The finite-time blowup of the solution of an initial boundary-value problem for the nonlinear equation of ion sound waves

We obtain blowup conditions for the solutions of initial boundary-value problems for the nonlinear equation of ion sound waves in a hydrogen plasma in the approximation of “hot” electrons and “heavy” ions. A specific characteristic of this nonlinear equation is the noncoercive nonlinearity of the form ?_t|?u|², which complicates its study by any energy method. We solve this problem by the Mitidieri–Pohozaev method of nonlinear capacity.