The blow-up curve of solutions to one dimensional nonlinear wave equations with the Dirichlet boundary conditions

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 ∂t2u-∂x2u=|u|p-1u 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 ∂t2u-∂x2u=|∂tu|p-1∂tu 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.