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


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 ut + ux - uxxt + P(u, v)x = 0, vt + vx - vxxt + Q(u, v)x = 0, 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 ≥ 0, t ≥ 0}. Under certain restrictions on the coefficients of the nonlinearities P and Q, global well posedness is also shown to obtain.

Publication Title

Advances in Differential Equations

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