The Blow-up Surface for Nonlinear Wave Equations with Small Spatial Velocity

Abstract

Consider the Cauchy problem for ${u_{tt}} - {\varepsilon ^2}\Delta u = f(u)$ in space dimension $\leqslant 3$ where $f(u)$ is superlinear and nonnegative. The solution blows up on a surface $t = {\phi _\varepsilon }(x)$. Denote by $t = \phi (x)$ the blow-up surface corresponding to $v'' = f(v)$. It is proved that $|{\phi _\varepsilon }(x) - \phi (x)| \leqslant C{\varepsilon ^2}$, $|\nabla ({\phi _\varepsilon }(x) - \phi (x))| \leqslant C{\varepsilon ^2}$ in a neighborhood of any point ${x_0}$ where $\phi ({x_0}) < \infty$.