##
**Global well-posedness and singularity propagation for the BBM-BBM system on a quarter plane.**
*(English)*
Zbl 1382.35207

Summary: 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 \[ u_t + u_x -u_{xxt} + P(u,v)_x \, = \, 0, \]

\[ v_t + v_x - v_{xxt} + Q(u,v)_x \, = \, 0, \] 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 \(\{(x,t): x \geq 0, \, t \geq 0 \}\). Under certain restrictions on the coefficients of the nonlinearities \(P\) and \(Q\), global well posedness is also shown to obtain.

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 \[ u_t + u_x -u_{xxt} + P(u,v)_x \, = \, 0, \]

\[ v_t + v_x - v_{xxt} + Q(u,v)_x \, = \, 0, \] 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 \(\{(x,t): x \geq 0, \, t \geq 0 \}\). Under certain restrictions on the coefficients of the nonlinearities \(P\) and \(Q\), global well posedness is also shown to obtain.

### MSC:

35Q35 | PDEs in connection with fluid mechanics |

35M33 | Initial-boundary value problems for mixed-type systems of PDEs |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

35G61 | Initial-boundary value problems for systems of nonlinear higher-order PDEs |

45G15 | Systems of nonlinear integral equations |

76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |

35Q51 | Soliton equations |

35Q53 | KdV equations (Korteweg-de Vries equations) |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

76B25 | Solitary waves for incompressible inviscid fluids |

76B55 | Internal waves for incompressible inviscid fluids |