreover, assume that

ξ tr · ∂t

(A

12

)A

12 ξ ≤ 0, ∀ξ ∈ R

n and ∂t c(·, t) ≤ 0 ∀t ∈ [0, T ].Let u0, ψT ∈ L2(�) and g′ > 0. Then, there exists at most one spacewise-dependentsource f ∈ L2(�) such that (3) together with the condition (2) hold.

Proof Let us have two solutions 〈u1, f1〉 and 〈u2, f2〉 to (3) and (2). Put u = u1 − u2 andf = f1 − f2. Please note that u(x, 0) = 0, ut (x, 0) = 0 and u(x, T ) = 0. We will showthat u = 0 and f = 0.

The relation (7) holds true. The last two terms on the left-hand side of (7) arenon-negative, due to the same reasoning as in Theorem 2.2. Thus, we see that

1

2‖ut (T )‖2 +

∫ T

0(g(∂t u1)− g(∂t u2), ∂t u1 − ∂t u2)dt ≤ 0.

According to the strict monotonicity of the function g, we obtain that ut = 0, i.e. u isconstant in time. Looking at u(x, 0) = 0 we get u(x, t) = 0, Inspecting the relation (6), weconclude that f = 0. �

4. Conclusions

In this work, we have studied the uniqueness of two inverse source problems: parabolic(1) and (2); and hyperbolic (3) and (2). We have showed that under reasonable physical

Dow

nloa

ded

by [

Port

land

Sta

te U

nive

rsity

] at

13:

54 1

5 O

ctob

er 2

014

Inverse Problems in Science and Engineering 9

conditions on the data functions, the final over-determination is sufficient for the uniquenessof a space dependent source.

AcknowledgementThis work was supported by the BOF/GOA-project No. 01G006B7 of Ghent University.

Note1. The symbol Atr denotes the transpose of A.

References

[1] Delleur JW. The handbook of groundwater engineering. Heidelberg: Springer Verlag GmbH &Co. KG; 1999. ISBN 3-540-64745-7.

[2] Cannon JR. Determination of an unknown heat source from overspecified boundary data. SIAMJ. Numer. Anal. 1968;5:275–286.

[3] Farcas A, Lesnic D. The boundary-element method for the determination of a heat sourcedependent on one variable. J. Eng. Math. 2006;54:375–388.

[4] Hasanov A. Simultaneous determination of source terms in a linear parabolic problem from thefinal overdetermination: weak solution approach. J. Math. Anal. Appl. 2007;330:766–779.

[5] Isakov V. Inverse source problems. Mathematical Surveys and Monographs, 34. Providence(RI): American Mathematical Society (AMS); 1990. xiv, p. 193.

[6] Johansson T, Lesnic D. A variational method for identifying a spacewise dependent heat source.IMA J. Appl. Math. 2007;72:748–760.

[7] Prilepko AI, Solov’ev VV. Solvability theorems and Rothe’s method for inverse problems for aparabolic equation. Differ. Equ. 1988;23:1341–1349.

[8] Rundell W. Determination of an unknown non-homogeneous term in a linear partial differentialequation from overspecified boundary data. Appl. Anal. 1980;10:231–242.

[9] Solov’ev VV. Solvability of the inverse problems of finding a source, using overdeterminationon the upper base for a parabolic equation. Differ. Equ. 1990;25:1114–1119.

[10] Johansson T, Lesnic D. Determination of a spacewise dependent heat source. J. Comput. Appl.Math. 2007;209:66–80.

[11] D’haeyer S, Johansson BT, Slodicka M. Reconstruction of a spacewise dependent heat source in atime-dependent heat diffusion process. IMAJ.Appl. Math. in press. doi: 10.1093/imamat/hxs038

[12] Pazy A. Semigroups of linear operators and applications to partial differential equations. Vol.44, Applied Mathematical Sciences. New York (NY): Springer; 1983.

[13] Bogdanov YS, Chebotarev GN. Über Matrizen, die mit ihrerAbleitung kommutieren. Izv. Vyssh.Uchebn. Zaved. Mat. 1959;4:27–37. ISSN:0021–3446.

[14] Martin JFP. Some results on matrices which commute with their derivatives. SIAM J. Appl.Math. 1967;15:1171–1183.

Dow

nloa

ded

by [

Port

land

Sta

te U

nive

rsity

] at

13:

54 1

5 O

ctob

er 2

014