Unique solvability of some nonlinear partial differential equations with Fuchsian and irregular singularities. (English) Zbl 1302.35010

Summary: 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 \(F(0,x,0,0)=0\), \(\mathrm{Re} F_u(0,0,0,0) < 0\), 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 \(x=0\).


35A20 Analyticity in context of PDEs
35A10 Cauchy-Kovalevskaya theorems
35F20 Nonlinear first-order PDEs
Full Text: DOI Euclid


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