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

Abstract

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.