Reconstructing The Moore-Gibson-Thompson Equation

from one observation of the solution at a single point, i.e. u(b, t) for t > 0. Here Ω ⊂ Rd(d ≥ 1) is a bounded domain with a smooth boundary, say C2. The model in (1) describes the nonlinear propagation of acoustic waves (such as high frequency ultrasound waves) in a viscoelastic material, whose behavior and e ect are governed by the convolution "memory" term ∫ t 0 g(t − s)∆u(s)ds. We refer to [6] for the derivation of the MGT equation and the list of references therein. These equations have been the focus of several recent research due to their wide range of applications in themedical and industrial elds of high intensity ultrasoundwaves.


Introduction
The objective of this note is to reconstruct the positive constants α, β, γ and the relaxation function g of the following Moore-Gibson-Thompson (MGT) equation from one observation of the solution at a single point, i.e. u(b, t) for t > .
Here Ω ⊂ R d (d ≥ ) is a bounded domain with a smooth boundary, say C . The model in (1) describes the nonlinear propagation of acoustic waves (such as high frequency ultrasound waves) in a viscoelastic material, whose behavior and e ect are governed by the convolution "memory" term t g(t − s)∆u(s)ds. We refer to [6] for the derivation of the MGT equation and the list of references therein. These equations have been the focus of several recent research due to their wide range of applications in the medical and industrial elds of high intensity ultrasound waves.
The direct problem of (1) has received a lot attention. First Lasiecka and Wang established its wellposedness and showed exponential decay of the energy [6] and more general decay results followed, for example [7] for the subcritical case (i.e. α − γ β > ). These have been made more precise and optimal decay rates was obtained by Liu et al. [11]. For the critical case (α − γ β = ) Dell'Oro et al. [4] proved a polynomial decay rate under more regular initial data.
The inverse problem for (1) with g = but α being a function of x was treated by Liu and Triggiani [9,10], who proved uniqueness of the coe cient α(x) − γ β and also showed stability by Carleman estimates. However no reconstruction procedure were given. We recall that the well-posedness in this case was proven  [5] and Marchand et al. [12].
Our aim in this note is to give an explicit reconstruction procedure of the coe cients α, β, γ and the kernel function g found in (1) from a single observation of the solution at one point. To achieve this, we use the idea developed in [1], taking into consideration the nature of MGT equation under the subcritical case, and establish a simple computational method to reconstruct the sought parameters. Vespri et al. [3] considered a parabolic functional equation, where they showed that two measurements are enough to yield uniqueness.

Preliminaries
We now state su cient assumptions that ensure global existence of the solution on (1).
As the unique global solution u(., t) ∈ L (Ω), means that it can be written as where The initial conditions translate into, for all n ≥ ,

Multiply (1) by φn(x) and integrate over Ω to get
which reduces to the functional equation, thanks to (3) and (5), with initial conditions Since all sought parameters appear in (6), it is easy to extract some from it. For example since λ = , we get from (6) c as long as c ′′ is not trivial, i.e. u is not orthogonal to φn. Hence by taking the limit, since the solution is C [ , ∞).
We now show that a non trivial c contains all the information required to reconstruct the sought coe cients. From (6) and by taking the limit as t → + , we deduce that c satis es, Similarly di erentiating (6)and taking the limit as t → + we deduce the above system reduces to a lower triangular matrix equation, which is easily solved by the Gaussian method Since the determinant of A is 1, the system above has the unique solution (α, βλ , γλ ) T .

Remark:
The solution u(x, t), is known to be globally C in time, and so (1) implies that u ttt is also locally continuous it t. Thus its Fourier coe cients c ( ) n (t), are well de ned and continuous in time. Therefore (6) holds in the classical sense and as the integral term is C , we obtain that c ( ) n ∈ C , i.e. cn ∈ C .
Once we have obtained the values (α, β, γ), we apply the Laplace transform,ĝ(s) = L(g)(s), to (6) cn(s) s + αs + βλn s + γλn = c ′′ n ( ) + (s + α)c ′ n ( ) + (s + αs + βλn)cn( ) + λnĝ(s)ĉn(s), (8) and so, for n ≥ , And if we use the boundary conditions in (7), the previous equation reduces tô Note that it follows from condition (A ) that g ∈ L ( , ∞), since Next recall that the Laplace transform is a bounded and invertible operator in L ( , ∞), and so not onlŷ g ∈ L ( , ∞), it is also in the range of the Laplace transform, [2]. Thus its inverse Laplace transform exists and would yield back g. It remains to see that we can observe c from the solution u(x, t), see (7) Proof. With the given initial condition, the unique solution can be observed at x = b is which exists for t ≥ . It remains to avoid nodal curves, so the observation is not trivial.