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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 \(u\in\bigcap^2_{j=0} C^j([0, T], H^{s+2-j})\).

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
Full Text: DOI
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