Cappiello M. - Evolution operators on Gevrey and Gelfand-Shilov spaces - Bando "Grant for Internationalization - GFI" - 2022

Introduction. Gevrey classes are among the most popular function spaces in the study of partial differential equations. They appeared for the first time in the analysis of solutions of the heat equation and acquired more and more relevance along the years. They form a scale of spaces between the class of analytic functions and the class of smooth (C^∞) functions. For this reason, it is extremely interesting to understand if and to what extent a certain property, which holds for one of the two classes and not for the other, extends to Gevrey spaces. Another class of function spaces strictly related to Gevrey spaces is represented by Gelfand-Shilov spaces of type S, introduced in [14] as an alternative to the Schwartz space of smooth and rapidly decreasing functions for the study of partial differential equations. Broadly speaking, elements of these spaces are Gevrey functions which exhibit also an exponential decay at infinity. The Fourier transform has convenient mapping properties on these spaces and this makes them a suitable functional setting for Fourier and microlocal analysis. Pseudo-differential and Fourier integral operators have been studied on these spaces. Thanks to these tools, several results concerning well-posedness of initial value problems and propagation of singularities for hyperbolic equations have been proved in Gevrey and Gelfand-Shilov spaces, see for instance [5,9,10,11,16]. However, for other classes of evolution equations, the above-mentioned problems have been only partially explored. For instance, this is the case of the so-called p-evolution equations. The Cauchy problem for p-evolution equations: state of art. The class of p-evolution equations, p being a positive integer, is a wide class of linear and semilinear equations involving operators L whose principal part is of the form D_t+a_p(t)P(x,D_x), where D =-i∂, a_p(t) is a real-valued coefficient and P is a differential operator of order p. This class includes as particular cases strictly hyperbolic equations (p=1), Schrödinger-type equations (p=2), dispersive equations such as Korteweg-de Vries equation and its generalizations (p=3). A particularly interesting case is when the lower order terms of L have complex-valued coefficients. In this case, it is necessary to assume suitable decay conditions for |x| tending to infinity on the imaginary parts of these coefficients to obtain well-posedness of the Cauchy problem in Sobolev spaces, in general with a loss of derivatives with respect to the initial data. In the last thirty years, several authors developed a method for studying this class of equations. This method is based on suitable changes of variables defined in terms of pseudo-differential operators, on a suitable partition of the phase space and on an iterative application of lower bound estimates such as sharp Gårding and Fefferman-Phong inequalities. Thanks to these tools, it has been possible to study the well-posedness of the Cauchy problem in Sobolev spaces [3] and, only for p=2,3, in Gevrey spaces [1,12,15]. Also the effect of the presence of initial data with a suitable decay at infinity on the solution has been investigated, see [2,6,7]. Roughly speaking, the choice of initial data which decay at infinity allows to obtain solutions which do not lose regularity with respect to the data. Finally, necessary conditions for the well-posedness of the Cauchy problem in Sobolev spaces have been stated in [4] for arbitrary p, whereas for Gevrey spaces, the only results concern the case of Schrödinger-type equations (p=2), see [13]. Main objective of the project and problems proposed. The aim of this project is to address the study of the Cauchy problem for p-evolution equations with initial data in Gevrey and Gelfand-Shilov spaces. Namely, our purpose is to focus on some of the following items: a) Linear p-evolution equations for any p>3; b) Semilinear p-evolution equations (with