### Risolubilità e ipoellitticità per equazioni differenziali a derivate parziali semilineari con caratteristiche multiple

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We prove local solvability in Gevrey spaces for a class of semilinear partial differential equations. The linear part admits characteristics of multiplicity k ≥ 2 and data are fixed in ${G}^{\sigma}$, 1 < σ < k/(k-1). The nonlinearity, containing derivatives of lower order, is assumed of class ${G}^{\sigma}$ with respect to all variables.

We propose a unified approach, based on methods from microlocal analysis, for characterizing the hypoellipticity and the local solvability in ${C}^{\mathrm{\infty}}$ and Gevrey ${G}^{\lambda}$ classes of semilinear anisotropic partial differential operators with Gevrey nonlinear perturbations, in dimension $n\ge 3$. The conditions for our results are imposed on the sign of the lower order terms of the linear part of the operator, see Theorem 1.1 and Theorem 1.3 below.

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