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

In this paper, we study the loss of regularity for solutions to one-dimensional degenerate wave equations. We first consider the linear equation $u_{tt}=(c(t,x{)}^2{u}_x{)}_x$ with $c(t,x)\sim x^p$, and then the quasilinear equation $u_{tt}=(u^{2a}{u}_x{)}_x\mathrm{.}$ 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 $\beta \ge \alpha$. In the present paper, we show that this condition is also necessary. More precisely, if $\beta <\alpha$, then $C_{1}t{x}^{\beta }\le u(t,x)\le C_2{x}^{\beta }$ 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.