Blow-up of solutions of a full non-linear equation of ion-sound waves in a plasma with non-coercive non-linearities

We consider a series of initial-boundary value problems for the equation of ion-sound waves in a plasma. For each of them we prove the local (in time) solubility and perform an analytical-numerical study of the blow-up of solutions. We use the method of test functions to obtain sufficient conditions for finite-time blow-up and an upper bound for the blow-up time. In concrete numerical examples we improve these bounds numerically using the mesh refinement method. Thus the analytical and numerical parts of the investigation complement each other. The time interval for the numerical modelling is chosen in accordance with the analytically obtained upper bound for the blow-up time. In return, numerical calculations specify the moment and pattern of this blow-up.