Remarks on Formal Solution and Genuine Solutions for Some Nonlinear Partial Differential Equations

Abstract

Ōuchi ([2], [3]) found a formal solution $\widetilde{u}(t,x)=\sum_{k\geq 0}u_k(x)t^k$ with \begin{equation*} |u_k(x)|\leq AB^k\Gamma\Bigg(\frac{k}{\gamma_*}+1\Bigg) \quad 0<\gamma_*\leq\infty \end{equation*} for some class of nonlinear partial differential equations. For these equations he showed that there exists a genuine solution $u_{S}(t,x)$ on a sector $S$ with asymptotic expansion $u_{S}(t,x)\sim \widetilde{u}(t,x)$ as $t\rightarrow 0$ in the sector $S$. These equations have polynomial type nonlinear terms. In this paper we study a similar class of equations with the following nonlinear terms \begin{equation*} \sum_{|q|\geq1}t^{\sigma_q}c_q(t,x)\prod_{j+|\alpha|\leq m} \Bigg\{\Bigg(t\frac{\partial}{\partial t}\Bigg)^j\Bigg(\frac{\partial}{\partial x}\Bigg)^{\alpha}u(t,x)\Bigg\}^{q_{j,\alpha}}\,. \end{equation*} It is main purpose to get a solvability of the equation in a category $u_S(t,x)\sim 0$ as $t\rightarrow 0$ in a sector $S$. We give a proof by the method that is a little different from that in [3]. Further we give a remark that the similar class of equations has a genuine solution $u_{S}(t,x)$ with $u_{S}(t,x)\sim \widetilde{u}(t,x)$ as $t\rightarrow 0$ in the sector $S$.