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This work is devoted to prove the exponential decay for the energy of solutions of the Korteweg-de Vries equation in a bounded interval with a localized damping term. Following the method in Menzala (2002) which combines energy estimates, multipliers and compactness arguments the problem is reduced to prove the unique continuation of weak solutions. In Menzala (2002) the case where solutions vanish on a neighborhood of both extremes of the bounded interval where equation holds was solved combining...
This work is devoted to prove the exponential decay for the energy
of solutions of the Korteweg-de Vries equation in a bounded interval
with a localized damping term. Following the method in Menzala (2002)
which combines energy estimates, multipliers and compactness
arguments the problem is reduced to prove the unique continuation of
weak solutions. In Menzala (2002) the case where solutions vanish on a
neighborhood of both extremes of the bounded interval where equation
holds was solved combining...
We study here the water waves problem for uneven bottoms in a highly nonlinear regime where
the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known
that, for such regimes, a generalization of the KdV equation (somehow linked to
the Camassa-Holm equation) can be derived and justified [Constantin and Lannes,
Arch. Ration. Mech. Anal. 192 (2009) 165–186] when the bottom is
flat. We generalize here this result
with a new class of equations taking into account...
The existence of a traveling wave with special properties to modified KdV and BKdV equations is proved. Nonlinear terms in the equations are defined by means of a function f of an unknown u satisfying some conditions.
In this paper, travelling wave solutions for the Zakharov equation in plasmas with power law nonlinearity are studied by using the Weierstrass elliptic function method. As a result, some previously known solutions are recovered, and at the same time some new ones are also given.
In this paper, we study a class of Initial-Boundary Value Problems proposed by Colin and Ghidaglia for the Korteweg-de Vries equation posed on a bounded domain (0,L). We show that this class of Initial-Boundary Value Problems is locally well-posed in the classical Sobolev space Hs(0,L) for s > -3/4, which provides a positive answer to one of the open questions of Colin and Ghidaglia [Adv. Differ. Equ. 6 (2001) 1463–1492].
Evolution equations featuring nonlinearity, dispersion and
dissipation are considered here. For classes of such equations
that include the Korteweg-de Vries-Burgers equation and the
BBM-Burgers equation, the zero dissipation limit is studied.
Uniform bounds independent of the dissipation coefficient are derived
and zero dissipation limit results with
optimal convergence rates are established.
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