Periodic problem for a model nonlinear evolution equation
The paper is a study of the asymptotic behavior for large time of the solutions of the initial problem for a nonlinear and, in general, nonlocal evolution equation in the interval \([0,2\pi]\) with periodic boundary conditions. Both the linear and the nonlinear part of the equation are pseudodifferential operators defined by the Fourier series. The model equation includes many equations of mathematical physics related to wave processes in periodic media, namely, among others, the Fisher, the Korteweg-de Vries-Burgers, the Kuramoto-Sivashinsky, the Landau-Ginzburg and the derivative nonlinear Schrödinger partial differential equations, as well as non-local PDE's as the Otto-Sudan-Ostrovsky equation. Unlike some previous work by some of the authors, the approach used here to prove existence and uniqueness of solutions and to obtain asymptotic formulas for the solutions is based in a ``variation of constants'' integral equation and the contraction mapping principle. Under dissipation conditions on the linear term and bounding estimates on the nonlinearities, exponential decay is proven for small initial data whereas a further symmetry condition on the nonlinear term allows to obtain global existence and large time asymptotics of the solutions for large initial data, via an ``energy inequality''. Depending on the linear part and on the structure of the nonlinearities, the solutions can exponentially decay with time, grow or oscillate.
scientific article; zbMATH DE number 1983412