Year: 2026
Type: preprint
Abstract: We study the large-time behavior of the displacement for wave equations with non-integrable speeds and zero initial displacement. For initial velocities in $L^2\cap L^1$, we derive an $L^2$ upper bound under a weighted Poincaré inequality and symbol estimates up to order two on the propagation speed. The rate is determined by the speed and its time integral. For these speeds, an additional symbol estimate for the third derivative and the nonzero mean condition on the initial velocity give a lower bound of the same order, obtained without using an explicit formula for the Fourier multiplier. At the endpoint $a(t)=(1+t)^{-1}$, the sharp rate is $t(\ln t)^{-\frac{n}{4}}$. The estimates apply to polynomial, logarithmically modified, oscillating, and exponential-type speeds.
Source: arXiv (Cornell University)
Authors: Aslan, Halit Sevki, Chen, Wenhui
Institutions: Universidade de São Paulo, Guangzhou University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2469264322, G7952596114
OpenAlex ID: W7214047070