Abstract
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 $x=0$.
Citation
Dennis B. BACANI. Hidetoshi TAHARA. "Unique solvability of some nonlinear partial differential equations with Fuchsian and irregular singularities." J. Math. Soc. Japan 66 (3) 1017 - 1042, July, 2014. https://doi.org/10.2969/jmsj/06631017
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