Calderón problem for nonlocal viscous wave equations: Unique determination of linear and nonlinear perturbations
Philipp Zimmermann
TL;DR
The paper proves a Calderón-type inverse result for nonlocal viscous wave equations, showing that partial Dirichlet-to-Neumann data uniquely determine both linear potentials and homogeneous nonlinearities under precise growth assumptions. The main strategy combines well-posedness theory for the forward problem, a Runge approximation in $L^2(0,T;\widetilde{H}^s(\Omega))$, an Alessandrini-type integral identity, and a differentiable (Fréchet) dependence of the nonlinear solution map to linearize the inverse problem. For the linear case, uniqueness follows from DN-map data and Runge approximation; for the nonlinear case, a suitable integral identity together with a first-order expansion of the DN map yields the unique determination of the nonlinearity $f$ under the stated homogeneity and growth hypotheses. Overall, the work extends Calderón-type recovery to nonlocal viscous dynamics, enabling stable identification of both coefficients and nonlinearities from boundary measurements in a time-dependent, nonlocal setting.
Abstract
The main goal of this article is to study a Calderón type inverse problem for certain viscous nonlocal wave equations. We show that the partial Dirichlet to Neumann map uniquely determines on the one hand linear perturbations and on the other hand homogeneous nonlinearities $f(u)$ whenever the latter satisfy a certain growth assumption. As a preliminary step we discuss the well-posedness in each case, where for the nonlinear setting we invoke the implicit function theorem after establishing the differentiability of the associated Nemytskii operator $f(u)$. In the linear case we establish a Runge approximation theorem in $L^2(0,T;\widetilde{H}^{s}(Ω))$, which allows us to uniquely determine potentials that belong only to $L^{\infty}(0,T;L^p(Ω))$ for some $1<p\leq \infty$ satisfying suitable restrictions. In the nonlinear case, we first derive an appropriate integral identity and combine this with the differentiability of the solution map around zero to show that the nonlinearity is uniquely determined by the Dirichlet to Neumann map. To make this linearization technique work, it is essential that we have a Runge approximation in $L^2(0,T;\widetilde{H}^s(Ω))$ instead of $L^2(Ω_T)$ at our disposal.