Local solvability and solution blowup for the Benjamin–Bona–Mahony–Burgers equation with a nonlocal boundary condition

We consider a model initial–boundary value problem for the Benjamin–Bona–Mahony–Burgers equation with initial conditions having a physical meaning. We prove the unique local solvability in the classical sense and obtain sufficient conditions for blowup and an estimate of the blowup time. To prove the blowup, we use the known test function method developed in papers by V. A. Galaktionov, E. L. Mitidieri, and S. I. Pohozaev. We note that this is one of the first results toward the blowup for the considered equation.