Exact $L^2$-growth profiles for the free wave equation: Sharp constants, critical regularity, and local energy decay

Abstract

We study the large-time behavior of the $L^2$-norm of solutions to the free wave equation on ${\mathbb{R}}^N$ 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 $u_1\in L^1({\mathbb{R}}^N)$, we derive exact first-order asymptotics: when $P={\int }_{{\mathbb{R}}^N}{u}_1(x)dx\ne 0$, 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; $u_1\in L^1\cap L^2$ suffices. More generally, we prove that the $L^2$-norm is bounded, and hence convergent, if and only if $u_1\in {\dot{H}}^{-1}({\mathbb{R}}^N)$. 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 $N\ge 2$. 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.

33 pages, comments are welcome