On the Radius of Spatial Analyticity for the Quartic Generalized KdV Equation

Selberg, Sigmund; Tesfahun, Achenef

doi:10.1007/s00023-017-0605-y

Cited by 31 publications

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References 14 publications

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“…It was applied to the on line KdV equation in improving an earlier result of Bona et al. , to the dispersion‐generalized periodic KdV equation in and to the quartic generalized KdV equation on the line in .…”

Section: Resultsmentioning

confidence: 99%

“…The method used here for proving lower bounds on the radius of analyticity was introduced in [18] in the context of the 1D Dirac-Klein-Gordon equations. It was applied to the on line KdV equation in [16] improving an earlier result of Bona et al [2], to the dispersion-generalized periodic KdV equation in [8] and to the quartic generalized KdV equation on the line in [17].…”

Section: )mentioning

confidence: 97%

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On the radius of spatial analyticity for the modified Kawahara equation on the line

Petronilho

Silva

2019

Mathematische Nachrichten

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First, by using linear and trilinear estimates in Bourgain type analytic and Gevrey spaces, the local well‐posedness of the Cauchy problem for the modified Kawahara equation on the line is established for analytic initial data u0false(xfalse) that can be extended as holomorphic functions in a strip around the x‐axis. Next we use this local result and a Gevrey approximate conservation law to prove that global solutions exist. Furthermore, we obtain explicit lower bounds for the radius of spatial analyticity r(t) given by rfalse(tfalse)≥ct−(4+δ), where δ>0 can be taken arbitrarily small and c is a positive constant.

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Section: Resultsmentioning

confidence: 99%

Section: )mentioning

confidence: 97%

On the radius of spatial analyticity for the modified Kawahara equation on the line

Petronilho

Silva

2019

Mathematische Nachrichten

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“…The main ingredients in our proof are almost conservation law for the solution to the KdV equation in spaces of analytic functions and spacetime dyadic bilinear estimates associated with the KdV equation. For similar studies for the Dirac-Klein-Gordon system, generalized KdV and cubic NLS see [32,16,31,34]. For studies on related issues for nonlinear partial differential equations see for instance [5,6,7,9,14,17,8,15,18,28,19,29,26].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic lower bound for the radius of spatial analyticity to solutions of KdV equation

Tesfahun

2019

Commun. Contemp. Math.

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It is shown that the uniform radius of spatial analyticity σ(t ) of solutions at time t to the KdV equation cannot decay faster than |t | −4/3 as |t | → ∞ given initial data that is analytic with fixed radius σ 0 . This improves a recent result of Selberg and Da Silva, where they proved a decay rate of |t | −(4/3+ε) for arbitrarily small positive ε. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear L 2 estimates associated with the KdV equation. 1991 Mathematics Subject Classification. 35Q53; 35L30. Key words and phrases. KdV equation; Radius of spatial analticity; Asymptotic lower bound .1 4.3. Local well-posedness in Gevery class. Define the map Φ(u)(t ) = χ(t /4)W (t ) f + B(u, u)(t ). Let s ∈ [−3/4, 0] and J = [−1, 1]. By Lemma 7(ii), (iii) and Corollary 2 we have Φ(u) X σ,s J

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“…Finally, we mention some references devoted to the uniform radius of analyticity for other partial differential equations. We refer the readers to [4,25,32] for generalized KdV equations, to [3,5,33] for Schrödinger equations, to [17,18,19] for Euler equations, to [12,28,31] for Klein-Gordon equations, and to [8] for the cubic Szegő equation.…”

Section: Introductionmentioning

confidence: 99%

New lower bounds on the radius of spatial analyticity for the KdV equation

Huang

Wang

2019

Journal of Differential Equations

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The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than t −1/4 as time t goes to infinity. This improves the works [Selberg, da Silva, Lower bounds on the radius of spatial analyticity for the KdV equation, Annales Henri Poincaré, 2017, 18(3): 1009-1023] and [Tesfahun, Asymptotic lower bound for the radius of spatial analtyicity to solutions of KdV equation, arXiv preprint arXiv:1707.07810, 2017]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspired by the I−method.

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On the Radius of Spatial Analyticity for the Quartic Generalized KdV Equation

Abstract: Lower bound on the rate of decrease in time of the uniform radius of spatial analyticity of solutions to the quartic generalized KdV equation is derived, which improves an earlier result by Bona, Grujić and Kalisch.

Cited by 31 publications

References 14 publications

On the radius of spatial analyticity for the modified Kawahara equation on the line

On the radius of spatial analyticity for the modified Kawahara equation on the line

Asymptotic lower bound for the radius of spatial analyticity to solutions of KdV equation

New lower bounds on the radius of spatial analyticity for the KdV equation

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