Stability for solutions to nonlinear nonautonomous evolution equations
Consider the autonomous problem \[ {\dot{u}(t)+Au(t)\ni 0\text{, }t\geq 0 }\quad u(0)=x \] where \(A\) is an accretive operator in a real Banach space \(X\). In [\textit{M. Pierre}, Nonlinear Anal., Theory, Methods Appl. 2, 107--117 (1978; Zbl 0371.47056)], the author derived an upper bound for the expression d\(\left( u(t),F\right) \), where \(F\subseteq X\) is closed. The goal of this article is to extend the ideas of Pierre's article to the nonautonomous problem \[ \dot{u}(t)+A(t)u(t)\ni 0\text{, }0\leq s\leq t\quad u(s)=x \] where \(F\) is replaced by a time-dependent closed set \(K(t)\). One aim is to find a nonautonomous extension of a certain technical assumption made by Pierre. In addition to this assumption, it is assumed that \(A(t)\subseteq X\times X\), \(t\geq 0\), is a family of nonlinear, multivalued, \(\omega \)-accretive operators, and \(t_{n}\uparrow t,x_{n}\in K\left( t_{n}\right) \Longrightarrow \lim\limits_{n\rightarrow \infty }\)d\(\left( x_{n},K(t)\right) =0\). The inequality derived is of the form \[ {\text{d}\left( u(t),K(t)\right) \leq e^{\alpha (t-s)}\left[ \text{d}\left( x,K(s)\right) +\beta (t-s)\right] \text{}} \] for certain nonnegative real constants \(\alpha \) and \(\beta \), where \(u\) is a mild solution to the nonautonomous problem. Finally, the result is applied to a nonautonomous nonlinear partial differential delay equation.
scientific article; zbMATH DE number 6345569