Gevrey regularity of the solutions of inhomogeneous nonlinear partial differential equations

Abstract

In this article, we are interested in the Gevrey properties of the formal power series solutions in time of some inhomogeneous nonlinear partial differential equations with analytic coefficients at the origin of Cn+1. We systematically examine the cases where the inhomogeneity is s-Gevrey for any s≥0, in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy with respect to a nonnegative rational number sc fully determined by the Newton polygon of a convenient associated linear partial differential equation: for any s≥sc, the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s<sc, the formal solutions are generically sc-Gevrey. In the latter case, we give an explicit example in which the solution is s'-Gevrey for no s'<sc. As a practical illustration, we apply our results to the generalized Burgers-Korteweg-de Vries equation.