The initial-and boundary-value problem for the Benjamin-Bona-Mahony (BBM) equation is studied in this paper. The goal is to understand the periodic behavior (termed as eventual periodicity) of its solutions corresponding to periodic boundary condition or periodic forcing. To this aim, we derive a new formula representing solutions of this initial-and boundary-value problem by inverting the operator ∂ t + α∂ x − γ∂ xxt defined in the space-time quarter plane. The eventual periodicity of the linearized BBM equation with periodic boundary data and forcing term is established by combining this new representation formula and the method of stationary phase. AMS (MOS) Numbers35Q53, 35B40, 76B15, 37K40

This paper studies solutions of the generalized Buckley-Leverett (BL) equation with variable porosity and solutions of the regularized BL equation with the Burgers-Benjamin-Bona-Mahony-type regularization. We construct global in time weak solutions to the Cauchy problem and to the initial- and boundary-value problem for the generalized BL equation and global classical solutions for the regularized BL equation. Solutions of the regularized BL equation are shown to converge to the corresponding solution of the generalized BL equation when the coefficient γ of the BBM term and the coefficient ν of the Burgers term obey γ = O(ν2).

Applying the concept of anti-integrable limit to coupled map lattices originated from space-time discretized nonlinear wave equations, we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions. In particular, the coupled map lattices display spatio-temporal chaos on the horseshoes.

This paper focuses on the two-dimensional Benjamin-Bona-Mahony and Benjamin-Bona-Mahony-Burgers equations with a general flux function. The aim is at the global (in time) well-posedness of the initial-and boundary-value problem for these equations defined in the upper half-plane. Under suitable growth conditions on the flux function, we are able to establish the global well-posedness in a Sobolev class. When the initial-and boundarydata become more regular, the corresponding solutions are shown to be classical. In addition, the continuous dependence on the data is also obtained.