WEAKLY VISCOUS NONLOCAL WAVE EQUATIONS WITH VARIABLE COEFFICIENTS: WELL-POSEDNESS, RUNGE APPROXIMATION, AND INVERSE PROBLEMS
Abstract
We study weakly viscous nonlocal wave equations with space-and time-dependent inertia, viscosity, and potential. The elastic and viscous terms involve fractional Laplacians of orders 2s and 2r, respectively, where 0 ≤ r < s, and the viscosity is allowed to degenerate. We establish well-posedness for weak finite energy solutions in weighted spaces adapted to the dissipation. For uniformly elliptic viscosities, we also develop a transposition theory for rough data under a quantitative smallness condition. Under suitable exterior localization, decay, and exponent assumptions, we prove Runge approximation in L^2 for possibly degenerate viscosities and stronger approximation results in the uniformly elliptic setting, including density in the full energy space. The proofs combine parabolic regularization, weighted energy estimates, duality, and the entanglement principle for fractional Laplacians. As an application, we show that an exterior Dirichlet-to-Neumann map determines the inertia, viscosity, and potential simultaneously. The recovery of nonlocal viscosity uses exterior data concentrated in short time intervals to separate the prescribed exterior velocity from the interior response.
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BibTeX
@misc{zimmermann2026weakly,
title = {{WEAKLY VISCOUS NONLOCAL WAVE EQUATIONS WITH VARIABLE COEFFICIENTS: WELL-POSEDNESS, RUNGE APPROXIMATION, AND INVERSE PROBLEMS}},
author = {Philipp Zimmermann},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05839107v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - WEAKLY VISCOUS NONLOCAL WAVE EQUATIONS WITH VARIABLE COEFFICIENTS: WELL-POSEDNESS, RUNGE APPROXIMATION, AND INVERSE PROBLEMS
AU - Philipp Zimmermann
PY - 2026
DA - 2026/10/26
PB - HAL
UR - https://hal.science/hal-05839107v1
AB - We study weakly viscous nonlocal wave equations with space-and time-dependent inertia, viscosity, and potential. The elastic and viscous terms involve fractional Laplacians of orders 2s and 2r, respectively, where 0 ≤ r < s, and the viscosity is allowed to degenerate. We establish well-posedness for weak finite energy solutions in weighted spaces adapted to the dissipation. For uniformly elliptic viscosities, we also develop a transposition theory for rough data under a quantitative smallness condition. Under suitable exterior localization, decay, and exponent assumptions, we prove Runge approximation in L^2 for possibly degenerate viscosities and stronger approximation results in the uniformly elliptic setting, including density in the full energy space. The proofs combine parabolic regularization, weighted energy estimates, duality, and the entanglement principle for fractional Laplacians. As an application, we show that an exterior Dirichlet-to-Neumann map determines the inertia, viscosity, and potential simultaneously. The recovery of nonlocal viscosity uses exterior data concentrated in short time intervals to separate the prescribed exterior velocity from the interior response.
ER -