SPACE-TIME RUNGE APPROXIMATION FOR WEAKLY VISCOUS NONLOCAL WAVES AND IDENTIFICATION OF COEFFICIENTS
Abstract
We study very weak solutions and Runge approximation for weakly viscous nonlocal wave equations with elastic term (-∆)^s u and dissipative term (-∆)^r \partial_t u, where 0 ≤ r< s. For time-dependent potentials satisfying a smallness condition, we establish a transposition theory for sources in the dual of the natural space-time energy space. Under explicit nonlocality and nonresonance assumptions, we prove Runge approximation in L^2 (0, T ; \widetilde{H}^s (Ω)) and in L^2 (0, T ; \widetilde{H}^s (Ω)) ∩ H^1_0 (0, T ; \widetilde{H}^r (Ω)). As an application, partial exterior Dirichlet-to-Neumann measurements uniquely determine a bounded, nonnegative, time-dependent coefficient in the nonlinear velocity damping term q|\partial_t u|^{p-2} \partial_t u for 0\leq r < s and 2 < p ≤ 2 + 4r/n. The proof establishes the small-amplitude response by energy estimates, weighted L^p monotonicity convergence, and a linear Green identity, then applies adjoint Runge approximation followed by a forward velocity approximation in L^2 . No smallness or positive lower bound is imposed on the nonlinear damping coefficient.
Related
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BibTeX
@misc{zimmermann2026spacetime,
title = {{SPACE-TIME RUNGE APPROXIMATION FOR WEAKLY VISCOUS NONLOCAL WAVES AND IDENTIFICATION OF COEFFICIENTS}},
author = {Philipp Zimmermann},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2611.04790},
eprint = {2611.04790},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05839102v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - SPACE-TIME RUNGE APPROXIMATION FOR WEAKLY VISCOUS NONLOCAL WAVES AND IDENTIFICATION OF COEFFICIENTS
AU - Philipp Zimmermann
PY - 2026
DA - 2026/10/26
PB - HAL
DO - 10.48550/arxiv.2611.04790
UR - https://hal.science/hal-05839102v1
AB - We study very weak solutions and Runge approximation for weakly viscous nonlocal wave equations with elastic term (-∆)^s u and dissipative term (-∆)^r \partial_t u, where 0 ≤ r< s. For time-dependent potentials satisfying a smallness condition, we establish a transposition theory for sources in the dual of the natural space-time energy space. Under explicit nonlocality and nonresonance assumptions, we prove Runge approximation in L^2 (0, T ; \widetilde{H}^s (Ω)) and in L^2 (0, T ; \widetilde{H}^s (Ω)) ∩ H^1_0 (0, T ; \widetilde{H}^r (Ω)). As an application, partial exterior Dirichlet-to-Neumann measurements uniquely determine a bounded, nonnegative, time-dependent coefficient in the nonlinear velocity damping term q|\partial_t u|^{p-2} \partial_t u for 0\leq r < s and 2 < p ≤ 2 + 4r/n. The proof establishes the small-amplitude response by energy estimates, weighted L^p monotonicity convergence, and a linear Green identity, then applies adjoint Runge approximation followed by a forward velocity approximation in L^2 . No smallness or positive lower bound is imposed on the nonlinear damping coefficient.
ER -