Blow-up of weak solutions of the Cauchy problem for $(3+1)$-dimensional equation of plasma drift waves

he 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 $|u|^q$. It is shown that, for $1<q\le3$, 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 $3<q\le5$, 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 $g>4$, the existence of a unique local-in-time weak solution is proved using results of distribution theory and the contraction mapping principle.