Differential and Integral Equations

On second order weakly hyperbolic equations with oscillating coefficients

Tamotu Kinoshita

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Abstract

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.

Article information

Source
Differential Integral Equations, Volume 28, Number 5/6 (2015), 581-600.

Dates
First available in Project Euclid: 30 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.die/1427744102

Mathematical Reviews number (MathSciNet)
MR3328135

Zentralblatt MATH identifier
1340.35190

Subjects
Primary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35B44: Blow-up 35L15: Initial value problems for second-order hyperbolic equations

Citation

Kinoshita, Tamotu. On second order weakly hyperbolic equations with oscillating coefficients. Differential Integral Equations 28 (2015), no. 5/6, 581--600. https://projecteuclid.org/euclid.die/1427744102


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