Asymptotic decomposition of nonlinear, dispersive wave equations with dissipation.

*(English)*Zbl 0982.35018Provided $\nu >0$, solutions of the generalized regularized long wave-Burgers equation

$${u}_{t}+{u}_{x}+P{\left(u\right)}_{x}-\nu {u}_{xx}-{u}_{xxt}=0\phantom{\rule{2.em}{0ex}}(*)$$

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

$${u}_{t}+{u}_{x}+P{\left(u\right)}_{x}-\nu {u}_{xx}+{u}_{xxx}=0\xb7$$

Reviewer: Eryk Infeld (Warszawa)