Artículos de revistas
On The Critical Kdv Equation With Time-oscillating Nonlinearity
Registro en:
Differential And Integral Equations. , v. 24, n. 05/06/15, p. 541 - 567, 2011.
8934983
2-s2.0-84864837343
Autor
Carvajal X.
Panthee M.
Scialom M.
Institución
Resumen
We investigate the initial-value problem (IVP) associated with the equation {equation presented} where g is a periodic function. We prove that, for given initial data φ ∈ H1(R), as |ω| → ∞, the solution uω converges to the solution U of the initial-value problem associated with {equation presented} 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 {equation presented} then we prove that the solution u ω is also global provided |ω| is suffciently large. 24 05/06/15 541 567 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 (2003) Phys. Rev. A, 67, p. 012605 Angulo, J., Bona, J.L., Linares, F., Scialom, M., Scaling, stability and singularities for nonlinear, dispersive wave equations: The critical case (2002) Nonlinearity, 15 (3), pp. 759-786. , DOI 10.1088/0951-7715/15/3/315, PII S0951771502250872 Bona, J.L., Souganidis, P.E., Strauss, W.A., Stability and instability of solitary waves of korteweg-de vries type (1987) Proceedings of The Royal Society of London, Series A: Mathematical and Physical Sciences, 411 (1841), pp. 395-412 Bourgain, J., Refinements of Strichartz' Inequality and Applications to 2D-NLS with Critical Nonlinearity (1998) International Mathematics Research Notices, (5), pp. 253-283 Cazenave, T., Scialom, M., A schrödinger equation with time-oscillating nonlinear-ity (2010) Revista Matemática Complutense, 23, pp. 321-339 Damergi, I., Goubet, O., Blow-up solutions to the nonlinear schrödinger equation with oscillating nonlinearities (2009) J. Math. Anal. Appl, 352, pp. 336-344 Farah, L.G., Global rough solutions to the critical generalized kdv equation (2010) J. Diff. Equations, 249, pp. 1968-1985 Fonseca, G., Linares, F., Ponce, G., Global existence for the critical generalized KDV equation (2003) Proceedings of the American Mathematical Society, 131 (6), pp. 1847-1855 Friedman, A., (1969) Partial Differential Equations, , Holt, Rinehart and Winston, Inc Grillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry i (1987) J. Funct. Anal, 74, pp. 160-197 Kato, T., On the cauchy problem for the (generalized), korteweg-de vries equation, advances in mathematics supplementary studies (1983) Studies in Appl. Math, 8, pp. 93-128 Kenig, C.E., Ponce, G., Vega, L., On the (generalized), korteweg-de vries equation (1989) Duke Math. J, 59, pp. 585-610 Kenig, 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. 323-347 Kenig, C.E., Ponce, G., Vega, L., Oscillatory integrals and regularity of dispersive equations (1991) Indiana University Math. J, 40 (1), pp. 33-69 Kenig, C.E., Ponce, G., Vega, L., Well-posedness and scattering results for the generalized korteweg-de vries equation via the contraction principle (1993) Comm. Pure Appl. Math, 46, pp. 527-620 Kenig, C.E., Ponce, G., Vega, L., On the concentration of blow up solutions for the generalized kdv equation critical in l2 (2000) Nonlinear Wave Equations (Providence, RI, 1998), 263, pp. 131-156. , Contemp. Math., Amer. Math. Soc., Providence, RI Kenig, C.E., Ruiz, A., A strong type (2,2), estimate for a maximal operator associated to the schrödinger equation (1983) Trans. Amer. Math. Soc, 230, pp. 239-246 Knickerbocker, C.J., Newell, A.C., Internal solitary waves near a turning point (1980) Phys. Lett, 75 A, pp. 326-330 Konotop, V.V., Pacciani, P., Collapse of solutions of the nonlinear schrödinger equation with a time dependent nonlinearity: Application to the bose-einstein condensates (2005) Phys. Rev. Lett, 94, p. 240405 Martel, Y., Merle, F., Blow up in finite time and dynamics of blow up solutions for the critical generalized kdv equation (2002) J. Amer. Math. Soc, 15, pp. 617-664 Martel, Y., Merle, F., Stability of blowup profile and lower bounds on blowup rate for the critical generalized kdv equation (2002) Ann. of Math, 155, pp. 235-280 Merle, F., Existence of blow-up solutions in the energy space for the critical generalized kdv equation (2001) J. Amer. Math. Soc, 14, pp. 555-578 Nunes, W.V.L., Global well-posedness for the transitional Korteweg-de Vries equation (1998) Applied Mathematics Letters, 11 (5), pp. 15-20. , PII S089396599800072X Nunes, W.V.L., On the well-posedness and scattering for the transitional benjamin-ono equation (1992) Mat. Contemp, 3, pp. 127-148 Stein, E., Weiss, G., (1971) Introduction to Fourier Analysis on Euclidean Spaces, , Princeton University Press Weinstein, M., Lyapunov stability of ground states of nonlinear dispersive evolution equations (1986) Comm. Pure Appl. Math, 39, pp. 51-67