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The Neumann problem for nonstationary systems in tubular domains. (English. Russian original) Zbl 0898.35020
Math. Notes 60, No. 4, 475-478 (1996); translation from Mat. Zametki 60, No. 4, 631-634 (1996).
We consider the system \[ (-1)^{m-1} \left[\sum^m_{| p|,| q|=1} D^p \bigl(a_{pq} (x,t)D^q u\bigr) +\sum^m_{| p|=1} a_p(x,t) D^pu+ a(x,t)u \right]- u_{tt} =f, \] where \((-1)^{| p| +| q|} \overline a_{pq} =a_{qp}\), \(a_p\), and \(a\) are bounded measurable complex-valued \(r\times r\) matrices on the tubular domain \(Q_T= \{(x,t): x\in \Omega\), \(t\in [0,T]\}\), and \(\Omega \subset \mathbb{R}^n\) is a bounded domain whose boundary satisfies the Lipschitz condition. We prove smoothness with respect to time of generalized solutions.
MSC:
35D05 Existence of generalized solutions of PDE (MSC2000)
35L55 Higher-order hyperbolic systems
35L20 Initial-boundary value problems for second-order hyperbolic equations
74B05 Classical linear elasticity
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