Global existence for the critical generalized KdV equation

Abstract

Abstract. We discuss results regarding global existence of solutions for the critical generalized Korteweg-de Vries equation,The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space H 1 (R), i.e. solutions corresponding to datawhere Q is the solitary wave solution of the equation.

Decay of Small Solutions for the Zakharov-Kuznetsov Equation posed on a half-strip

“…We call (1.1) a critical ZK equation by analogy with the critical KdV equation (1.2) for k = 4. It means that we could not prove the existence and uniqueness of global regular solutions without smallness restrictions for initial data similarly to the critical case for the KdV equation [8,10,11,24,31,32]. As far as the ZK equation is concerned, the results on both IVP and IBVP can be found in [5,6,8,27,28,30].…”

Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation

“…The method we use to prove Theorem 1.5 is that one developed in [9] and [10], which combines the smoothing effects for the solution of the linear problem with the iteration process introduced by Bourgain [5]. Since we are in the critical case, as in [10], controlling the L 2 -norm of the initial data could bring some difficult. Nevertheless, with a suitable decomposition of the initial data into low and high frequencies, we are able to handle this.…”

Global solutions for the generalized Benney-Lin equation posed on bounded intervals and on a half-line

“…In [4,13,21,38], Kuramoto-Sivashinsky type equations have been considered which included u xxx (KdV) term. What concerns (1.1) with k > 1, η = β = γ = 0; α = 1, called generalized Korteweg-de Vries equations, the Cauchy problem for (1.1) has been studied in [11,14,15,29,30], where it has been proved that for k = 4, called the critical case, the initial problem is well-posed for small initial data, whereas for arbitrary initial data, solutions may blow-up in a finite time. The generalized KdV equation was intensively studied in order to understand the interaction between the dispersive term and nonlinearity in the context of the theory of nonlinear dispersive evolution equations [16,17].…”