## 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

### Keywords:

Levi condition; log-Lipschitz continuity
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### References:

  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  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  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  Colombini, F.; Lerner, N., Hyperbolic operators with non-Lipschitz coefficients, Duke math. J., 77, 657-698, (1995) · Zbl 0840.35067  Colombini, F.; Spagnolo, S., On the convergence of solutions of hyperbolic equations, Comm. partial differential equations, 3, 77-103, (1978) · Zbl 0375.35034  Hörmander, L., Linear partial differential operators, (1963), Springer Berlin, Göttingen, Heidelberg · Zbl 0171.06802  Mizohata, S., The theory of partial differential equations, (1973), Cambridge University Press Cambridge · Zbl 0263.35001  Reissig, M., Hyperbolic equations with non-Lipschitz coefficients, Rend. sem. mat. univ. politec. Torino, 61, 135-181, (2003) · Zbl 1182.35155  Reissig, M., About strictly hyperbolic operators with non-regular coefficients, Pliska stud. math., 15, 105-130, (2003)  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
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