An Inverse Problem for a Semilinear Wave Equation

For the equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{u}_{{tt}}} - \Delta u - f(x,u) = 0, (x,t) \in {{\mathbb{R}}^{4}},$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x,u)$$\end{document} is a smooth function of its variables and is compact in x, the inverse problem of recovering this function from given information on solutions of Cauchy problems for the differential equation is studied. Plane waves with a strong front that propagate in a homogeneous medium in the direction of the unit vector ν and then impinge on an inhomogeneity localized inside some ball B(R) are considered. It is supposed that the solutions of the Cauchy problems can be measured on the boundary of this ball for all ν at times close to the arriving time of the front. The forward Cauchy problem is studied, and the existence of a unique bounded solution in a neighborhood of a characteristic wedge is stated. An amplitude formula for the derivative of the solution with respect to t on the front of the wave is derived. It is demonstrated that the solution of the inverse problem reduces to a series of X-ray tomography problems.

− Δ − ∈ R 4 ( , ) = 0, ( , ) , Consider the Cauchy problem (1) where is a smooth function of x and u that is compactly supported with respect to and g(t) has a discontinuity at t = 0 such that and for Additionally, we assume that the structure of g(t) is such that for , where while, for , g(t) is arbitrary (specifically, it is possible that for ). The parameter α can vary, running over a set of values. In (1) is a vector belonging to the unit sphere . The parameter t 0 will be interpreted later. In problem (1), and α are parameters. Accordingly, its solution is denoted by to emphasize its dependence on these parameters. However, in the study of problem (1), the dependence of the solution on and α will be omitted for brevity.
In what follows, we consider the problem of determining the function from some information on the solutions of problem (1). In this context, we make some assumptions about to be used in the subsequent consideration.
(i) For any the support of the function is contained in the ball B(R) = , R > 0.
(ii) and are continuous functions for , .
(iv) For any there exists a positive constant such that Define . In (1) we set . The equation describes a plane wave propagating in the direction of the vector ν through homogeneous space (for ). At the time t = 0, the front of this wave touches the domain occupied by an inhomogeneity.
where is a given function and ε is an arbitrary small positive number. Inverse problems of determining coefficients in nonlinear hyperbolic equations have been intensively studied in recent years (see [1][2][3][4][5][6][7][8][9]). This work is based on the idea of expanding the solution in terms of singularities in a neighborhood of the wavefront; this idea was used, for example, in [10][11][12][13].

Theorem 1. Suppose that
, , and the functions and g(t) satisfy the assumptions made above. Then, near the characteristic wedge t = , problem (1) has a unique weak solution and it can be represented in the form (4) where H(t) is the Heaviside step function defined as for and for t < 0, H 1 (t) = , and the function is given by the formula

Here, ds is the element of the Euclidean length and the function in (4) is continuous in its arguments and infinitesimal as
. This theorem is proved using a series of lemmas. In a homogeneous medium (i.e., for ), the solution of problem (1) has the form u(x, t) = . The Kirchhoff formula for an inhomogeneous wave equation implies that the solution of problem (1) satisfies the integral equation (6) Since for and = 0, it follows that = 0 for · νt 0 . Therefore, (6) implies the equation (7) where is the domain bounded by the axisymmetric paraboloid with the central axis passing through in the direction of the vector -ν.
Consider the family of paraboloids for .
Along with the Cartesian coordinates , we consider a system of coordinates with the origin placed at the point and with basis vectors : In these formulas, and . Additionally, we introduce cylindrical coordinates related to by the equalities , , and , where . Then (8) and the equation defining the paraboloid becomes or (9) Therefore, as , the paraboloid degenerates into the ray L(x, ν) =: . In Eq. (7), instead of the variables of integration , we introduce curvilinear coordinates . Then Therefore, Eq. (7) becomes (10) where the variable ξ is defined by formulas (8) and (9).
Continuing the process of estimating the differences yields (17) Estimate (17) (4) and (5), information (3) determines the integrals (24) in which the function is given by the formula Thus, for any fixed , the integrals over all straight lines crossing the domain B(R) are known. As a result, the problem of determining for every fixed α from information (3) is reduced to an X-ray tomography problem (see, e.g., [14]). This problem is known to be uniquely solvable. Accordingly, the following uniqueness theorem holds.
Theorem 2. Suppose that the conditions of Theorem 1 are satisfied. Then the inverse problem has a unique solution.