×

Weighted Sobolev spaces and rapidly decreasing solutions of some nonlinear dispersive wave equations. (English) Zbl 0488.35071


MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35B45 A priori estimates in context of PDEs
35K30 Initial value problems for higher-order parabolic equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Asano, K., On semi-linear parabolic partial differential equations, Publ. Res. Inst. Math. Sci., 1, 67-98 (1964) · Zbl 0192.19801
[2] Benjamin, T. B.; Bona, J. L.; Mahony, J. J., Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272, 47-78 (1972) · Zbl 0229.35013
[3] Ginibre, J.; Velo, G., On a class of nonlinear Schrödinger equations, II, J. Funct. Anal., 32, 33-71 (1979) · Zbl 0396.35029
[4] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Dunod-Gauthier-Villars: Dunod-Gauthier-Villars Paris · Zbl 0189.40603
[5] Medeiros, L. A.; Miranda, M. M., Weak solutions for a nonlinear dispersive equation, J. Math. Anal. Appl., 59, 432-441 (1977) · Zbl 0376.35011
[6] Pecher, H.; von Wahl, W., Time dependent nonlinear Schrödinger equations, Manuscripta Math., 27, 125-157 (1979) · Zbl 0399.35030
[7] Saut, J. C.; Temam, R., Remarks on the Korteweg-de Vries equation, Israel J. Math., 24, 78-87 (1976) · Zbl 0334.35062
[8] Saut, J. C., Sur quelques généralisations de l’équation de Korteweg-de Vries (I), J. Math. Pures Appl., 58, 21-61 (1978) · Zbl 0449.35083
[9] Showalter, R. E., Well-posed problems for some nonlinear dispersive waves, J. Math. Pures Appl., 56, 123-135 (1977) · Zbl 0362.35032
[10] Sobolevskii, P. E., On equations of parabolic type in a Banach space, Trudy Moscov Mat. Onšč., 10, 297-350 (1961)
[11] Strauss, W. A., Dispersion of low-energy waves for two conservative equations, Arch. Rational Mech. Anal., 55, 86-92 (1974) · Zbl 0289.35048
[12] Strauss, W. A.; Lin, J. E., Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal., 30, 245-263 (1978) · Zbl 0395.35070
[13] Tanaka, S., Korteweg-de Vries equation. Construction of solutions in terms of scattering data, Osaka J. Math., 11, 49-59 (1974) · Zbl 0283.35063
[14] Temam, R., Sur un problème non linéaire, J. Math. Pures Appl., 48, 159-172 (1969) · Zbl 0187.03902
[15] Triebel, H., Spaces of distributions with weights. Multipliers in \(L_p\)-spaces with weights, Math. Nachr., 00, 339-355 (1977) · Zbl 0376.46020
[16] M. Tsutsumi, The space of rapidly decreasing functions and the weighted Sobolev spaces, to appear.; M. Tsutsumi, The space of rapidly decreasing functions and the weighted Sobolev spaces, to appear. · Zbl 0488.35071
[17] M. Tsutsumi and I. Fukuda, On solutions of some nonlinear dispersive wave equations, to appear.; M. Tsutsumi and I. Fukuda, On solutions of some nonlinear dispersive wave equations, to appear.
[18] Tsutsumi, M.; Mukasa, T., Parabolic regularizations for the generalized Korteweg-de Vries equation, Funkcial. Ekvac., 14, 89-110 (1971) · Zbl 0228.35077
[19] Tsutsumi, M.; Matahashi, T., On some nonlinear pseudoparabolic equations, J. Differential Equations, 32, 65-75 (1979) · Zbl 0372.34042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.