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On a weakly hyperbolic equation with a term of order zero. (English) Zbl 0895.35057

Summary: We study a weakly hyperbolic equation of second order, supposing that the coefficients of the principal part are analytic and depend only on time. We prove that if we add a term of order zero \(c(t,x)u\), with \(c\in C(\mathbb{R}^{n+1})\), the Cauchy problem remains well-posed in \(C^\infty\).

MSC:

35L15 Initial value problems for second-order hyperbolic equations
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References:

[1] Colombini, F.), De Giorgi, E.) and Spagnolo, S.) .- Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps, Ann. Scuola Norm. Sup. Pisa, 6 (1979), pp. 511-559. · Zbl 0417.35049
[2] Colombini, F.), Jannelli, E.) and Spagnolo, S.) .- Well-posedness in the Gevrey classes of the Cauchy problem for a non-strictly hyperbolic equation with coefficients depending on time, Ann. Scuola Norm. Sup. Pisa, 10 (1983), pp. 291-312. · Zbl 0543.35056
[3] Hörmander, L.) .- Linear Partial Differential Operators, Springer, Berlin (1963). · Zbl 0108.09301
[4] Kajitani, K.) .- The well posed Cauchy problem for hyperbolic operators, Exposé au Séminaire Vaillant (1989).
[5] Oleinik, O.A.) .- On the Cauchy problem for weakly hyperbolic equations of second order, Comm. Pure Appl. Math.23 (1970), pp. 569-586.
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