The blow-up curve of solutions for semilinear wave equations with Dirichlet boundary conditions in one space dimension

Abstract

We consider a blow-up curve for the one dimensional wave equation. Merle–Zaag [5] showed that there is a possibility that the blow-up curve for $\partial_t^2 u - \partial_x^2 u = |u|^{p-1}u$ is not differentiable if the sign of the solution changes. To show the result, they use the variational structure of the equation. In this paper, we study the blow-up curve $\partial_t^2 u - \partial_x^2 u = |\partial_t u|^{p-1}\partial_tu $ which does not have the variational structure. We state that the blow-up curve is not differentiable if the initial data is an odd function which satisfy some conditions. Next, we show the key of the proof. This paper is an announcement of our forthcoming paper.