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: u ˙ = A ( t ) u + F ( t , u ) + b ( t ) , t ≥ 0 ; u ( 0 ) = u 0 . (*) Here u ˙ : = d u d t , 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.