Abstract: | For the motion equations of Kelvin-Voight fluids one proves: 1) a global theorem for the existence and uniqueness of a solution (v;{ue}) of the initial-boundary value problem on the semiaxis t R+ from the class W
1
(R+); W
2
2
( ) H( )) with initial condition vo(x) W
2
2
( ) H( ) when the right-hand side f(x, t) L (R
+; L2( )); 2) a global theorem for the existence and uniqueness of a solution (v; {ul}) on the entire axisR from the classW
1
(R; W
2
2
( ) H( )) when the right-hand side f(x, t) L (R; L2( )); 3) a global theorem for the existence of at least one solution (v; {ul}), periodic with respect to t with period , from the class W
1
(R
+; W
2
2
( ) H( )) when the right-hand side f(x, t) L (R
+; L2( )) is periodic with respect to t with period , and a local uniqueness theorem for such a solution; 4) a theorem for the existence and uniqueness in the small of a solution (v; {ul}), almost periodic with respect to t R, from V. V. Stepanov's class S
1
(R; W
2
2
( ) H( )) when the right-hand side f(x, t) S (R; L2( )) is almost periodic with respect to t; 5) the linearization principle (Lyapunov's first method) is justified in the theory of the exponential stability of the solutions of an initial-boundary value problem in the space H( ) and conditions are given for the exponential stability of a stationary and periodic solution, with respect to t R, of the system (1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 146–185, 1990. |