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Asymptotic decomposition of nonlinear, dispersive wave equations with dissipation. (English) Zbl 0982.35018
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 $$u_t+ u_x+ P(u)_x- \nu u_{xx}+ u_{xxx}= 0.$$

MSC:
 35B40 Asymptotic behavior of solutions of PDE 35Q53 KdV-like (Korteweg-de Vries) equations
Full Text:
References:
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