We introduce some general classes of pseudodifferential operators with symbols admitting exponential type growth at infinity and we prove mapping properties for these operators on Gelfand–Shilov spaces. Moreover, we deduce composition and certain invariance properties of these classes.

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We consider the Cauchy problem for linear and quasilinear symmetrizable hyperbolic systems with coefficients depending on time and space, not smooth in t and growing at infinity with respect to x. We discuss well-posedness in weighted Sobolev spaces, showing that the non-Lipschitz regularity in t has an influence not only on the loss of derivatives of the solution but also on its behaviour for |x| → ∞. We provide examples to prove that the latter phenomenon cannot be avoided.

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.