Stability of Solutions to Evolution Problems

Abstract

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: u ˙ = A(t)u + F(t, u) + b(t), t ≥ 0; u(0) = u 0 . (*) Here u ˙ := 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) ‖ ≤ c 0 ‖ u ‖ p , p > 1, c 0 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.