Large time behavior of solutions to abstract differential equations is
studied. The corresponding evolution problem is: u˙=A(t)u+F(t,u)+b(t),t≥0;u(0)=u0.(∗) Here u˙:=dtdu,
u=u(t)∈H, t∈R+:=[0,∞), A(t) is a linear dissipative
operator: Re(A(t)u,u)≤−γ(t)(u,u), γ(t)≥0, F(t,u) is a
nonlinear operator, ∥F(t,u)∥≤c0∥u∥p, p>1, c0,p are constants,
∥b(t)∥≤β(t),β(t)≥0 is a continuous function. Sufficient
conditions are given for the solution u(t) to problem (*) to exist for all
t≥0, to be bounded uniformly on R+, and a bound on ∥u(t)∥ is given.
This bound implies the relation limt→∞∥u(t)∥=0 under suitable
conditions on γ(t) and β(t). The basic technical tool in this work
is the following nonlinear inequality: \dot{g}(t)\leq
-\gamma(t)g(t)+\alpha(t,g(t))+\beta(t),\ t\geq 0;\quad g(0)=g_0. $