Asymptotic Stability of Solutions to Abstract Differential Equations
An evolution problem for abstract differential equations is studied. The typical problem is: ˙ u = A(t)u+F(t,u), t 0; u(0) = u0; ˙ u = du dt () Here A(t) is a linear bounded operator in a Hilbert space H, and F is a nonlinear operator, kF(t,u)k c0kukp, p > 1, c0, p = const > 0. It is assumed that Re(A(t)u,u) −g(t)kuk2 8u 2 H, where g(t) > 0, and the case when limt!¥ g(t) = 0 is also considered. An estimate of the rate of decay of solutions to problem (*) is given. The derivation of this estimate uses a nonlinear differential inequality.