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July  2019, 18(4): 1891-1919. doi: 10.3934/cpaa.2019088

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

 1 School of Mathematical Sciences and Mathematical Institute, Nanjing Normal University, Nanjing, 210023, China 2 School of Mathematics and Physics, Anhui University of Technology, Maanshan 243032, China

* Corresponding author

Received  August 2018 Revised  November 2018 Published  January 2019

Fund Project: The first author and the second author are supported by the NSFC (No.11571177, No.11731007) and the NSF of the Jiangsu Higher Education Institutions of China (17KJA110002).

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.

Citation: Wenming Hu, Huicheng Yin. Well-posedness of low regularity solutions to the second order strictly hyperbolic equations with non-Lipschitzian coefficients. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1891-1919. doi: 10.3934/cpaa.2019088
##### References:
 [1] H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Velag, Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. [2] J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, (French) [Symbolic calculus and propagation of singularities for non-linear partial differential equations], Ann.Sci. École Norm. Sup., 14 (1981), 209-246. [3] M. Cicognani and F. Colombini, Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem, J. Diff. Eqs., 221 (2006), 143-157.  doi: 10.1016/j.jde.2005.06.019. [4] F. Colombini, E. De Giorgi and S. Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne ${\acute{\rm{q}}}$ue du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 6 (1979), 511-559. [5] F. Colombini, E. Jannelli and S. Spagnolo, Nonuniqueness in hyperbolic cauchy problems, Ann. of Math., 126 (1987), 495-524.  doi: 10.2307/1971359. [6] F. Colombini and N. Lerner, Operators with non-Lipschitz coefficients, Duke Math. J., 77 (1995), 657-698.  doi: 10.1215/S0012-7094-95-07721-7. [7] F. Colombini and G. Métivier, The cauchy problem for wave equations with non lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations, Ann. Sci. Éc. Norm. Supér.(4) , 41 (2008), 177–220. doi: 10.24033/asens.2066. [8] F. Colombini and D. D. Santo, A note on hyperbolic operators with log-zygmund coefficients, J. Math. Sci. Univ. Tokyo, 16 (2009), 95-111. [9] F. Colombini, D.D. Santo, F. Fanelli and G. Métivier, Time-dependent loss of derivatives for hyperbolic operators with non regular coefficients, Comm. Partial Differential Equations, 38 (2013), 1791-1817.  doi: 10.1080/03605302.2013.795968. [10] F. Colombini and S. Spagnolo, Some examples of hyperbolic equations without local solvability, Ann, Sci. Ecole Norm. Suph.(4), 22 (1989), 109–125. [11] L. Hörmander, Linear Partial Differential Operators, Springer-Velag, Berlin Heidelberg, 1976. [12] A. E. Hurd and D. H. Sattinger, Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients, Trans. Amer. Math. Soc., 132 (1968), 159-174.  doi: 10.2307/1994888. [13] E. Jannelli, Regularly hyperbolic systems and Gevrey classes, Ann. Mat. Pura Appl., 140 (1985), 133-145.  doi: 10.1007/BF01776846. [14] T. Nishitani, Sur les équations hyperboliques á coefficients höldériens en t et de classe de Gevrey en x, Bull. Sci. Math., 107 (1983), 113–138., [15] J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, Graduate Studies in Mathematics, 133, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/133. [16] M. E. Taylor, Tools for PDE : Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, 81, American Mathematical Society, Providence, RI, 2000.

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##### References:
 [1] H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Velag, Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. [2] J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, (French) [Symbolic calculus and propagation of singularities for non-linear partial differential equations], Ann.Sci. École Norm. Sup., 14 (1981), 209-246. [3] M. Cicognani and F. Colombini, Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem, J. Diff. Eqs., 221 (2006), 143-157.  doi: 10.1016/j.jde.2005.06.019. [4] F. Colombini, E. De Giorgi and S. Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne ${\acute{\rm{q}}}$ue du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 6 (1979), 511-559. [5] F. Colombini, E. Jannelli and S. Spagnolo, Nonuniqueness in hyperbolic cauchy problems, Ann. of Math., 126 (1987), 495-524.  doi: 10.2307/1971359. [6] F. Colombini and N. Lerner, Operators with non-Lipschitz coefficients, Duke Math. J., 77 (1995), 657-698.  doi: 10.1215/S0012-7094-95-07721-7. [7] F. Colombini and G. Métivier, The cauchy problem for wave equations with non lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations, Ann. Sci. Éc. Norm. Supér.(4) , 41 (2008), 177–220. doi: 10.24033/asens.2066. [8] F. Colombini and D. D. Santo, A note on hyperbolic operators with log-zygmund coefficients, J. Math. Sci. Univ. Tokyo, 16 (2009), 95-111. [9] F. Colombini, D.D. Santo, F. Fanelli and G. Métivier, Time-dependent loss of derivatives for hyperbolic operators with non regular coefficients, Comm. Partial Differential Equations, 38 (2013), 1791-1817.  doi: 10.1080/03605302.2013.795968. [10] F. Colombini and S. Spagnolo, Some examples of hyperbolic equations without local solvability, Ann, Sci. Ecole Norm. Suph.(4), 22 (1989), 109–125. [11] L. Hörmander, Linear Partial Differential Operators, Springer-Velag, Berlin Heidelberg, 1976. [12] A. E. Hurd and D. H. Sattinger, Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients, Trans. Amer. Math. Soc., 132 (1968), 159-174.  doi: 10.2307/1994888. [13] E. Jannelli, Regularly hyperbolic systems and Gevrey classes, Ann. Mat. Pura Appl., 140 (1985), 133-145.  doi: 10.1007/BF01776846. [14] T. Nishitani, Sur les équations hyperboliques á coefficients höldériens en t et de classe de Gevrey en x, Bull. Sci. Math., 107 (1983), 113–138., [15] J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, Graduate Studies in Mathematics, 133, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/133. [16] M. E. Taylor, Tools for PDE : Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, 81, American Mathematical Society, Providence, RI, 2000.
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