Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line
نویسنده
چکیده
Studied here is the large-time behavior of solutions of the Korteweg-de Vries equation posed on the right half-line under the effect of a localized damping. Assuming as in [20] that the damping is active on a set (a0,+∞) with a0 > 0, we establish the exponential decay of the solutions in the weighted spaces L((x + 1)dx) for m ∈ N∗ and L(edx) for b > 0 by a Lyapunov approach. The decay of the spatial derivatives of the solution is also derived. MSC: Primary: 93D15, 35Q53; Secondary: 93B05.
منابع مشابه
Global Well-posedness for Kdv in Sobolev Spaces of Negative Index
The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H(R) for −3/10 < s.
متن کاملSharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case
We complete the known results on the Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in H−1(R) with a solution-map that is analytic from H−1(R) to C([0, T ];H−1(R)) whereas it is ill-posed in Hs(R), as soon as s < −1, in the sense that the flow-map u0 7→ u(t) cannot be continuous from H s(R) to even D′(R) at any fixed t > 0 small enough....
متن کاملPersistence of Solutions to Nonlinear Evolution Equations in Weighted Sobolev Spaces
In this article, we prove that the initial value problem associated with the Korteweg-de Vries equation is well-posed in weighted Sobolev spaces X s,θ, for s ≥ 2θ ≥ 2 and the initial value problem associated with the nonlinear Schrödinger equation is well-posed in weighted Sobolev spaces X s,θ, for s ≥ θ ≥ 1. Persistence property has been proved by approximation of the solutions and using a pri...
متن کاملThe low regularity global solutions for the critical generalized KdV equation
We prove that the Cauchy problem of the mass-critical generalized KdV equation is globally well-posed in Sobolev spaces Hs(R) for s > 6/13; we require that the mass is strictly less than that of the ground state in the focusing case. The main approach is the “I-method” together with some multilinear correction analysis. The result improves the previous works of Fonseca, Linares, Ponce (2003) an...
متن کاملSharp Global Well - Posedness for Kdv and Modified Kdv On
The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all L 2-based Sobolev spaces H s where local well-posedness is presently known, apart from the H 1 4 (R) endpoint for mKdV. The result for KdV relies on a new method for co...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010