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Notes on Boussinesq Equations: Felipe Linares
The purpose of these notes is to give a brief introduction to nonlinear dispersive equations and the issues regarding solutions of these equations.
We have chosen the well know Boussinesq equation utt ¡ uxx + uxxxx + (u2)xx = 0 (0.1) and one of its generalizations utt ¡ uxx + uxxxx + (Ã(u))xx = 0 (0.2) as models to present the theory.
We will present techniques to deal with problems regarding properties of solutions to the associated initial value problem (IVP). We will discuss the so-called smoothing e®ect properties of solutions of the linear IVP.
Then we will use them to obtain local and global results for solutions of the nonlinear IVP. We next will study the regularity of these solutions.
We will also comment on results regarding decay and nonlinear scattering.
Finally we will discuss some blow-up results. The material presented here is an ampli¯ed version of the contents of the articles [31], [32] and [2].
We have included an appendix containing basic facts from Fourier Analysis, such as, Fourier transform, interpolation, Sobolev spaces and the fractional integral theorem. Most of the content of the appendix was taken from the notes by F. Linares and G. Ponce [33].
These notes were prepared to give an introductory minicourse on top- ics related to nonlinear dispersive equations in the Ponti¯cia Universidad Catolica (PUC) de Lima, Peru on June 2005.