论文标题
1D巡演方程家族的奇异性形成
Formation of singularities for a family of 1D quasilinear wave equations
论文作者
论文摘要
我们考虑将解决方案吹入以下参数化的非线性波方程：$ u_ {tt} = c（u）^{2} u_ {xx} +λc（u）c'（u）（u）（u_x）^2 $带有真实参数$λ$。在以前的工作中，据报道，存在$λ= 1 $和$ 2 $的有限时间爆破解决方案。但是，爆破解决方案的构建取决于方程的对称结构（例如，节能定律）。在本文中，我们通过使用新的$ l^{2/λ} $估算，将$λ= 1 $的爆炸结果扩展到$λ\ in（0,1] $。
We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + λc(u)c'(u)( u_x)^2$ with the real parameter $λ$. In previous works, it was reported that there exist finite time blow-up solutions with $λ=1$ and $2$. However, the construction of a blow-up solution depends on the symmetric structure of the equation (e.g., the energy conservation law). In the present paper, we extend the blow-up result with $λ=1$ to the case with $λ\in (0,1]$ by using a new $L^{2/λ}$ estimate. Moreover, some properties for the blow-up solution including the Hölder continuity are also discussed.
