Abstract: This thesis deals with the qualitative and quantitative properties of solutions to some wave equations in dispersive or dispersive-dissipative media.
In the first part, we study the Cauchy problem for the generalized Benjamin-Ono equations. By means of gauge transforms combined with some harmonic analysis tools, we prove some local well-posedness results for initial data with minimal regularity in Sobolev spaces.
In the second part, we study the Cauchy problem for some dissipative versions of the Benjamin-Ono and Korteweg-de Vries equations. We show the influence of the dissipative effects and prove sharp well and ill-posedness results. This is obtained by working in suitable Bourgain's spaces, adapted to the dispersive-dissipative part of the equation.
Finally, we study the asymptotic behavior of solutions to the dissipative KdV equations. We explicitly compute the first terms of the asymptotic expansion in Sobolev spaces.
Keywords: Dispersive and dissipative equation, local and global existence, asymptotic behavior, gauge transform, Bourgain's spaces.