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On second order weakly hyperbolic equations with oscillating coefficients. (English) Zbl 1340.35190

Summary: We study well-posedness issues in Gevrey classes for the Cauchy problem for wave equations of the form \(\partial_t^2u-a(t)\partial_x^2u=0\). In the strictly hyperbolic case \(a(t)\geq c(> 0)\), Colombini, De Giorgi and Spagnolo have shown that it is sufficient to assume Hölder regularity of the coefficients in order to prove Gevrey well-posedness. Recently, assumptions bearing on the oscillations of the coefficient have been imposed in the literature in order to guarantee well-posedness. In the weakly hyperbolic case \(a(t) \geq 0\), Colombini, Jannelli and Spagnolo proved well-posedness in Gevrey classes of order \(1\leq s < s_0=1+(k+\alpha)/2\). In this paper, we put forward condition \(\left| a'(t)\right| \leq Ca(t)^p/t^q\) that bears both on the oscillations and the degree of degeneracy of the coefficient. We show that under such a condition, Gevrey well-posedness holds for \(1\leq s < qs_0/\{k+\alpha)(1-p)+q-1\}\) if \(q\geq 3-2p\). In particular, this improves on the result (corresponding to the case \(p=0\)) of Colombini, Del Santo and Kinoshita.

MSC:

35L15 Initial value problems for second-order hyperbolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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