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Asymptotic decomposition of nonlinear, dispersive wave equations with dissipation. (English) Zbl 0982.35018

Provided ν>0, solutions of the generalized regularized long wave-Burgers equation

u t +u x +P(u) x -νu xx -u xxt =0(*)

that begin with finite energy decay to zero as t becomes unboundedly large. Consideration is given here to the case where P vanishes at least cubically at the origin. In this case, solutions of (*) may be decomposed exactly as the sum of a solution of the corresponding linear equation and a higher-order correction term. An explicit asymptotic form for the L 2 -norm of the higher-order correction is presented here. The effect of the nonlinearity is felt only in the higher-order term. A similar decomposition is given for the generalized Korteweg-de Vries-Burgers equation

u t +u x +P(u) x -νu xx +u xxx =0·

MSC:
35B40Asymptotic behavior of solutions of PDE
35Q53KdV-like (Korteweg-de Vries) equations