×

zbMATH — the first resource for mathematics

Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem. (English) Zbl 1097.35092
Let the operator \(P\) in \([0, T]\times\mathbb{R}^n_x\) be defined by
\[ P= \partial^2_t+ 2iB(t,x,D_x)\partial_t+ A(t,x,D_x), \]
\(D_x= -i\nabla_x\), with real symbols \[ A(t,x,\xi)= \sum^n_{j,k=1} a_{jk}(t, x)\xi_j \xi_k,\;B(t,x,\xi)= \sum^n_{j=1} b_j(t, x)\xi_j \] that satisfy the condition of strict hyperbolicity \[ B^2(t,x,\xi)+ A(t,x,\xi)\geq c_0|\xi|^2\quad (c_0> 0) \] for \(t\in [0, T]\), \(x,\xi\in\mathbb{R}^n\). Given a continuous function \(\omega(\theta)\) such that \(\omega(\theta)\downarrow 0\), for \(\theta\to 0^+\). Denote by \(M^\omega([0, T])\) the space of all functions \(a(t)\) such that \[ |a(t+\tau)- a(t)|\leq C|\tau||\log|\tau|)\quad (C> 0) \] for \(t,t+ \tau\in [0, T]\), \(0\leq|\tau|\leq{1\over 2}\). Let the coefficients of \(P\) be \(a_{jk}\), \(b_j\in M^\omega([0,T]:{\mathfrak B}^\infty)\). Then it proved that for every \(s,\delta> 0\) there is \(C_{s,\delta}> 0\) such that \[ \| u(t)\|_{H^{s+1-\delta}}+ \|\partial_t u(t)\|_{H^{s-\delta}}\leq C\Biggl(\| u(0)\|_{H^{s+1}}+ \|\partial_t u(0)\|_{H^s}+ \int^t_0\| Pu(\tau)\,d\tau\Biggr) \] for \(t\in[0, T]\), and for all \(u\in\bigcap^2_{j=0} C^j([0, T], H^{s+2-j})\).

MSC:
35L15 Initial value problems for second-order hyperbolic equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agliardi, R.; Cicognani, M., The Cauchy problem for a class of Kovalevskian pseudo-differential operators, Proc. amer. math. soc., 132, 841-845, (2004) · Zbl 1054.35154
[2] Agliardi, R.; Cicognani, M., Operators of \(p\)-evolution with non regular coefficients in the time variable, J. differential equations, 202, 143-157, (2004) · Zbl 1065.35094
[3] Colombini, F.; De Giorgi, E.; Spagnolo, S., Sur LES équations hyperboliques avec des coefficients qui ne dépendent que du temps, Ann. scuola norm. sup. Pisa cl. sci., 6, 511-559, (1979) · Zbl 0417.35049
[4] Colombini, F.; Lerner, N., Hyperbolic operators with non-Lipschitz coefficients, Duke math. J., 77, 657-698, (1995) · Zbl 0840.35067
[5] Colombini, F.; Spagnolo, S., On the convergence of solutions of hyperbolic equations, Comm. partial differential equations, 3, 77-103, (1978) · Zbl 0375.35034
[6] Hörmander, L., Linear partial differential operators, (1963), Springer Berlin, Göttingen, Heidelberg · Zbl 0171.06802
[7] Mizohata, S., The theory of partial differential equations, (1973), Cambridge University Press Cambridge · Zbl 0263.35001
[8] Reissig, M., Hyperbolic equations with non-Lipschitz coefficients, Rend. sem. mat. univ. politec. Torino, 61, 135-181, (2003) · Zbl 1182.35155
[9] Reissig, M., About strictly hyperbolic operators with non-regular coefficients, Pliska stud. math., 15, 105-130, (2003)
[10] Tarama, S., Local uniqueness in the Cauchy problem for second order elliptic equations with non-Lipschitzian coefficients, Publ. res. inst. math. sci. Kyoto university, 33, 167-188, (1997) · Zbl 0882.35034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.