Blow-up of Weak Solutions of the Cauchy Problem for (3+1)-Dimensional Equation of Plasma Drift Waves

The Cauchy problem for a new equation describing drift waves in a magnetoactive plasma is considered. The existence and uniqueness of a local-in-time weak solution of the Cauchy problem are proved. The considered equation contains the power-law nonlinearity $${\text{|}}u{\kern 1pt} {{{\text{|}}}^{q}}$$ . It is shown that, for $$1 < q \leqslant 3,$$ a weak solution $$u(x,t)$$ does not exist even locally in time for a wide class of initial functions $${{u}_{0}}(x)$$ , while, for $$3 < q \leqslant 5,$$ global-in-time weak solutions of the Cauchy problem do not exist for a wide class of initial functions independent of the initial function value, i.e., for “small” initial functions as well. For $$q > 4$$ , the existence of a unique local-in-time weak solution is proved using results of distribution theory and the contraction mapping principle.