### Biholomorphic maps with linear parts having Jordan blocks: linearization and resonance type phenomena.

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This paper is devoted to the effective stability estimates (of Nekhoroshev’s type) of the billiard flow for strictly convex bounded domains with analytic boundaries in any dimensions. The main result is that any billiard trajectory with initial data which are $\delta $ - close to the glancing manifold remains close to the glancing manifold in an exponentially large time interval with respect to $1/\delta $. The proof is based on a normal form of the billiard ball map in Gevrey classes. More generally, we prove effective...

Vengono considerate equazioni alle derivate parziali semilineari con caratteristiche multiple. Si studia in particolare la loro risolubilità locale e la buona positura del problema di Cauchy nell'ambito delle classi di Gevrey.

We study the simultaneous linearizability of $d$–actions (and the corresponding $d$-dimensional Lie algebras) defined by commuting singular vector fields in ${\u2102}^{n}$ fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of $d$ vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators occur, then...

We study the Gevrey regularity down to t = 0 of solutions to the initial value problem for a semilinear heat equation ${\partial}_{t}u-\Delta u={u}^{M}$. The approach is based on suitable iterative fixed point methods in ${L}^{p}$ based Banach spaces with anisotropic Gevrey norms with respect to the time and the space variables. We also construct explicit solutions uniformly analytic in t ≥ 0 and x ∈ ℝⁿ for some conservative nonlinear terms with symmetries.

We investigate the propagation of the uniform spatial Gevrey ${G}^{\sigma}$, σ ≥ 1, regularity for t → +∞ of solutions to evolution equations like generalizations of the Euler equation and the semilinear Schrödinger equation with polynomial nonlinearities. The proofs are based on direct iterative arguments and nonlinear Gevrey estimates.

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