Korteweg-de Vries Equation in Bounded Domains

where μ, ν are positive constants. This equation, in the case μ = 0, was derived independently by Sivashinsky [1] and Kuramoto [2] with the purpose to model amplitude and phase expansion of pattern formations in different physical situations, for example, in the theory of a flame propagation in turbulent flows of gaseous combustible mixtures, see Sivashinsky [1], and in the theory of turbulence of wave fronts in reaction-diffusion systems, Kuramoto [2]. The generalized KdV-KS equation (1.1) arises in modeling of long waves in a viscous fluid flowing down on an inclined plane. When ν = 0, we have the KdV equation studied by various authors [6-12]. From the mathematical point of a view, the history of the KdV equation is much longer than the one of the KS equation. Well-posedness of the Cauchy problem for the KdV equation in various classes of solutions was studied in [6-9]. Solvability of mixed problems for the KdV equation and for the KdV equation with dissipation in bounded domains studied Bubnov [11], Hublov [12], see also