Stability of Solutions to Evolution Problems

Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a finite limit as t → ∞, in particular, sufficient conditions for this limit to be zero. The evolution problem is: u̇ = A(t)u+ F (t, u) + b(t), t ≥ 0; u(0) = u0. (∗) Here u̇ := du dt , u = u(t) ∈ H , H is a Hilbert space, t ∈ R+ := [0,∞), A(t) is a linear dissipative operator: Re(A(t)u, u) ≤ −γ(t)(u, u), where F (t, u) is a nonlinear operator, ‖F (t, u)‖ ≤ c0‖u‖, p > 1, c0 and p are positive constants, ‖b(t)‖ ≤ β(t), and β(t) ≥ 0 is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The non-classical case γ(t) ≤ 0 is also treated.