Well-posedness of low regularity solutions to the second order strictly hyperbolic equations with non-Lipschitzian coefficients

Abstract

In this paper, we establish the local well-posedness of low regularity solutions to the general second order strictly hyperbolic equation of divergence form \begin{document}$ \partial _t(a_0 \partial _t u)+ \mathop \sum \limits_{j = 1}^n [ \partial _t(a_j \partial _j u)+ \partial _j(a_j \partial _t u)] -\mathop \sum \limits_{j,k = 1}^n \partial _j(a_{jk} \partial _k u) $\end{document} \begin{document}$ +b_0 \partial _t u+ \partial _t(c_0u)+ \mathop \sum \limits_{j = 1}^n [b_j \partial _ju+ \partial _j(c_ju)] +du = f $\end{document} in domain \begin{document}$ \Omega = (0, T_0)\times \mathbb R ^n $\end{document} , where the coefficients \begin{document}$ a_0, a_j, a_{jk}\in L^\infty( \Omega )\cap LL(\bar\Omega) $\end{document} \begin{document}$ (1\le j, k\le n) $\end{document} , \begin{document}$ b_0, c_0, b_j, c_j\in L^\infty( \Omega )\cap C^ \alpha (\bar\Omega) $\end{document} \begin{document}$ (1\le j\le n) $\end{document} for \begin{document}$ \alpha \in(\frac{1}{2},1) $\end{document} , \begin{document}$ d\in L^\infty(\Omega) $\end{document} , \begin{document}$ (u(0,x), Xu(0,x))\in (H^{1- \theta +\beta \log}, H^{- \theta +\beta \log}) $\end{document} with \begin{document}$ \theta\in (1- \alpha , \alpha ) $\end{document} , \begin{document}$ \beta\in\Bbb R $\end{document} , and \begin{document}$ Xu = a_0 \partial _tu+ \mathop \sum \limits_{j = 1}^n a_j \partial _ju $\end{document} . Compared with previous references, except a little more general initial data in the space \begin{document}$ (H^{1- \theta +\beta \log}, H^{- \theta +\beta \log}) $\end{document} (only \begin{document}$ \beta = 0 $\end{document} is considered as before), we improve both the lifespan of \begin{document}$ u $\end{document} up to the precise number \begin{document}$ T^* $\end{document} and the range of \begin{document}$ \theta $\end{document} to the left endpoint \begin{document}$ 1- \alpha $\end{document} under some suitable conditions.