On totally characteristic type nonlinear partial differential equations in the complex domain
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.
scientific article; zbMATH DE number 1437024