In this paper, we consider the blow-up curve of semilinear wave equations. Merle and Zaag (Am J Math 134:581–648, 2012) considered the blow-up curve for $$\partial _t^2 u- \partial _x^2 u = |u|^{p-1}u$$ ∂ t 2 u - ∂ x 2 u = | u | p - 1 u and showed that there is the case that the blow-up curve is not differentiable at some points when the initial value changes its sign. Their analysis depends on the variational structure of the problem. In this paper, we consider the blow-up curve for $$\partial _t^2 u- \partial _x^2 u = |\partial _t u|^{p-1}\partial _t u$$ ∂ t 2 u - ∂ x 2 u = | ∂ t u | p - 1 ∂ t u which does not have the variational structure. Nevertheless, we prove that the blow-up curve is not differentiable if the initial data changes its sign and satisfies some conditions.

中文翻译：

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具有狄利克雷边界条件的一维非线性波动方程解的膨胀曲线

在本文中，我们考虑半线性波动方程的爆破曲线。Merle 和 Zaag (Am J Math 134:581–648, 2012) 考虑了 $$\partial _t^2 u- \partial _x^2 u = |u|^{p-1}u$$ 的爆炸曲线∂ t 2 u - ∂ x 2 u = | 你| p - 1 u 并表明当初始值改变其符号时，膨胀曲线在某些点是不可微的。他们的分析取决于问题的变分结构。在本文中，我们考虑了 $$\partial _t^2 u- \partial _x^2 u = |\partial _t u|^{p-1}\partial _t u$$ ∂ t 2 u 的爆破曲线- ∂ x 2 u = | ∂ tu | p - 1 ∂ tu 没有变分结构。尽管如此，我们证明如果初始数据改变其符号并满足某些条件，则爆破曲线是不可微的。