Initial boundary value problem for one class of system of multidimensional inhomogeneous GBBM equations (Q914076)

\textit{J. A. Goldstein} and \textit{B. J. Wichnoski} [Nonlinear Anal., Theory Methods Appl. 4, 665-675 (1980; Zbl 0447.35068)] and \textit{F. Bampi} and \textit{A. Morro} [J. Math. Phys. 23, 2312-2321 (1982; Zbl 0502.73019)] studied the generalized Benjamin-Bona-Mahony (GBBM) equation in higher dimensions \[ (1)\quad u_ t-\Delta u_ t+\sum^{n}_{i=1}(\partial /\partial x_ i)\text{grad} \phi (u)=f(u), \] \[ u|_{t=0}=u_ 0(x),\quad x\in \Omega;\quad u|_{\partial \Omega}=0,\quad t\geq 0, \] where \(u(x,t)=(u_ 1(x,t),...,u_ N(x,t))\), \(f(u)=(f_ 1(u_ 1,...,u_ N),...,f_ N(u_ 1,...,u_ N))\), \(\phi (u)=\phi (u_ 1,...,u_ N).\) In this paper, using the Galerkin approximation method, the author proves the existence and the uniqueness of the global solution of the initial boundary value problem for the system of GBBM equations (1) and studies the regularity and ``blow up'' of the solution for (1).