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 —> oo, in particular, sufficient conditions for this limit to be zero. The evolution problem is: it = A(t)u + F (t , u) + b(t) , t > 0; u(0) = uo.？？？？？？？？ (*) Here U := 2, u = u(t) E H, H is a Hilbert space, t E IL F := [0, oo), A(t) is a linear dissipative operator: Re(A(t)u, u) < —7 (t)(u , u), where F(t, u) is a nonlinear operator, 11 F (t , u)11 < collul I P' P > 1, co and p are positive constants, Ilb(t) 11 < 13 (t) , and 1 3 (t) > 0 is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The non-classical case -y (t) < 0 is also treated.