On the Cauchy problem for nonlinear evolution equations and regularity of solutions

Abstract

In some previous works a generalized implicit function theorem of Nash-Moser type has been applied to prove the local well posedness for the Cauchy problem for several types of nonlinear evolution equations. For istance, applications of this new method have been given to ordinary differential equations in Fréchet spaces,to nonlinear parabolic partial differential equations and to some specific nonlinear Schrödinger type equations. All these results take $C¹([a,b],E)$ as the basic function space where E is a general Fréchet space or $E=H^∈fty({R}ⁿ)$, respectively. The purpose of this note is to show that a similar approach based on Nash-Moser techniques works with the function space $C^∈fty([a,b],E)$ as well providing the existence of $C^∈fty$-solutions smoothhly depending on the initial value. In particular, sufficient conditions on E are given suche that $C^∈fty ([a,b],E)$ satisfies a required smoothing property.