caa a22 4500 388108983 CHVBK 20180307125347.0 cr unu---uuuuu 161130s1999 xx s 000 0 eng 10.1017/S0308210500021478 doi S0308210500021478 pii (NATIONALLICENCE)cambridge-10.1017/S0308210500021478 Fourier splitting and dissipation of nonlinear dispersive waves [Elektronische Daten] Presented herein is a new method for analysing the long-time behaviour of solutions of nonlinear, dispersive, dissipative wave equations. The method is applied to the generalized Korteweg-de Vries equation posed on the entire real axis, with a homogeneous dissipative mechanism included. Solutions of such equations that commence with finite energy decay to zero as time becomes unboundedly large. In circumstances to be spelled out presently, we establish the existence of a universal asymptotic structure that governs the final stages of decay of solutions. The method entails a splitting of Fourier modes into long and short wavelengths which permits the exploitation of the Hamiltonian structure of the equation obtained by ignoring dissipation. We also develop a helpful enhancement of Schwartz's inequality. This approach applies particularly well to cases where the damping increases in strength sublinearly with wavenumber. Thus the present theory complements earlier work using centre-manifold and group-renormalization ideas to tackle the situation wherein the nonlinearity is quasilinear with regard to the dissipative mechanism. Copyright © Royal Society of Edinburgh 1999 Bona J. L. Department of Mathematics and Texas Institute for Computational and Applied Mathematics, University of Texas, Austin, TX 78712, USA Demengel F. Department de Mathematique, University de Cergy-Pontoise, 2 Ave. Adophe Chauvin, Site de St. Martin, 95302 Cergy-Pontoise, France Promislow K. Department of Mathematics and Statistics, Simon Fraser University, Burnaby BC, Canada V5A 1S6 Proceedings of the Royal Society of Edinburgh: Section A Mathematics Royal Society of Edinburgh Scotland Foundation 129/3(1999), 477-502 0308-2105 129:3<477 1999 129 PRM https://doi.org/10.1017/S0308210500021478 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0308210500021478 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bona J. L. Department of Mathematics and Texas Institute for Computational and Applied Mathematics, University of Texas, Austin, TX 78712, USA NATIONALLICENCE 700 1- Demengel F. Department de Mathematique, University de Cergy-Pontoise, 2 Ave. Adophe Chauvin, Site de St. Martin, 95302 Cergy-Pontoise, France NATIONALLICENCE 700 1- Promislow K. Department of Mathematics and Statistics, Simon Fraser University, Burnaby BC, Canada V5A 1S6 NATIONALLICENCE 773 0- Proceedings of the Royal Society of Edinburgh: Section A Mathematics Royal Society of Edinburgh Scotland Foundation 129/3(1999), 477-502 0308-2105 129:3<477 1999 129 PRM CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge