On the supercritical KDV equation with time-oscillating nonlinearity

For the initial value problem (IVP) associated to the generalized Korteweg-de Vries (gKdV) equation with supercritical nonlinearity, \begin{equation*} u_{t}+\partial_x^3u+\partial_x(u^{k+1}) =0,\qquad k\geq 5, \end{equation*} numerical evidence [Bona J.L., Dougalis V.A., Karakashian O.A., McKinney W.R.: Conservative, high-order numerical schemes for the generalized Korteweg–de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 351, 107–164 (1995) ] shows that, there are initial data $\phi\in H^1(\mathbb{R})$ such that the corresponding solution may blow-up in finite time. Also, with the evidence from numerical simulation [Abdullaev F.K., Caputo J.G., Kraenkel R.A., Malomed B.A.: Controlling collapse in Bose–Einstein condensates by temporal modulation of the scattering length. Phys. Rev. A 67, 012605 (2003) and Konotop V.V., Pacciani P.: Collapse of solutions of the nonlinear Schrödinger equation with a time dependent nonlinearity: application to the Bose–Einstein condensates. Phys. Rev. Lett. 94, 240405 (2005) ], it has been claimed that a periodic time dependent coefficient in the nonlinearity would disturb the blow-up solution, either accelerating or delaying it. In this work, we investigate the IVP associated to the gKdV equation \begin{equation*} u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^{k+1}) =0, \end{equation*} where $g$ is a periodic function and $k\geq 5$ is an integer. We prove that, for given initial data $\phi \in H^1(\mathbb{R})$, as $|\omega|\to \infty$, the solution $u_{\omega}$ converges to the solution $U$ of the initial value problem associated to \begin{equation*} U_{t}+\partial_x^3U+m(g)\partial_x(U^{k+1}) =0, \end{equation*} with the same initial data, where $m(g)$ is the average of the periodic function $g$. Moreover, if the solution $U$ is global and satisfies $\|U\|_{L_x^5L_t^{10}}<\infty$, then we prove that the solution $u_{\omega}$ is also global provided $|\omega|$ is sufficiently large.M. P. was partially supported by the Research Center of Mathematics of the University of Minho, Portugal through the FCT Pluriannual Funding Program, and through the project PTDC/MAT/109844/2009, and M. S. was partially supported by FAPESP Brazil

Global well-posedness for a coupled modified kdv system

We prove the sharp global well-posedness result for the initial value problem (IVP) associated to the system of the modi ed Korteweg-de Vries (mKdV) equation. For the single mKdV equation such result has been obtained by using Mirura's Transform that takes the KdV equation to the mKdV equation [8]. We do not know the existence of Miura's Transform that takes a KdV system to the system we are considering. To overcome this di culty we developed a new proof of the sharp global well-posedness result for the single mKdV equation without using Miura's Transform. We could successfully apply this technique in the case of the mKdV system to obtain the desired result.Fundação para a Ciência e a Tecnologia (FCT

Evidence for covert attention switching from eye-movements. Reply to commentaries on Liechty et al., 2003

We argue that our research objectives in Liechty, Pieters, and Wedel (2003) are to provide generalizable insights into covert visual attention to complex, multimodal stimuli in their natural context, through inverse inference from eye-movement data. We discuss the most important issues raised by Feng (2003) and Reichle and Nelson (2003), in particular the task definition, inclusion of ad features, object-based versus space-based attention and the evidence for the where and what streams.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45760/1/11336_2005_Article_BF02295611.pd

Global Well-posedness For The K-dispersion Generalized Benjamin-ono Equation

We consider the k-dispersion generalized Benjamin-Ono equation in the supercritical case. We establish sharp conditions on the data to show global well-posedness in the energy space for this family of nonlinear dispersive equations. We also prove similar results for the generalized Benjamin-Ono equation.2707/08/15601612Angulo, J., Bona, J.L., Linares, F., Scialom, M., Scaling, stability and singularities for nonlinear, dispersive wave equations: The critical case (2002) Nonlinearity, 15, pp. 759-786Amick, C.J., Toland, J.F., Uniqueness of benjamin's solitary-wave solution of the benjamin-ono equation (1991) IMA J. Appl. Math., 46, pp. 21-28Christ, M., Kiselev, A., Maximal functions associated to filtrations (2001) J. Funct. Anal., 179, pp. 409-425Farah, L.G., Linares, F., Pastor, A., The supercritical generalized KdV equation: Global well-posedness in the energy space and below (2011) Mathematical Research Letters, 18, pp. 357-377Fonseca, G., Linares, F., Ponce, G., The IVP for the dispersion generalized benjamin-ono equation in weighted sobolev spaces (2013) Ann. Inst. H. Poincaré Anal. Non Linéaire, 30, pp. 763-790Fonseca, G., Linares, F., Ponce, G., The I.V.P for the benjamin-ono equation in weighted sobolev spaces II (2012) J. Funct. Anal., 262, pp. 2031-2049Fonseca, G., Ponce, G., The I.V.P. for the benjamin-ono equation in weighted sobolev spaces (2011) J. Funct. Anal., 260, pp. 436-459Frank, R.L., Lenzmann, E., Uniqueness and nondegenerancy of ground states for (-Δ) sQ + Q - Qα+1 = 0 in ℝ (2013) Acta Math., 210, pp. 261-318Herr, S., Ionescu, A., Kenig, C.E., Koch, H., A para-differential renormalization technique for nonlinear dispersive equations (2010) Comm. Pde, 35, pp. 1827-1875Holmer, J., Roudenko, S., A sharp condition for scattering of the radial 3D cubic nonlinear schrödinger equation (2008) Commun. Math. Phys., 282, pp. 435-467Ionescu, A.D., Kenig, C.E., Global well-posedness of the benjamin-ono equation on low-regularity spaces (2007) J. Amer. Math. Soc, 20, pp. 753-798Iorio, R.J., On the cauchy problem for the benjamin-ono equation (1986) Comm. Pde, 11, pp. 1031-1081Kenig, C.E., Martel, Y., Robbiano, L., Local well-posedness and blow up in the energy space for a class of L2 critical dispersion generalized benjamin-ono equation (2011) Ann. Inst. H. Poincaré Anal. Non Linéaire, 28, pp. 853-887Kenig, C.E., Merle, F., Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear schrödinger equation in the radial case (2006) Invent. Math., 166, pp. 645-675Kenig, C.E., Ponce, G., Vega, L., Well-posedness of the initial value problem for the korteweg-de vries equation (1991) J. Amer. Math. Soc., 4, pp. 323-347Kenig, C.E., Takaoka, H., Global wellposedness of the modifed benjamin-ono equation with initial data in H1/2 (2006) Int. Math. Res. Not., , Art. ID 95702Koch, H., Tzvetkov, N., On the local well-posedness of the benjamin-ono equation on H'(R) (2003) Int. Math. Res. Not., 26, pp. 1449-1464Martel, Y., Merle, F., Blow up in finite time and dynamics of blow up solutions for the L 2-critical generalized KdV equation (2002) J. Amer. Math. Soc, 15, pp. 617-664Merle, F., Existence of blow-up solutions in the energy space for the critical generalized korteweg-de vries equation (2001) J. Amer. Math. Soc, 14, pp. 555-578Molinet, L., Pilod, D., The cauchy problem for the benjamin-ono equation in L 2revisited (2012) Analysis and PDE, 5, pp. 365-395Molinet, L., Ribaud, F., Well-posedness results for the generalized benjamin-ono equation with small initial data (2004) J. Math. Pures Appl., 83, pp. 277-311Molinet, L., Ribaud, F., Well-posedness results for the benjamin-ono equation with arbitrary large initial data (2004) Int. Math. Res. Not., 70, pp. 3757-3795Molinet, L., Ribaud, F., On global well-posedness for a class of nonlocal dispersive wave equations (2006) Discrete Contin. Dyn. Syst., 15, pp. 657-668Molinet, L., Saut, J.-C., Tzvetkov, N., Ill-posedness issues for the benjamin-ono and related equations (2001) SIAM J. Math. Anal., 33, pp. 982-988Ponce, G., On the global well-posedness of the benjamin-ono equation (1991) Diff. & Int. Eqs., 4, pp. 527-542Saut, J.-C., Sur quelques généralisations de V équations de Korteweg-de Vries (1979) J. Math. Pures Appl., 58, pp. 21-61Tao, T., Global well-posedness of the benjamin-ono equation on H1 (2004) Journal Hyp. Diff. Eqs., 1, pp. 27-49(2006) Int. Math. Res. Not., pp. 1-44. , Art. ID 95702Vento, S., Well-posedness for the generalized benjamin-ono equations with arbitrary large initial data in the critical space (2010) Int. Math. Res. Not. IMRN, pp. 297-319Weinstein, M.I., Nonlinear schrödinger equations and sharp interpolation estimates (1983) Comm. Math. Phys., 87, pp. 567-57

Phenolic profile and antioxidant activity from non-toxic Mexican Jatropha curcas L. shell methanolic extracts

Jatropha curcas seed shells are the by-product obtained during oil extraction process. Recently, its chemical composition has gained attention since its potential applications. The aim of this study was to identify phenolic compounds profile from a non-toxic J. curcas shell from Mexico, besides, evaluate J. curcas shell methanolic extract (JcSME) antioxidant activity. Free, conjugate and bound phenolics were fractionated and quantified (606.7, 193.32 and 909.59 μg/g shell, respectively) and 13 individual phenolic compounds were detected by HPLC. The radical-scavenging activity of JcSME was similar to Trolox and ascorbic acid by DPPH assay whi