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June 2013 Remarks on Formal Solution and Genuine Solutions for Some Nonlinear Partial Differential Equations
Hiroshi YAMAZAWA
Tokyo J. Math. 36(1): 215-237 (June 2013). DOI: 10.3836/tjm/1374497521

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 $S$.

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Hiroshi YAMAZAWA. "Remarks on Formal Solution and Genuine Solutions for Some Nonlinear Partial Differential Equations." Tokyo J. Math. 36 (1) 215 - 237, June 2013. https://doi.org/10.3836/tjm/1374497521

Information

Published: June 2013
First available in Project Euclid: 22 July 2013

zbMATH: 1276.35054
MathSciNet: MR3112385
Digital Object Identifier: 10.3836/tjm/1374497521

Subjects:
Primary: 34M30
Secondary: 34M25

Rights: Copyright © 2013 Publication Committee for the Tokyo Journal of Mathematics

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Vol.36 • No. 1 • June 2013
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