Well-posedness for Fractional Growth-Dissipative Benjamin-Ono Equations

Abstract

This paper is devoted to study the Cauchy problem for the fractional growth-dissipative BO equations ut + Huxx − (D α x − D β x )u + uux = 0. For a wide class of parameter β > 1 and 0 < α < β, taking into account dispersive and dissipative effects, we establish sharp well-posedness results in Sobolev spaces H s (R) and H s (T) which yield new well-posedness conclusions for some physical relevant equations. In addition, we study the behavior of solutions as α → β.