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Unique continuation and decay for the Korteweg-de Vries equation with localized damping

Ademir Fernando Pazoto (2005)

ESAIM: Control, Optimisation and Calculus of Variations

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...

Unique continuation and decay for the Korteweg-de Vries equation with localized damping

Ademir Fernando Pazoto (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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...

Uniqueness of the boundary behavior for large solutions to a degenerate elliptic equation involving the ∞-Laplacian.

Gregorio Díaz, Jesús Ildefonso Díaz (2003)

RACSAM

En esta nota estimamos la tasa máxima de crecimiento en la frontera de las soluciones de viscosidad de -Δ∞u + λ|u|m-1u = f en Ω (λ > 0, m > 3).De hecho, mostramos que sólo hay una única tasa de explosión en la frontera para esas soluciones explosivas. También obtenemos una versión del Teorema de Liouville para el caso Ω = RN.

Universal solutions of a nonlinear heat equation on N

Thierry Cazenave, Flávio Dickstein, Fred B. Weissler (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper, we study the relationship between the long time behavior of a solution u ( t , x ) of the nonlinear heat equation u t - Δ u + | u | α u = 0 on N (where α > 0 ) and the asymptotic behavior as | x | of its initial value u 0 . In particular, we show that if the sequence of dilations λ n 2 / α u 0 ( λ n · ) converges weakly to z ( · ) as λ n , then the rescaled solution t 1 / α u ( t , · t ) converges uniformly on N to 𝒰 ( 1 ) z along the subsequence t n = λ n 2 , where 𝒰 ( t ) is an appropriate flow. Moreover, we show there exists an initial value U 0 such that the set of all possible z attainable in this...

Upper Hausdorff dimension estimates for invariant sets of evolutionary systems on Hilbert manifolds

Kruck, Amina, Reitmann, Volker (2017)

Proceedings of Equadiff 14

We prove a generalization of the Douady-Oesterlé theorem on the upper bound of the Hausdorff dimension of an invariant set of a smooth map on an infinite dimensional manifold. It is assumed that the linearization of this map is a noncompact linear operator. A similar estimate is given for the Hausdorff dimension of an invariant set of a dynamical system generated by a differential equation on a Hilbert manifold.

Variable depth KdV equations and generalizations to more nonlinear regimes

Samer Israwi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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...

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