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June 2018 On the Unique Solvability of Nonlinear Fuchsian Partial Differential Equations
Dennis B. BACANI, Jose Ernie C. LOPE, Hidetoshi TAHARA
Tokyo J. Math. 41(1): 225-239 (June 2018). DOI: 10.3836/tjm/1502179268

Abstract

We consider a singular nonlinear partial differential equation of the form $$ (t\partial_t)^mu= F \Bigl( t,x,\bigl\{(t\partial_t)^j \partial_x^{\alpha}u \bigr\}_{(j,\alpha) \in I_m} \Bigr) $$ with arbitrary order $m$ and $I_m=\{(j,\alpha) \in \mathbb{N} \times \mathbb{N}^n \,;\, j+|\alpha| \leq m, j<m \}$ under the condition that $F(t,x,\{z_{j,\alpha} \}_{(j,\alpha) \in I_m})$ is continuous in $t$ and holomorphic in the other variables, and it satisfies $F(0,x,0) \equiv 0$ and $(\partial F/\partial z_{j,\alpha})(0,x,0) \equiv 0$ for any $(j,\alpha) \in I_m \cap \{|\alpha|>0 \}$. In this case, the equation is said to be a nonlinear Fuchsian partial differential equation. We show that if $F(t,x,0)$ vanishes at a certain order as $t$ tends to $0$ then the equation has a unique solution with the same decay order.

Citation

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Dennis B. BACANI. Jose Ernie C. LOPE. Hidetoshi TAHARA. "On the Unique Solvability of Nonlinear Fuchsian Partial Differential Equations." Tokyo J. Math. 41 (1) 225 - 239, June 2018. https://doi.org/10.3836/tjm/1502179268

Information

Published: June 2018
First available in Project Euclid: 26 January 2018

zbMATH: 06966866
MathSciNet: MR3830816
Digital Object Identifier: 10.3836/tjm/1502179268

Subjects:
Primary: 35A01, 35A10
Secondary: 35A20, 35G20

Rights: Copyright © 2018 Publication Committee for the Tokyo Journal of Mathematics

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Vol.41 • No. 1 • June 2018
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