Exact L2-growth profiles for the free wave equation: Sharp constants, critical regularity, and local energy decay
Abstract
We study the large-time behavior of the L2-norm of solutions to the free wave equation on ℝN and, more generally, their homogeneous Sobolev norms of negative order. We refine and extend the results of R. Ikehata [J. Hyperbolic Differ. Equ. 20 (2023)] who identified and quantified the infinite-time L2-growth of free waves in dimensions N=1,2. For finite-energy data with u1∈ L1(ℝN), we derive exact first-order asymptotics: when P=∫ℝNu1(x) dx≠0, the squared L2-norm grows linearly for N=1 and logarithmically for N=2, with a leading coefficient depending only on P. In particular, no weighted moment assumption on u1 is required; u1∈ L1∩ L2 suffices. More generally, we prove that the L2-norm is bounded, and hence convergent, if and only if u1∈Ḣ−1(ℝN). We further establish first-order asymptotic profiles in negative-order homogeneous Sobolev spaces, covering polynomial growth, logarithmic growth, and convergence to a finite limit according to the Sobolev index. These results reveal a unified low-frequency mechanism underlying the low-dimensional L2-growth. We also derive second-order asymptotic expansions in dimension one and at the critical Sobolev index in dimensions N≥2. Finally, in dimension two, the exact L2-asymptotics yield a sharper local energy decay estimate with an explicit upper asymptotic constant independent of the localization radius.
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BibTeX
@misc{kumar2026exact,
title = {{Exact $L^2$-growth profiles for the free wave equation: Sharp constants, critical regularity, and local energy decay}},
author = {Vishvesh Kumar and Mohamed Majdoub},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15086},
eprint = {2610.15086},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15086},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact L^2-growth profiles for the free wave equation: Sharp constants, critical regularity, and local energy decay
AU - Vishvesh Kumar
AU - Mohamed Majdoub
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15086
UR - https://arxiv.org/abs/2610.15086
AB - We study the large-time behavior of the L^2-norm of solutions to the free wave equation on ℝ^N and, more generally, their homogeneous Sobolev norms of negative order. We refine and extend the results of R. Ikehata [J. Hyperbolic Differ. Equ. 20 (2023)] who identified and quantified the infinite-time L^2-growth of free waves in dimensions N=1,2. For finite-energy data with u_1∈ L^1(ℝ^N), we derive exact first-order asymptotics: when P=∫_ℝ^Nu_1(x) dx≠0, the squared L^2-norm grows linearly for N=1 and logarithmically for N=2, with a leading coefficient depending only on P. In particular, no weighted moment assumption on u_1 is required; u_1∈ L^1∩ L^2 suffices. More generally, we prove that the L^2-norm is bounded, and hence convergent, if and only if u_1∈Ḣ^−1(ℝ^N). We further establish first-order asymptotic profiles in negative-order homogeneous Sobolev spaces, covering polynomial growth, logarithmic growth, and convergence to a finite limit according to the Sobolev index. These results reveal a unified low-frequency mechanism underlying the low-dimensional L^2-growth. We also derive second-order asymptotic expansions in dimension one and at the critical Sobolev index in dimensions N≥2. Finally, in dimension two, the exact L^2-asymptotics yield a sharper local energy decay estimate with an explicit upper asymptotic constant independent of the localization radius.
ER -