Local solvability and decay of the solution of an equation with quadratic non-coercive nonlineatity

An initial-boundary value problem for plasma ion-sound wave equation is considered. Boltzmann distribution is approximated by a quadratic function. The local (in time) solvability is proved and the analitycal-numerical investigation of the solution's decay is performed for the considered problem. The sufficient conditions for solution's decay and an upper bound of the decay moment are obtained by the test function method. In some numerical examples, the estimation is specified by Richardson's mesh refinement method. The time interval for numerical modelling is chosen according to the decay moment's analytical upper bound. In return, numerical calculations refine the moment and the space-time pattern of the decay. Thus, analytical and numerical parts of the investigation amplify each other.