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

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