Abstract:
Two Cauchy problems for the nonlinear Sobolev equations $\frac{\partial^2}{\partial t^2}\frac{\partial^2u}{\partial x^2_3}+\Delta u=|u|^q$ and $\frac{\partial^2}{\partial t^2}\Delta_\perp u+\Delta u=|u|^q$ are investigated. Conditions are found under which the Cauchy problems have weak generalized local-in-time solutions, and the blow-up conditions for weak solutions of these problems are determined.
Key words:nonlinear Sobolev-type equations, blow-up, local solvability, nonlinear capacity.
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