Semilinear evolution equations in Frechet spaces, Quaderno 23/1999, Dipartimento di Matematica dell'Universita' di Milano

This work is devoted to the Cauchy problem for a class of semilinear evolution equations in abstract Frechet spaces. Existence of the solution is derived with the Peano-Picard method; uniqueness and smoothness of the flow are also proved. Criteria are given for the existence of global solutions, defined for all positive times. Also, approximate solutions and the related errors are discussed. The proposed framework can be applied to the nonlinear Schroedinger, Klein-Gordon and heat equations, formulated in appropriate spaces of smooth functions; the general theory controls the derivatives of any order of the solutions.