Journal of the Mathematical Society of Japan

Unique solvability of some nonlinear partial differential equations with Fuchsian and irregular singularities

Dennis B. BACANI and Hidetoshi TAHARA

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The paper considers nonlinear partial differential equations of the form $t (\partial u/\partial t) = F(t,x,u, \partial u/\partial x)$, with independent variables $(t,x) \in \mathbb{R} \times \mathbb{C}$, and where $F(t,x,u,v)$ is a function continuous in $t$ and holomorphic in the other variables. It is shown that the equation has a unique solution in a sectorial domain centered at the origin under the condition that $F(0,x,0,0)=0$, Re$F_u(0,0,0,0)$ < 0, and $F_v(0,x,0,0)=x^{p+1}\gamma(x)$, where $\gamma(0) \neq 0$ and $p$ is any positive integer. In this case, the equation has a Fuchsian singularity at $t=0$ and an irregular singularity at $x=0$.

Article information

J. Math. Soc. Japan, Volume 66, Number 3 (2014), 1017-1042.

First available in Project Euclid: 24 July 2014

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Zentralblatt MATH identifier

Primary: 35A01: Existence problems: global existence, local existence, non-existence
Secondary: 35A10: Cauchy-Kovalevskaya theorems 35A20: Analytic methods, singularities 35F20: Nonlinear first-order equations

nonlinear partial differential equations Fuchsian singularity irregular singularity existence and uniqueness of solutions


BACANI, Dennis B.; TAHARA, Hidetoshi. Unique solvability of some nonlinear partial differential equations with Fuchsian and irregular singularities. J. Math. Soc. Japan 66 (2014), no. 3, 1017--1042. doi:10.2969/jmsj/06631017.

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