# Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data

###### Abstract.

We consider the inverse problem of determining a time-dependent damping coefficient and a time-dependent potential , appearing in the wave equation in , with and a bounded domain of , , from partial observations of the solutions on . More precisely, we look for observations on that allow to determine uniquely a large class of time-dependent damping coefficients and time-dependent potentials without involving an important set of data. Assuming that is known on , we prove global unique determination of , with , and from partial observations on . Our problem is related to the determination of nonlinear terms appearing in nonlinear wave equations.

Keywords: Inverse problems, wave equation, time-dependent damping coefficient, time-dependent potential, uniqueness, Carleman estimates, partial data.

Mathematics subject classification 2010 : 35R30, 35L05.

## 1. Introduction

### 1.1. Statement of the problem

Let be a bounded domain of , , and fix , with . We consider the wave equation

(1.1) |

where the damping coefficient and the potential are real valued. In the present paper we seek unique determination of both and from observations of solutions of (1.1) on .

Let be the outward unit normal vector to , the normal derivative and from now on let and be the differential operators , . It has been proved by [41], that for the data

(1.2) |

determines uniquely a time-independent potential when . The result of [41] has been extended to the recovery of a time-independent damping coefficient by [24]. Contrary to time-independent coefficients, due to domain of dependence arguments there is no hope to recover the restriction of a general time-dependent coefficient to the set

from the data (see [30, Subsection 1.1]). On the other hand, according to [23, Theorem 4.2], for , the extended set of data

(1.3) |

determines uniquely a time-dependent potential . Taking into account the obstruction to the unique determination from the data and the result of [23], the goal of the present paper is to determine a general time-dependent damping coefficient and a general time-dependent potential from partial knowledge of the important set of data .

### 1.2. Physical and mathematical motivations

In practice, our inverse problem consists of determining physical properties such as the time evolving damping force and the density of an inhomogeneous medium by probing it with disturbances generated on the boundary and at initial time and by measuring the response to these disturbances on some parts of the boundary and at the end of the experiment. The goal is to determine the functions and which measure the damping force and the property of the medium. The determination of such a time-dependent coefficients can also correspond to the recovery of some time evolving properties that can not be modeled by time-independent coefficients.

As mentioned in [30, 31], following the strategy set in [25] for parabolic equations, the recovery of nonlinear terms, appearing in some suitable nonlinear wave equations, from observations on can be reduced to the determination of time-dependent coefficients, with weak regularity, appearing in a linear wave equation. In this context, the regularity of the time-dependent coefficients depends on the regularity of the solutions of the nonlinear equations. Thus, for this application of our problem it is important to weaken as much as possible the regularity of the admissible time-dependent coefficients.

### 1.3. State of the art

The determination of coefficients for hyperbolic equations from boundary measurements has attracted many attention in recent years. Many authors considered the recovery of time-independent potentials from observations given by the set defined by (1.2) for . In [41], the authors proved that, for , determines uniquely a time-independent potential . The uniqueness by partial boundary observations has been considered in [16]. We also precise that the stability issue for this problem has been studied by [4, 5, 29, 38, 46, 47].

Some authors treated also the recovery of both time-independent damping coefficients and potentials from boundary measurements. In [24], Isakov extended the result of [41], to the recovery of both damping coefficients and potentials from the data . For , [26] proved stable recovery of the restriction of both time-independent damping coefficients and potentials on the intersection of the domain and a half-space from measurements on the intersection of the boundary of the domain and the same half-space. Following the strategy set by [8], [7, 36, 37] proved uniqueness and stability in the recovery of both damping coefficients and potentials from a single boundary measurements. In some recent work, [1] proved a log-type stability estimate in the recovery of time-independent damping coefficients and potentials appearing in a dissipative wave equation from the initial boundary map.

All the above mentioned results are concerned with time-independent coefficients. Several authors considered the problem of determining time-dependent coefficients for hyperbolic equations. In [45], Stefanov proved the recovery of a time-dependent potential appearing in the wave equation from the knowledge of scattering data by using some properties of the light-ray transform. In [42], Ramm and Sjöstrand considered the determination of a time-dependent potential from the data of forward solutions of (1.1) with on the infinite time-space cylindrical domain instead of ( instead of ). Rakesh and Ramm [40] treated this problem at finite time on , with , and they determined uniquely restricted to some subset of from with . Isakov established in [23, Theorem 4.2] unique determination of general time-dependent potentials on the whole domain from the extended data given by (1.3) with . Using a result of unique continuation borrowed from [49], Eskin [17] proved unique recovery of time-dependent coefficients analytic with respect to the time variable from partial knowledge of the data . Salazar [43] extended the result of [42] to more general coefficients. Moreover, [50] stated stability in the recovery of X-ray transforms of time-dependent potentials on a manifold and [6] proved log-type stability in the determination of time-dependent potentials from the data considered by [23] and [40]. We mention also the recent work of [2] where the authors have extended the results of [6] to the recovery of both time-dependent damping coefficients and potentials. In [30, 31], the author considered both uniqueness and stability in the recovery of some general time dependent potential from (roughly speaking) half of the data considered by [23]. To our best knowledge the results of [30, 31] are stated with the weakest conditions so far that guaranty determination of a general time dependent potential, appearing in a wave equation, at finite time. We also mention that [10, 11, 12, 19, 20] examined the determination of time-dependent coefficients for fractional diffusion, parabolic and Schrödinger equations and proved stability estimate for these problems.

### 1.4. Main result

To state our main result, we first introduce some intermediate tools and notations. For all we introduce the -shadowed and -illuminated faces

of . Here, for all , corresponds to the scalar product in defined by

We define also the parts of the lateral boundary given by . We fix and we consider with a closed neighborhood of in .

The main purpose of this paper is to prove the unique global determination of time-dependent and real valued damping coefficient and from the data

We refer to Section 2 for the definition of this set. Our main result can be stated as follows.

###### Theorem 1.1.

Let and let with . Assume that

(1.4) |

Then, the condition

(1.5) |

implies that and .

To our best knowledge this paper is the first treating uniqueness in the recovery of time-dependent damping coefficients. Moreover, it seems that with [17, 18, 43] this paper is the first considering recovery of time-dependent coefficients of order one and it appears that this work is the first treating this problem for general coefficients at finite time ([17, 18] proved recovery of coefficients analytic with respect to the time variable , [43] considered the problem for all time ). We point out that our uniqueness result is stated for general coefficients with observations close to the one considered by [30, 31], where recovery of time-dependent potentials is proved with conditions that seems to be one of the weakest so far for a general class of time-dependent coefficients. Indeed, the only difference between [30, 31] and the present paper comes from the restriction on the Dirichlet boundary condition and the initial value ([30, 31] consider Dirichlet boundary condition supported on a neighborhood of the -shadowed face and vanishing at , where here we do not make restriction on the support of the Dirichlet boundary condition and at ). We also mention that in contrast to [17], we do not apply results of unique continuation that require the analyticity with respect to the time variable of the coefficients.

Let us observe that even for large, according to the obstruction to uniqueness given by domain of dependence arguments (see [30, Subsection 1.1]), there is no hope to remove all the information on and for the global recovery of general time-dependent coefficients. Therefore, for our problem the extra information on and , of solutions of (1.1), can not be completely removed.

The main tools in our analysis are Carleman estimates with linear weight and geometric optics (GO in short) solutions suitably designed for our inverse problem. In a similar way to [5, 30, 31], we use GO solutions taking the form of exponentially growing and exponentially decaying solutions in accordance with our Carleman estimate in order to both recover the coefficients and restrict the observations. Our GO solutions differ from the one of [16, 17, 24, 41, 42, 43] and, combined with our Carleman estimate, they make it possible to prove global recovery of time-dependent coefficients from partial knowledge of the set without using additional smoothness or geometrical assumptions. Even if this strategy is inspired by [5, 30, 31] (see also [9, 28] for the original idea in the case of elliptic equations), due to the presence of a variable coefficient of order one in (1.1), our approach differs from [5, 30, 31] in many aspects. Indeed, to prove our Carleman estimate we perturb the linear weight and we prove this estimate by using a convexity argument that allows us to absorb the damping coefficient. Moreover, in contrast to [30, 31] our GO are designed for the recovery of the damping coefficient and we can not construct them by applying properties of solutions of PDEs with constant coefficients. We remedy to this by considering Carleman estimates in Sobolev space of negative order and by using these estimates to build our GO solutions. This construction is inspired by the one used in [15, 28] for the recovery of Schrödinger operators from partial boundary measurements.

Note that condition (1.4) is meaningful for damping coefficients that actually depend on the time variable (, ). Indeed, for time-independent damping coefficients , , (1.4) implies that . However, by modifying the form of the principal part of the GO given in Section 4 in accordance with [24], for we believe that we can restrict condition (1.4) to the knowledge of time-independent damping coefficients on ( on instead of (1.4)). In order to avoid the inadequate expense of the size of the paper we will not treat that case.

We believe that, with some suitable modifications, the approach developed in the present paper can be used for proving recovery of more general time-dependent coefficients of order one including a magnetic field associated to a time-dependent magnetic potential.

### 1.5. Outline

This paper is organized as follows. In Section 2 we introduce some tools and we define the set of data . In Section 3, we prove our first Carleman estimate which will play an important role in our analysis. In Section 4 we extend and mollify the damping coefficient and we introduce the principal part of our GO solutions. In Section 5, we derive a Carleman estimate in Sobolev space of negative order. Then, using this estimate, we build suitable GO solutions associated to (1.1). Finally in Section 6, we combine the GO solutions of Section 5 with the Carleman estimate of Section 3 to prove Theorem 1.1.

## 2. Preliminary results

In the present section we define the set of data and we recall some properties of the solutions of (1.1) for any . For this purpose, in a similar way to [30], we will introduce some preliminary tools. We define the space

with the norm

We consider also the space

and topologize it as a closed subset of . Indeed, let be a sequence lying in that converge to in . Then, converge to in the sense of and in the same way converge to in the sense of . Now using the fact that for all , we deduce that . This proves that and that is a closed subspace of .

In view of [30, Proposition 4], the maps

can be extended continuously to , . Here for all we set

where

Therefore, we can introduce

Following [30] (see also [9, 13, 14, 39] in the case of elliptic equations), in order to define an appropriate topology on we consider the restriction of to the space . Indeed, by repeating the arguments used in [30, Proposition 1], one can check that the restriction of to is one to one and onto. Thus, we can use to define the norm of by

with considered as its restriction to . Let us introduce the IBVP

(2.1) |

We are now in position to state existence and uniqueness of solutions of this IBVP for .

###### Proposition 2.1.

Let , and . Then, the IBVP (2.1) admits a unique weak solution satisfying

(2.2) |

and the boundary operator is a bounded operator from to

.

###### Proof.

We split into two terms where solves

(2.3) |

Since , from the theory developed in [34, Chapter 3, Section 8], one can check that the IBVP (2.3) admits a unique solution satisfying

(2.4) | ||||

Therefore, is the unique solution of (2.1) and estimate (2.4) implies (2.2). Now let us show the last part of the proposition. For this purpose fix and consider the solution of (2.1). Note first that . Thus, and , with

Combining this with (2.2) we deduce that is a bounded operator from to .∎

From now on we consider the set to be the graph of the boundary operator given by

## 3. Carleman estimates

This section will be devoted to the proof of a Carleman estimate with linear weight associated with (1.1) which will be one of the main tools in our analysis. More precisely, we will consider the following.

###### Theorem 3.1.

Let , and . If satisfies the condition

(3.1) |

then there exists depending only on , and such that the estimate

(3.2) |

holds true for with depending only on , and .

For , the Carleman estimate (3.2) has already been established in [30, Theorem 2] by applying some results of [5]. In contrast to the equation without the damping coefficient (), due to the presence of a variable coefficient of order one, we can not derive (3.2) from [30, Theorem 2]. In order to establish this Carleman estimate, in a similar way to [15, 28], we need to perturb our linear weight in order to absorb the damping coefficient. More precisely, we introduce a new parameter independent of that will be precised later, we consider, for , the perturbed weight

(3.3) |

and we define

Proof of Theorem 3.1. We fix such that

In the remaining part of this proof we will systematically omit the subscripts in and without lost of generality we will assume that is real valued. Our first goal will be to establish for sufficiently large and for suitable fixed value of depending on , and , the estimate

(3.4) | ||||

Then, we will deduce (3.2). We decompose into three terms

with

Recall that

We have

For we obtain

(3.5) |

We have also

and using the fact that we get

(3.6) |

In a similar way, we find

and using the formula

we obtain

(3.7) |

Moreover, in a similar way to [9, Lemma 2.1], we get

(3.8) |

Now note that

Then, we have

Choosing such that

we get

(3.9) |

Combining estimates (3.5)-(3.9), we obtain

(3.10) | ||||

On the other hand, we have

Fixing and , we deduce (3.4) from (3.10). Armed with (3.4), we will complete the proof of the Carleman estimate (3.2). Note that, condition (3.1) implies and we deduce that

(3.11) |

Moreover, using the fact that

we obtain

Combining these estimates with (3.4), (3.11), we get

(3.12) |

From this last estimate we deduce (3.2). ∎

Now that our Carleman estimate is proved we will extend it into Sobolev space of negative order and apply it to construct our GO solutions. But first let us define the principal part of our GO solutions that will allow us to recover the damping coefficient. For this purpose we need some suitable approximation of our damping coefficients.

## 4. Approximation of the damping coefficients

Let us first remark that the GO solutions that we use for the recovery of the damping coefficients should depend explicitly on the damping coefficients. In order to avoid additional smoothness assumptions on the class of admissible coefficients, in a similar way to [32, 44], we consider GO solutions depending on some smooth approximation of the damping coefficients instead of the damping coefficients themselves. The main purpose of this section is to define our choice for the smooth approximation of the damping coefficients and to introduce the part of our GO solutions that will be used for the recovery of the damping coefficient. From now on we fix the coefficients , with , and . Moreover, we assume that

(4.1) |

For all we define . Then, according to [48, Theorem 5, page 181], fixing