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

Abstract

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 $${\mathbb {C}}^{n+1}$$ . We prove in particular that the inhomogeneity of the equation and the formal solution are together s-Gevrey for any $$s\ge s_c$$ , where $$s_c$$ 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 $$s'<s_c$$ .