Abstract:
In this paper, two Cauchy problems that contain different nonlinearities $|u|^q$ and $(\partial/\partial t)|u|^q$ are studied. The differential operator in these problems is the same. It is defined by the formula $\mathfrak{M}_{x,t}:=(\partial^2/\partial t^2)\Delta_{\perp}+ \partial^2/\partial x_3^2$. 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.
Keywords:Sobolev-type nonlinear equations, blow-up, local solvability, nonlinear capacity.
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