The problem of determining source term in a kinetic equation in an unbounded domain

: In this paper, we deal with an inverse problem of determining the source function in a kinetic equation that is considered in an unbounded domain with Cauchy data. We prove the uniqueness of the solution of an inverse problem by means of a pointwise Carleman estimate. In recent years, kinetic equations have occurred in a variety of important fields and applications, such as aerospace engineering, semi-conductor technology, nuclear engineering


Introduction
The kinetic equation has become a powerful mathematical tool to describe the dynamics of many interacting particle systems, such as electrons, ions, stars, and galaxy or galactic aggregations.Since the nineteenth century, when Boltzmann formalized the concepts of kinetic equations, they have been used to model a variety of phenomena in different fields, such as rarefied gas dynamics, plasma physics, astrophysics, and socioeconomics.Particularly, in the life and social sciences, kinetic theory is used to model the dynamics of a large number of individuals, for example biological cells, animal flocks, pedestrians, or traders in large economic markets [1][2][3][4][5].Moreover, it has applications in aerospace engineering [6], semi-conductor technology [7], nuclear engineering [8], chemotaxis, and immunology [9].
In this article, we consider the kinetic equation ∂ t u(x, p, t) + n j=1 p j ∂ x j u(x, p, t) = f (t)g(x, p), (1.1) under the conditions: u(x, p, t)| x 1 ≤0 = u 1 (x, p, t), ( u(x, p, 0) = u 0 (x, p) (1.3) in the domain Ω = {(x, p, t) : x 1 > 0, x ∈ D, p ∈ R n , t ∈ R}, where D ⊂ R n , x = (x 1 , x) ∈ R n , x = (x 2 , ..., x n ) ∈ R n−1 .Throughout the paper, we used the following notations: In applications, u represents the number (or the mass) of particles in the unit volume element of the phase space in the neighbourhood of the point (x, p, t), where x, p and t are the space, momentum and time variables, respectively.

Making the change of variables
x − y = ξ, x + y = η and introducing the function ) such that z = 0 for x 1 − y 1 ≥ ξ 0 , and the Fourier transform of z with respect to t is finite.We assume that f ∈ C(R) ∩ H 2 (R).
We investigate uniqueness of solution of Problem 1.For the proof, we shall use the Fredholm alternative theorem, so we consider the related homogeneous problem.
Then, by the condition z(ξ, η, 0) = z 0 (ξ, η) = 0 from (1.8), we can write so we have The solvability of various inverse problems for kinetic equations was studied by Amirov [10] and Anikonov [11] on a bounded domain, where the problem is reduced to a Dirichlet problem for a thirdorder partial differential equation.See also [12] for an inverse problem for the transport equation, where the equation is reduced to a second-order differential equation with respect to the time variable t.Moreover, in [10,11], the problem of determining the potential in a quantum kinetic equation was discussed in an unbounded domain.Numerical algorithms to obtain the approximate solutions of some inverse problems were developed in [13,14].The main difference between the current work and the existing works is that here the problem of finding the source function is considered in an unbounded domain with the Cauchy data which is given on a planar part of the boundary.
The main result of this paper is given below: Theorem 1.1.Let ∂ η 1 a > 0 and f (0) 0.Then, Problem 1 has at most one solution (z, h) such that z ∈ Z and h ∈ L 1 (R 2n ).
We apply the Fourier transform to Eq (1.10) and for condition (1.9) with respect to (ξ, t), we get We write ẑ = z 1 + iz 2 and f = f 11), and so we obtain the following system of equations: where By (1.12), we have Thus, we shall show that this homogeneous problem has only the trivial solution.In the proof of Theorem 1.1, the main tool is a pointwise Carleman estimate, which will be presented in the next section.The proof of Theorem 1.1 will be given in the last section.
To establish a Carleman estimate we first present an auxiliary lemma.Lemma 2.1.The following equality holds for any function z k ∈ Z : where Proof of Lemma 2.1.Since P 0 z k = ∆ η + a −1 |s| 2 z k , we can write and using the equalities we obtain Proposition 2.1.Under the assumptions of Theorem 1.1, the following inequality is valid for all z k ∈ Z: where λ and ν are large parameters to be specified in the proof below.Moreover, d 2 (z k ) denotes the sum of divergence terms, which will be given explicitly later.
Proof of Proposition 2.1.We first define a new function w = φz k , and then we can write Now, we calculate the terms K j , 1 ≤ j ≤ 5 as follows: (2.12) Similarly, we can calculate (2.13) Next, we have As for the fourth term, Moreover, we have where we choose If we replace the functions w with z k , then from (2.12)-(2.16),we can write where Here, we can choose λ ≥ λ 0 such that and so from (2.17) we obtain Finally, multiplying inequality (2.4) by −2nλv and summing with (2.18), we get By choosing v ≥ v 1 , we obtain (2.10).Thus, the proof of Proposition 2.1 is complete.

Proof of Theorem 1.1
First, for the right-hand side of (1.13), we can write In (3.1), we used the following expressions: and Using (2.10), for k = 1, 2 we have If we multiply (3.3) by (1 + ω 2 ) 2 and integrate with respect to the parameters ω over R, we have where In inequality (3.4), we can choose the big parameter λ, such that all terms on the right-hand side can be absorbed into the left-hand side.Then, we have where div(ẑ) = (3.6) Integrating inequality (3.6) over Ω 0 and passing to the limit as λ → ∞, we have which means that ẑ = 0. Therefore, we conclude that z = 0. Finally, by (1.8) we obtain h = 0, which completes the proof of Theorem 1.1.

Conclusions
In this study, we considered an inverse problem for the kinetic equation in an unbounded domain.We reduced the equation to a second-order partial differential equation and proved the uniqueness of the solution of the problem by using the Carleman estimate.The method used in this paper can be applied to a variety of equations, including some first and second-order partial differential equations in mathematical physics, such as transport, ultrahyperbolic, and ultrahyperbolic Schrödinger equations.By similar arguments, a stability estimate can be obtained in an unbounded domain.

Use of AI tools declaration
The author declares that she did not use Artificial Intelligence (AI) tools in the creation of this article.