Solvability and Blow-Up of Weak Solutions of Cauchy Problems for (3+1)-Dimensional Equations of Drift Waves in a Plasma

In this paper, two Cauchy problems that contain different nonlinearities vertical bar u vertical bar(q) and (partial derivative/partial derivative t)vertical bar u vertical bar(q) are studied. The differential operator in these problems is the same. It is defined by the formula M-x,M-t := (partial derivative(2)/partial derivative t(2))Delta(perpendicular to) +partial derivative(2)/partial derivative x(3)(2). The problems have a concrete physical meaning, namely, they describe drift waves in a magnetically active plasma. Conditions are found under which weak generalized solutions of these Cauchy problems exist and also under which weak solutions of the same Cauchy problems blow up. However, the question of the uniqueness of weak generalized solutions of Cauchy problems remains open, because uniqueness conditions have not been found.