The IVP for the Benjamin–Ono equation in weighted Sobolev spaces

Fonseca, Guilherme D. da; Ponce, Gustavo

doi:10.1016/j.jfa.2010.09.010

Cited by 64 publications

(120 citation statements)

References 36 publications

(46 reference statements)

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“…Fonseca and Ponce [10] extended the results of Iorio from integers values to the continuum optimal range of indices (s, r). They proved the persistence of solutions in Z s,r for s 1, r ∈ [0, s] and r < 7/2.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 82%

“…In this work, we follow the ideas of Fonseca and Ponce in [10] and Fonseca, Linares, and Ponce [11] to prove some persistence properties of the solution of the IVP (1) in the weighted Sobolev spaces Z s,r and a unique continuation property for the Benjamin equation. More precisely we prove the following results.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…Following Fonseca and Ponce [10], to handle the second term on the left hand side (l.h.s.) of (34) we write…”

Section: Proofs Of Existence Resultsmentioning

confidence: 99%

“…Finally, combining the fact that for a fixed θ ∈ (−1, 1) the family of weights w θ N , N ∈ Z + satisfies the A 2 inequality in (32) with a constant C independent of N, and Theorem 8, can be prove the following proposition, see [10].…”

Section: Moreover This Inequality Is Still Valid With W N (X) Insteamentioning

confidence: 95%

“…The proof is a consequence of the three-lines theorem (see [18]), the properties of the Stein derivative and an argument of duality. For details we refer to [10]. 2…”

Section: Moreover This Inequality Is Still Valid With W N (X) Insteamentioning

confidence: 99%

See 4 more Smart Citations

The Cauchy problem associated to the Benjamin equation in weighted Sobolev spaces

Urrea

2013

Journal of Differential Equations

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 82%

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…Following Fonseca and Ponce [10], to handle the second term on the left hand side (l.h.s.) of (34) we write…”

Section: Proofs Of Existence Resultsmentioning

confidence: 99%

Section: Moreover This Inequality Is Still Valid With W N (X) Insteamentioning

confidence: 95%

“…The proof is a consequence of the three-lines theorem (see [18]), the properties of the Stein derivative and an argument of duality. For details we refer to [10]. 2…”

Section: Moreover This Inequality Is Still Valid With W N (X) Insteamentioning

confidence: 99%

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The Cauchy problem associated to the Benjamin equation in weighted Sobolev spaces

Urrea

2013

Journal of Differential Equations

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Properties of the support of solutions of a class of nonlinear evolution equations

Bustamante

Urrea

2022

Mathematische Nachrichten

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In this work we consider equations of the form 84.0pt∂tubadbreak+Ptrue(∂xtrue)ugoodbreak+Gtrue(u,∂xu,⋯,∂xlutrue)=0,$$\begin{equation*}\hskip7pc \partial _t u+P\big (\partial _x\big ) u+G\big (u,\partial _xu,\dots ,\partial _x^l u\big )=0, \end{equation*}$$where P is any polynomial without constant term, and G is any polynomial without constant or linear terms. We prove that if u is a sufficiently smooth solution of the equation, such that prefixsuppufalse(0false),prefixsuppufalse(Tfalse)⊂(−∞,B]$\operatorname{supp}u(0),\operatorname{supp}u(T)\subset { (-\infty ,B ]}$ for some B>0$B>0$, then there exists R0>0$R_0>0$ such that prefixsuppu(t)⊂(−∞,R0]$\operatorname{supp}u(t)\subset (-\infty ,R_0]$ for every t∈false[0,Tfalse]$t\in [0,T]$. Then, as an example of the application of this result, we employ it to show a unique continuation principle for the Kawahara equation, 108.0pt∂tubadbreak+∂x5ugoodbreak+∂x3ugoodbreak+u∂xugoodbreak=0,$$\begin{equation*}\hskip9pc \partial _t u+\partial _x^5 u+\partial _x^3 u+u\partial _x u=0, \end{equation*}$$and for the generalized KdV hierarchy 72.0pt∂tubadbreak+false(−1false)k+1∂x2k+1ugoodbreak+Gtrue(u,∂xu,⋯,∂x2kutrue)=0.$$\begin{equation*}\hskip6pc \partial _t u+ (-1)^{k+1}\partial _x^{2k+1} u+G\big (u,\partial _x u,\dots , \partial _x^{2k}u\big ) =0. \end{equation*}$$

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Asymptotic Behavior of Solutions for the NLS Equation

Linares

Ponce

2014

Universitext

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We consider solutions to the initial value problem associated to the intermediate long wave (ILW) equation. We establish persistence properties of the solution flow in weighted Sobolev spaces, and show that they are sharp. We also deal with the long time dynamics of large solutions to the ILW equation. Using virial techniques, we describe regions of space where the energy of the solution must decay to zero along sequences of times. Moreover, in the case of exterior regions, we prove complete decay for any sequence of times. The remaining regions not treated here are essentially the strong dispersion and soliton regions.

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The IVP for the Benjamin–Ono equation in weighted Sobolev spaces

Cited by 64 publications

References 36 publications

The Cauchy problem associated to the Benjamin equation in weighted Sobolev spaces

The Cauchy problem associated to the Benjamin equation in weighted Sobolev spaces

Properties of the support of solutions of a class of nonlinear evolution equations

Asymptotic Behavior of Solutions for the NLS Equation

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