On the determination of nonlinear terms appearing in semilinear hyperbolic equations

Abstract

We consider the inverse problem of determining the shape of a general nonlinear term appearing in a semilinear hyperbolic equation on a Riemannian manifold with boundary (M, g) of dimension n = 2, 3. We prove results of unique recovery of the nonlinear term F (t, x, u), appearing in the equation ∂ 2 t u − Δgu + F (t, x, u) = 0 on (0, T ) × M with T > 0, from partial knowledge of the solutions u on the lateral boundary (0, T ) × ∂M . We obtain, what seems to be, the first result of determination of the expression F (t, x, u) on the boundary x ∈ ∂M for such a general class of nonlinear terms. With additional assumptions on the manifold and some extended measurements at t = 0 and t = T , we prove also the recovery of F inside the manifold x ∈ M .

Recovery of Nonsmooth Coefficients Appearing in Anisotropic Wave Equations

“…The latter result is based on a reduction via linearization to the problem to recover the zeroth order coefficient in a linear wave equation. For this reason, contrary to our result, the geometric context in [9] in confined to the cases where results are available for linear wave equations.…”

“…We mention that the approach [11] has also been applied to recovery of coefficients appearing in non-linear terms [13], and to a problem arising in seismic imaging [6]. Recently two other approaches were used by Nakamura and Vashisth to recover time-independent leading order coefficients, as well as coefficients in nonlinear terms [15], and by Kian to recover a general function corresponding to the non-linearity and also including zeroth order coefficients [9]. The latter result is based on a reduction via linearization to the problem to recover the zeroth order coefficient in a linear wave equation.…”