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.
| Subjects: | Analysis of PDEs (math.AP) |
| MSC classes: | Primary 35L05, Secondary 35B40, 35B45 |
| Cite as: | arXiv:2605.20557 [math.AP] |
| (or arXiv:2605.20557v2 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.20557 arXiv-issued DOI via DataCite |
From: Hiroshi Takeda [view email]
[v1]
Tue, 19 May 2026 23:17:08 UTC (13 KB)
[v2]
Thu, 21 May 2026 23:50:06 UTC (13 KB)
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