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On totally characteristic type nonlinear partial differential equations in the complex domain. (English) Zbl 0961.35002
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.

##### MSC:
 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 35C10 Series solutions to PDEs 35A10 Cauchy-Kovalevskaya theorems 35A20 Analyticity in context of PDEs
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