Persistence properties for the dispersion generalized BO-ZK equation in weighted anisotropic Sobolev spaces

On decay of the solutions for the dispersion generalized-Benjamin-Ono and Benjamin-Ono equations

“…Proof. The proof follows from Lemma 2.5 and is similar to that of Lemma 2.13 in [6]; so we omit the details.…”

“…By the other hand, this also shows that the uniqueness properties obtained in Theorem B can not be extended to all pair of non-zero solutions. To achieve such results we use some ideas developed in [6](see also [4]).…”

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We show that uniqueness results of the kind those obtained for KdV and Schrödinger equations ([7], [28]), are not valid for the dispersion generalized-Benjamin-Ono equation in the weighted Sobolev spacesfor appropriated s and r.In particular, we obtain that the uniqueness result proved for the dispersion generalized-Benjamin-Ono equation ([13]), is not true for all pairs of solutions u 1 = 0 and u 2 = 0. To achieve these results we employ the techniques present in our recent work [6]. We also improve some Theorems established for the dispersion generalized-Benjamin-Ono equation and for the Benjamin-Ono equation ([13], [12]).

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“…In particular, we obtain that the uniqueness result proved for the dispersion generalized-Benjamin-Ono equation ([13]), is not true for all pairs of solutions u 1 = 0 and u 2 = 0. To achieve these results we employ the techniques present in our recent work [6]. We also improve some Theorems established for the dispersion generalized-Benjamin-Ono equation and for the Benjamin-Ono equation ([13], [12]).…”

On some regularity properties for the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov Equation

“…We remark that their results are based on the short-time Bourgain spaces approach developed by Ionescu, Kenig and Tataru [14], combined with localized Strichartz estimates and a modified energy technique. Very recently, Cunha and Pastor in [8] studied the Cauchy problem (1.1) in weighted anisotropic Sobolev spaces as well as some unique continuation principles, which establish optimal spatial decay in the x-spatial variable.…”