Loss of regularity for solutions to 1D degenerate quasilinear wave equations

Yanbo Hu; Yuusuke Sugiyama

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arXiv · 2608.14233

Loss of regularity for solutions to 1D degenerate quasilinear wave equations

Yanbo Hu·Yuusuke Sugiyama

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Abstract

In this paper, we study the loss of regularity for solutions to one-dimensional degenerate wave equations. We first consider the linear equation $ u_{tt}=\bigl(c(t,x)^2u_x\bigr)_x $ with $c(t,x)\sim x^p$, and then the quasilinear equation $ u_{tt}=(u^{2a}u_x)_x. $ The initial data are assumed to satisfy $ u_0(x)\sim x^\alpha $ and $ u_1(x)\sim x^\beta $ near the degenerate point $x=0$. In the linear problem, $p$ describes the strength of the degeneracy, while $\alpha$ and $\beta$ describe the regularity of the initial data. In the quasilinear problem, $\alpha$ also determines the initial degeneracy through $u_0(x)^a\sim x^{a\alpha}$. Our main concern is the actual occurrence of loss of regularity, namely, the phenomenon in which the regularity of $u(t,\cdot)$ near $x=0$ becomes lower than that of $u_0$ for $t>0$. Previously, such a loss had mainly been established for special linear equations with coefficients depending only on time. In our previous work on the quasilinear equation, we proved local well-posedness without loss of regularity when $\beta\geq\alpha$. In the present paper, we show that this condition is also necessary. More precisely, if $\beta<\alpha$, then $ C_1tx^\beta\leq u(t,x)\leq C_2x^\beta $ near $x=0$ for sufficiently small positive time. Thus the order of the solution changes from that of $u_0$ to that of $u_1$, and an actual loss of regularity occurs for both linear equations with time-space dependent coefficients and degenerate quasilinear equations.

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BibTeX RIS

Yanbo Hu, Yuusuke Sugiyama. 2026-08-14. Loss of regularity for solutions to 1D degenerate quasilinear wave equations. https://arxiv.org/abs/2608.14233

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