Inverse Initial Boundary Value Problem for a Non-linear Hyperbolic Partial Differential Equation

Abstract

In this article we are concerned with an inverse initial boundary value problem for a non-linear wave equation in space dimension n ⩾ 2. In particular we consider the so called interior determination problem. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear wave equation with a time independent potential. For any small solution u = u(t, x) of our non-linear wave equation which is the perturbation of linear wave equation with time-independent potential perturbed by a divergence with respect to (t, x) of a vector whose components are quadratics with respect to ∇ t,x u(t, x). By ignoring the terms with smallness O(|∇ t,x u(t, x)|3), we will show that we can uniquely determine the potential and the coefficients of these quadratics by many boundary measurements at the boundary of the spacial domain over finite time interval and the final overdetermination at t = T. In other words, our measurement is given by the so-called the input-output map (see ()).

On the determination of nonlinear terms appearing in semilinear hyperbolic equations

“…For hyperbolic equations we refer to the work of [28,29] dealing with the recovery of some linear and quadratic coefficients appearing in a nonlinear wave equation. We mention also the recent works of [8,22,23], who have considered inverse problems for semilinear hyperbolic equations on a general Lorentzian manifold.…”

Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{ö}dinger equation

“…The aim of this section is to demonstrate the unique solvability of magnetic Schrödinger equation (1.1) via ε-expansion technique. This expansion technique is quite instrumental in the hyperbolic and elliptic problems (see [38,39]) for nonlinear wave equation and (see [10,29]) for quasilinear elliptic equation. However, we hereby impose the technique in a nonlinear dynamical Schrödinger equation where the treatment differs significantly and hence, the complete details will be provided for existence, uniqueness and regularity of the solution to equation (1.1).…”

“…Proof. For the properties ( 1) and ( 2), one can use the elliptic regularity argument [39]. For the property (3), we have…”

“…Proof. We follow the arguments similar to the one used in [29] for elliptic case and in [39] for the wave equation. Let us begin with choosing ω j ∈ S n−1 for 1 ≤ j ≤ n such that the set {ω 1 , ω 2 , • • • , ω n } is a linearly independent subset of R n .…”

“…Recently in [35], the determination of time-dependent coefficients in a semi-linear dyanmical Schrödinger equation is considered where they showed that the electric potential and coefficient of non-lnearity can be determined from the knowledg of source-to-solution map. Inverse initial boundary value problem for a quasilinear hyperbolic equation in second or higher dimension was studied (refer to [39]), in which unique determination of time-dependent coefficients is proved from the measurement given by input-output map.…”