On the derivative loss for a class of abstract evolution equations with time dependent oscillating coefficients vanishing at initial time
Abstract
We consider an abstract wave-type evolution equation with a propagation speed depending only on time. We assume that this propagation speed is the product of a monotone, but possibly degenerate, shape function and an oscillating factor bounded from above and from below by positive constants. We investigate how degeneracy and oscillations interact in determining the derivative loss of solutions. After reducing the problem to a family of ordinary differential equations through spectral analysis, we derive frequency-dependent energy estimates. In the purely degenerate case, monotonicity of the shape function prevents excessive loss, and solutions lose at most one derivative. In the presence of oscillations, the resulting derivative loss is described by a minimization problem in which the degenerate and oscillatory effects genuinely interact. The proof is based on a decomposition of the time interval into frequency-dependent regions, where different adapted energies are used.
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BibTeX
@misc{ghisi2026derivative,
title = {{On the derivative loss for a class of abstract evolution equations with time dependent oscillating coefficients vanishing at initial time}},
author = {Marina Ghisi and Massimo Gobbino and Michael Reissig},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02915},
eprint = {2610.02915},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02915},
}
RIS
TY - UNPB
TI - On the derivative loss for a class of abstract evolution equations with time dependent oscillating coefficients vanishing at initial time
AU - Marina Ghisi
AU - Massimo Gobbino
AU - Michael Reissig
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02915
UR - https://arxiv.org/abs/2610.02915
AB - We consider an abstract wave-type evolution equation with a propagation speed depending only on time. We assume that this propagation speed is the product of a monotone, but possibly degenerate, shape function and an oscillating factor bounded from above and from below by positive constants. We investigate how degeneracy and oscillations interact in determining the derivative loss of solutions. After reducing the problem to a family of ordinary differential equations through spectral analysis, we derive frequency-dependent energy estimates. In the purely degenerate case, monotonicity of the shape function prevents excessive loss, and solutions lose at most one derivative. In the presence of oscillations, the resulting derivative loss is described by a minimization problem in which the degenerate and oscillatory effects genuinely interact. The proof is based on a decomposition of the time interval into frequency-dependent regions, where different adapted energies are used.
ER -