We prove that the Cauchy problem for a hyperbolic, homogeneous equation with C ∞ coefficients depending on time, is well posed in every Gevrey class, although in general it is not well-posed in C ∞ , provided the characteristic roots satisfy the condition
We consider the wave propagating in the Einstein and de Sitter space-time. The covariant d’Alembert’s operator in the Einstein and de Sitter space-time belongs to the family of the non-Fuchsian partial differential operators. We introduce the initial value problem for this equation and give the explicit representation formulas for the solutions. We also show the Lp−Lq estimates for solutions.
We study the Cauchy Problem for a hyperbolic system with multiple characteristics and non-smooth coefficients depending on time. We prove in particular that, if the leading coefficients are a-Ho¨lder continuous, and the system has size mp3; then the Problem is well posed in each Gevrey class of exponent so1 þ a=m: r 2004 Elsevier Inc. All rights reserved.
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