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Zbl 0982.35018
Bona, Jerry L.; Luo, Laihan
Asymptotic decomposition of nonlinear, dispersive wave equations with dissipation. (English)
[J] Physica D 152-153, 363-383 (2001). ISSN 0167-2789

Provided $\nu> 0$, solutions of the generalized regularized long wave-Burgers equation $$u_t+ u_x+ P(u)_x- \nu u_{xx}- u_{xxt}= 0\tag{$*$}$$ 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 $$u_t+ u_x+ P(u)_x- \nu u_{xx}+ u_{xxx}= 0.$$
[Eryk Infeld (Warszawa)]

MSC 2000:: *35B40 Asymptotic behavior of solutions of PDE
35Q53 KdV-like equations

Keywords: generalized regularized long wave-Burgers equation; generalized Korteweg-de Vries-Burgers equation

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