Well-posedness for the coupled BBM systems

Consideration is given to initial value problem for systems of two evolution equations of generalized BBM-type coupled through nonlinearity described in (1.3). It is shown that the problem is always locally well-posed in the L2-based Sobolev spaces Hs(ℝ) × Hs(ℝ) for s ≥ 0. Under exact conditions on A;… F; the local well-posedness theory extends globally, and bounds for the growth in time of relevant norms of solutions corresponding to very general auxiliary data are derived