On totally characteristic type nonlinear partial differential equations in the complex domain
type: Item
instance of: scholarly article
title: On totally characteristic type nonlinear partial differential equations in the complex domain
zbMATH Open document ID: 0961.35002
author: Hua Chen
author: Hidetoshi Tahara
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2001-05-28T00:00:00Z
review text: The authors consider nonlinear singular partial differential equations \(t\partial_tu= F(t, x,\partial_xu)\). Their goal is to construct holomorphic solutions in a neighbourhood of the origin in \(\mathbb{C}^2\). The assumption \(F(0,x,0,0)\equiv 0\) implies that these solutions satisfy \(u(0,x)\equiv 0\) near \(x=0\). The right-hand side can be developed into the series \[ F(t,x,u,v)= \alpha(x)t+ \beta(x)u+ \gamma(x) v+ \sum_{p+ q+\alpha\geq 2}a_{p, q,\alpha}(x) t^p u^q v^\alpha. \] Conditions with respect to \(\gamma= \gamma(x)\) determine the solvability behaviour. The authors are interested in the totally characteristic case, this means, \(\gamma(x)\not\equiv 0\) under the additional assumption \(\gamma(x)= xc(x)\) with \(c(0)\neq 0\). If a non-resonance condition of the form \[ |i-\beta(0)- jc(0)|\geq \sigma(j+1)\quad \text{for any }(i,j)\in \mathbb{N}\times \mathbb{N}_0 \] is satisfied for some positive \(\sigma\), then the starting equation has a unique holomorphic solution in a neighbourhood of the origin in \(\mathbb{C}^2\). The non-resonance condition is useful for the construction of formal solutions. Their convergence is shown by the method of majorant power series. At the end of the paper the authors study a special class of higher-order totally characteristic partial differential equations. After defining a suitable symbol they formulate a kind of non-resonance condition for this symbol. This condition guarantees a unique holomorphic solution in a neighbourhood of the origin in \(\mathbb{C}^2\), too.
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35C10
Mathematics Subject Classification ID: 35A10
Mathematics Subject Classification ID: 35A20
zbMATH DE Number: 1437024
zbMATH Keywords: holomorphic solutions
zbMATH Keywords: non-resonance condition
zbMATH Keywords: method of majorant power series
reviewed by: Michael Reissig
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195143496
OpenAlex ID: W2000628362
cites work: Cauchy problems with characteristic initial hypersurface
cites work: Holomorphic and singular solutions of nonlinear singular first order partial differential equations
cites work: Singular and holomorphic solutions of singular nonlinear partial differential equations.
cites work: Q4379396
cites work: Q3868136
cites work: Nonlinear singular first order partial differential equations whose characteristic exponent takes a positive integral value
DOI: 10.2977/PRIMS/1195143496
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description: scientific article; zbMATH DE number 1437024