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 formal power series solution time partial differential equations with a polynomial semilinearity and analytic coefficients at origin $${\mathbb {C}}^{n+1}$$ . We prove particular that inhomogeneity equation together s-Gevrey for any $$s\ge s_c$$ , where $$s_c$$ is nonnegative rational number fully determined by Newton polygon associated linear PDE. opposite case $$s<s_c$$ show generically -Gevrey while s-Gevrey, give an explicit example which $$s'$$ no $$s'<s_c$$