Inhomogeneous Gevrey ultradistributions and Cauchy problem

Daniela Calvo ; L. Rodino

Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 31 (2006) no. 1.

Résumé

After a short survey on Gevrey functions and ultradistributions, we present the inhomogeneous Gevrey ultradistributions introduced recently by the authors in collaboration with A. Morando, cf. [7]. Their definition depends on a given weight function $\lambda$ , satisfying suitable hypotheses, according to Liess-Rodino [16]. As an application, we define $(s,\lambda)$-hyperbolic partial differential operators with constant coefficients (for $s>1$), and prove for them the well-posedness of the Cauchy problem in the frame of the corresponding inhomogeneous ultradistributions. This sets in the dual spaces a similar result of Calvo [4] in the inhomogeneous Gevrey classes, that in turn extends a previous result of Larsson [14] for weakly hyperbolic operators in standard homogeneous Gevrey classes.