Initial-boundary value problems for the 2D Zakharov-Kuznetsov equation posed on bounded rectangles and on a strip are considered. Spectral properties of a linearized operator and critical sizes of domains are studied. Exponential decay of regular solutions for the original nonlinear problems is proved. 1991 Mathematics Subject Classification. 35M20, 35Q72.

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The initial-boundary value problem in a bounded domain with moving boundaries and nonhomogeneous boundary conditions for the Korteweg-de Vries equation is considered. Existence and uniqueness of global strong solutions are proved as well as the exponential decay of small solutions in asymptotically cylindrical domains.

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