一维退化拟线性波动方程解的正则性损失
Loss of regularity for solutions to 1D degenerate quasilinear wave equations
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中文总结 AI 辅助
本文研究一维退化波动方程解的正则性损失，证明当 \\(\beta<\alpha\\) 时，退化拟线性方程的正则性损失条件必要，解的阶从初始数据 \\(u_0\\) 的阶变为 \\(u_1\\) 的阶。
中文摘要 AI 辅助
本文研究一维退化波动方程解的正则性损失问题。我们首先考虑线性方程 \\( u_{tt}=\bigl(c(t,x)^2u_x\bigr)_x \\)，其中 \\(c(t,x)\sim x^p\\)，随后考虑拟线性方程 \\( u_{tt}=(u^{2a}u_x)_x \\)。假设初始数据在退化点 \\(x=0\\) 附近满足 \\( u_0(x)\sim x^\alpha \\) 和 \\( u_1(x)\sim x^\beta \\)。在线性问题中，\\(p\\) 描述退化强度，\\(\alpha\\) 和 \\(\beta\\) 描述初始数据的正则性；在拟线性问题中，\\(\alpha\\) 还通过 \\(u_0(x)^a\sim x^{a\alpha}\\) 决定初始退化程度。我们的核心关注点是正则性损失的实际发生，即当 \\(t>0\\) 时，\\(u(t,\cdot)\\) 在 \\(x=0\\) 附近的正则性低于 \\(u_0\\) 的正则性的现象。此前，这类损失主要针对仅依赖时间的特殊线性方程得到证明。在我们之前关于该拟线性方程的研究中，当 \\(\beta\geq\alpha\\) 时，我们证明了无正则性损失的局部适定性。本文表明该条件也是必要的：更准确地说，若 \\(\beta<\alpha\\)，则在足够小的正时间内，\\(x=0\\) 附近满足 \\( C_1tx^\beta\leq u(t,x)\leq C_2x^\beta \\)。因此，解的阶从 \\(u_0\\) 的阶变为 \\(u_1\\) 的阶，对于具有时空依赖系数的线性方程以及退化拟线性方程，均会发生实际的正则性损失。
英文摘要
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 \( u_0(x)\sim x^α\) and \( u_1(x)\sim x^β\) near the degenerate point \(x=0\). In the linear problem, \(p\) describes the strength of the degeneracy, while \(α\) and \(β\) describe the regularity of the initial data. In the quasilinear problem, \(α\) also determines the initial degeneracy through \(u_0(x)^a\sim x^{aα}\). 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 \(β\geqα\). In the present paper, we show that this condition is also necessary. More precisely, if \(β<α\), then \( C_1tx^β\leq u(t,x)\leq C_2x^β\) 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.
