May/June 2015 On second order weakly hyperbolic equations with oscillating coefficients
Tamotu Kinoshita
Differential Integral Equations 28(5/6): 581-600 (May/June 2015). DOI: 10.57262/die/1427744102


We study well-posedness issues in Gevrey classes for the Cauchy problem for wave equations of the form $\partial_t^2 u -a(t) \partial_x^2 u=0$. In the strictly hyperbolic case $a (t)\geq c(> 0)$, Colombini, De Giorgi and Spagnolo have shown that it is sufficient to assume H$\ddot { \rm o}$lder 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 $ |a'(t)| \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.


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Tamotu Kinoshita. "On second order weakly hyperbolic equations with oscillating coefficients." Differential Integral Equations 28 (5/6) 581 - 600, May/June 2015.


Published: May/June 2015
First available in Project Euclid: 30 March 2015

zbMATH: 1340.35190
MathSciNet: MR3328135
Digital Object Identifier: 10.57262/die/1427744102

Primary: 35B30 , 35B44 , 35L15

Rights: Copyright © 2015 Khayyam Publishing, Inc.


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Vol.28 • No. 5/6 • May/June 2015
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