Tokyo Journal of Mathematics

On the Unique Solvability of Nonlinear Fuchsian Partial Differential Equations

Dennis B. BACANI, Jose Ernie C. LOPE, and Hidetoshi TAHARA

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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.

Article information

Tokyo J. Math., Volume 41, Number 1 (2018), 225-239.

First available in Project Euclid: 26 January 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35A01: Existence problems: global existence, local existence, non-existence 35A10: Cauchy-Kovalevskaya theorems
Secondary: 35A20: Analytic methods, singularities 35G20: Nonlinear higher-order equations


BACANI, Dennis B.; LOPE, Jose Ernie C.; TAHARA, Hidetoshi. On the Unique Solvability of Nonlinear Fuchsian Partial Differential Equations. Tokyo J. Math. 41 (2018), no. 1, 225--239.

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