Convergence of a blow-up curve for a semilinear wave equation

Abstract

We consider a blow-up phenomenon for \begin{document}$ { \partial_t^2 u_ \varepsilon} $\end{document} \begin{document}$ {- \varepsilon^2 \partial_x^2u_ \varepsilon } $\end{document} \begin{document}$ { = F(\partial_t u_ \varepsilon)}. $\end{document} The derivative of the solution \begin{document}$ \partial_t u_ \varepsilon $\end{document} blows-up on a curve \begin{document}$ t = T_ \varepsilon(x) $\end{document} if we impose some conditions on the initial values and the nonlinear term \begin{document}$ F $\end{document} . We call \begin{document}$ T_ \varepsilon $\end{document} blow-up curve for \begin{document}$ { \partial_t^2 u_ \varepsilon} $\end{document} \begin{document}$ {- \varepsilon^2 \partial_x^2u_ \varepsilon } $\end{document} \begin{document}$ { = F(\partial_t u_ \varepsilon)}. $\end{document} In the same way, we consider the blow-up curve \begin{document}$ t = \tilde{T}(x) $\end{document} for \begin{document}$ {\partial_t^2 u} $\end{document} \begin{document}$ = $\end{document} \begin{document}$ {F(\partial_t u)}. $\end{document} The purpose of this paper is to show that, for each \begin{document}$ x $\end{document} , \begin{document}$ T_ \varepsilon(x) $\end{document} converges to \begin{document}$ \tilde{T}(x) $\end{document} as \begin{document}$ \varepsilon\rightarrow 0. $\end{document}