Asymptotic Behavior of Solutions of a Class of Nonlinear Differential Equations in Banach Space

A semi-linear ordinary differential equation in a Banach space is considered. The coefficient operator $A(t)$ has the domain $D(A)$ which is independent of t and not necessarily dense. It is shown that the evolution operator $U(t,s)$ corresponding to the unperturbed linear equation has an integrable singularity at $t = s$ and is strongly continuously differentiable in s on $D(A)$. Such examples have been obtained by W. von Wahl [13] and H. Kielhöfer [4], [5], and are also obtained in this paper. The nonlinear term satisfies either a uniform or local Lipschitz’s condition with respect to the unknown solution. The principal tools are the semigroup theory and integral inequalities. Several results on the asymptotic behavior of the solution of the semi-linear equation are obtained. These results are applicable to the problem of the stability of semi-linear parabolic initial boundary value problems within the framework of the $C^\alpha $-theory.