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We consider semilinear partial differential equations in R n of the formwhere k and m are given positive integers. Relevant examples are semilinear Schrödinger equations − u + V(x)u = F(u), where the potential V(x) is given by an elliptic polynomial. We propose techniques, based on anisotropic generalizations of the global ellipticity condition of M. Shubin and multiparameter Picard type schemes in spaces of entire functions, which lead to new results for entire extensions and asymptotic behaviour of the solutions. Namely, we study solutions (eigenfunctions and homoclinics) in the framework of the Gel ′ fand-Shilov spaces S µ ν (R n ). Critical thresholds are identified for the indices µ and ν, corresponding to analytic regularity and asymptotic decay, respectively. In the one-dimensional case −u ′′ + V(x)u = F(u), our results for linear equations link up with those given by the classical asymptotic theory and by the theory of ODE in the complex domain, whereas for homoclinics, new phenomena concerning analytic extensions are described.

We show that all eigenfunctions of linear partial differential operators in $R^n$ with polynomial coefficients. We also show that under semilinear\ud polynomial perturbations all nonzero homoclinics keep the super-exponential decay of the above type,\ud whereas a loss of the holomorphicity occurs. Our estimates on homoclinics are sharp.\ud of Shubin type are extended to entire functions in $C^n$ of finite exponential type 2 and decay like $exp(−|z|2)$\ud for $|z|\to \infty$ in conic neighbourhoods of the form $|Im z| \leq |Re z|$

This paper concerns perturbations of smooth vector fields on T n (constant if n 3) with zeroth-order C ∞ and Gevrey G σ , σ 1, pseudodifferential operators. Simultaneous resonance is introduced and simultaneous resonant normal forms are exhibited (via conjugation with an elliptic pseudodifferential operator) under optimal simultaneous Diophantine conditions outside the resonances. In the C ∞ category the results are complete, while in the Gevrey category the effect of the loss of the Gevrey regularity of the conjugating operators due to Diophantine conditions is encountered. The normal forms are used to study global hypoellipticity in C ∞ and Gevrey G σ . Finally, the exceptional sets associated with the simultaneous Diophantine conditions are studied. A generalized Hausdorff dimension is used to give precise estimates of the 'size' of different exceptional sets, including some inhomogeneous examples.

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