On the $L^{2}$ estimates of the diffusion waves

TL;DR

This paper analyzes the long-time behavior of solutions to the strongly damped wave equation in low dimensions, revealing when diffusion-wave profiles approximate solutions and when they fail due to low-frequency effects.

Contribution

It provides new $L^2$ estimates for the diffusion-wave profile and identifies dimension-dependent behaviors in low-dimensional cases.

Findings

In 1D, the difference operator is controlled by Ct^{1/4} times the initial data norm.

In 2D, the difference operator has a logarithmic lower bound if the initial velocity has nonzero mass.

The free wave remains an effective asymptotic profile in 1D but not in 2D.

Abstract

In this paper, we investigate the long-time behavior of the $L^2$-norm of solutions to the Cauchy problem for the strongly damped wave equation on $\mathbb{R}^n$, with particular focus on the low-dimensional cases $n=1$ and $n=2$. Although the energy is dissipative, the $L^2$-norm may grow because of low-frequency effects. We compare the diffusion-wave profile of the strongly damped equation with the corresponding free-wave evolution generated by the same initial velocity. Introducing the difference operator $D(t)$ between these two evolutions, we prove that in one dimension $D(t)$ is controlled by $Ct^{1/4}\|g\|_{L^1}$, showing that the free wave remains an effective asymptotic profile. In contrast, in two dimensions $D(t)$ has a logarithmic lower bound when the mass of the initial velocity is nonzero, implying that the wave approximation fails. Corresponding estimates for the original solution are also obtained.