Global well-posedness and singularity propagation for the BBM-BBM system on a quarter plane

Abstract

Nonlinear, dispersive wave equations arise as models of various physical phenomena. A major preoccupation on the mathematical side of the study of such equations has been to settle the fundamental issues of local and global well-posedness in Hadamard's classical sense. The development so far has been mostly for the initial-value problem for single equations.

However, systems of such equations have also received consideration, and there is now theory for pure initial-value problems where data are given on the entire space or on the torus. Here, consideration is given to non-homogeneous initial-boundary-value problems for a class of BBM-type systems having the form \begin{equation*} \begin{aligned} u_t + u_x -u_{xxt} + P(u,v)_x \, = \, 0, \\ v_t + v_x - v_{xxt} + Q(u,v)_x \, = \, 0, \end{aligned} \end{equation*} where $P$ and $Q$ are homogeneous, quadratic polynomials, $u$ and $v$ are real-valued functions of a spatial variable $x$ and a temporal variable $t$, and subscripts connote partial differentiation. Local in time well-posedness is established in the quarter plane $\{(x,t): x \geq 0, \, t \geq 0 \}$. Under certain restrictions on the coefficients of the nonlinearities $P$ and $Q$, global well posedness is also shown to obtain.