Gevrey regularity of the solutions of the inhomogeneous partial differential equations with a polynomial semilinearity

In this article, we are interested in the Gevrey properties of the formal power series solution in time of the partial differential equations with a polynomial semilinearity and with analytic coefficients at the origin of Cn+1. We prove in particular that the inhomogeneity of the equation and the formal solution are together s-Gevrey for any s >= s(c), where sc is a nonnegative rational number fully determined by the Newton polygon of the associated linear PDE. In the opposite case s < s(c), we show that the solution is generically s(c)-Gevrey while the inhomogeneity is s-Gevrey, and we give an explicit example in which the solution is s'-Gevrey for no s' < s(c).