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Benjamin-Bona-Mahony-Burgers type equations (BBMB)

The original Benjamin-Bona-Mahoni equation is given as follows:
\begin{eqnarray*}u_t-u_{xxt}-\nu u_{xx}+u_{x}+uu_x=0\end{eqnarray*}
This equation has been proposed as a model for propagation of long waves. Here $\nu$ is a positive constant and $u$ is typically the amplitude or velocity. $x$ describes the distance in the direction of propagation. This equation incorporates dispersive and dissipative effects. The dissipative term is just expressed by $-\nu u_{xx}$. The equation without dissipative term is called Benjamin-Bona-Mahony equation. The first paper in this field has been written by Benjamin, R.T., Bona J.L. and Mahony, J.J. in 1972 . The title of this paper was Model equations for long waves in nonlinear dispersiv systems (Philos. Trans. Roy. Soc. London, 272, 47-78). BBMB can be seen as an alternative model for the Korteweg-de Vries equation. Nowadays were studied generalizations as :
\begin{eqnarray*}u_t-u_{xxx}-\nu u_{xx}+ (f(u))_x=g(x)\end{eqnarray*}
where $f\in C^\infty(\R)$ and $g\in L_2(G), \;G\subset\R$ is a bounded interval. We have a initial condition
\begin{eqnarray*}u(x,0)=u_0(x) \quad (x\in G).\end{eqnarray*}
In many cases the boundary condition is assumed periodically. This equation belongs to the Sobolev-Galpern type equations.
 

A late Reference:: B. Wang: Attractors and Approximate Inertial Manifolds for the generalized BBME, MMAS, 20, 3 (1997)


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collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg