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 \(s'<s_c\).

在本文中，我们对具有多项式半线性且解析系数位于\({\mathbb {C}}^{n+1 }\)。我们特别证明方程的不均匀性和形式解对于任何\(s\ge s_c\)都是s -Gevrey ，其中\(s_c\)是一个非负有理数，完全由相关联的牛顿多边形确定线性偏微分方程。在相反的情况下\(s<s_c\)，我们表明解决方案一般是\(s_c\) -Gevrey 而不均匀性是s-Gevrey，我们给出了一个明确的例子，其中解决方案是\(s'\) -Gevrey 没有\(s'<s_c\)。