The “good” Boussinesq equation on the half-line

Himonas, A. Alexandrou; Mantzavinos, Dionyssios

doi:10.1016/j.jde.2015.01.005

Cited by 41 publications

(39 citation statements)

References 22 publications

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“…Our main theorem is below. Note that it extends the result in [12], which established well-posedness for s > 1 2 . In addition we prove that the nonlinear part of the solution is smoother that the initial data.…”

Section: Introductionsupporting

confidence: 82%

“…As we have already mentioned our result improves the result in [12]. The initial and boundary value problem (IVBP) for the "good" Boussinesq equation on the half line has also been considered in [18] and [19].…”

Section: Introductionsupporting

confidence: 59%

“…Moreover we prove that the nonlinear part of the solution on the half line is smoother than the initial data, obtaining half derivative smoothing of the nonlinear term in some cases. Our paper improves the result in [12], being the first result that constructs solutions for the initial and boundary value problem of the "good" Boussinesq equation below the L 2 space. Our theorems are sharp within the framework of the restricted norm method that we use and match the known results on the full line in [13] and [11].2010 Mathematics Subject Classification.…”

supporting

confidence: 53%

“…The uniqueness of the solutions thus constructed is not immediate since we do not know that the fixed points of the Duhamel operators have restrictions on the half line which are independent of the extension of the data. For the case of more regular data the uniqueness property of the solution is proved in [12]. For less regular data we take advantage of the smoothing estimate we establish in Theorem 1.1 to obtain uniqueness all the way down to the local theory threshold H − 1 4 + (R+) × H − 5 4 + (R+).…”

Section: Introductionmentioning

confidence: 99%

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Well-posedness and nonlinear smoothing for the “good” Boussinesq equation on the half-line

Compaan

Tzirakis

2017

Journal of Differential Equations

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Abstract. In this paper we study the regularity properties of the "good" Boussinesq equation on the half line. We obtain local existence, uniqueness and continuous dependence on initial data in low-regularity spaces. Moreover we prove that the nonlinear part of the solution on the half line is smoother than the initial data, obtaining half derivative smoothing of the nonlinear term in some cases. Our paper improves the result in [12], being the first result that constructs solutions for the initial and boundary value problem of the "good" Boussinesq equation below the L 2 space. Our theorems are sharp within the framework of the restricted norm method that we use and match the known results on the full line in [13] and [11].

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Section: Introductionsupporting

confidence: 82%

Section: Introductionsupporting

confidence: 59%

supporting

confidence: 53%

Section: Introductionmentioning

confidence: 99%

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Well-posedness and nonlinear smoothing for the “good” Boussinesq equation on the half-line

Compaan

Tzirakis

2017

Journal of Differential Equations

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“…This approach bypasses the absence of Fourier transform in the IBVP setting by utilizing the unified transform method (UTM) of Fokas for the explicit solution of forced linear evolution IBVPs [F1, F2]. The new approach has already been implemented for the nonlinear Schrödinger, the Korteweg-de Vries and the "good" Boussinesq equations on the half-line [FHM1,FHM2,HM]. These three IBVPs share two things in common: (i) they concern dispersive equations, and (ii) they are formulated on the half-line.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Initial-boundary value problems for a reaction-diffusion equation

Himonas

Mantzavinos

Yan

2019

Journal of Mathematical Physics

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A reaction-diffusion equation with power nonlinearity formulated either on the halfline or on the finite interval with nonzero boundary conditions is shown to be locally well-posed in the sense of Hadamard for data in Sobolev spaces. The result is established via a contraction mapping argument, taking advantage of a novel approach that utilizes the formula produced by the unified transform method of Fokas for the forced linear heat equation to obtain linear estimates analogous to those previously derived for the nonlinear Schrödinger, Korteweg-de Vries and "good" Boussinesq equations. Thus, the present work extends the recently introduced "unified transform method approach to well-posedness" from dispersive equations to diffusive ones. -boundary value problems, wellposedness in Sobolev spaces on the half-line and the finite interval, L 2 -boundedness of Laplace transform, unified transform method of Fokas. F. Weissler, Local existence and nonexistence for semilinear parabolic equations in L p . Indiana Univ. Math.

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