ABSTRACT
Large-time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the solution to an abstract dynamical system (evolution problem) not to exhibit chaotic behavior. A classical area of study is stability of solutions to evolution equations. The technical tool is a new nonlinear differential inequality. The results are stated in several theorems and illustrated by several examples. The literature on the stability of solutions to evolution problems and their behavior at large times is enormous. In the theory of chaos, one of the reasons for the chaotic behavior of a solution to an evolution problem to appear is the lack of stability of solutions to this problem. The chapter develops a method for a study of stability of solutions to the evolution problems described by the Cauchy problems for abstract differential equations with a dissipative bounded linear operator A(t) and a nonlinearity F(t, u) satisfying inequality.
