Exponential type Nagumo norms and summability of formal solutions of singular partial differential equations

In this paper, we study a class of first order nonlinear degenerated partial differential equations with singularity at $(t,x)=(0,0)\in {\text{CC}}^2$. By means of exponential type Nagumo norm approach, Gevrey asymptotic analysis extends to case of holomorphic parameters by a natural way. A sharp condition is then established to deduce the $k$-summability for the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.