Linearized stability for semilinear non-autonomous evolution equations with applications to retarded differential equations (Q5933774)

The authors consider the semilinear nonautonomous evolution equation NEWLINE\[NEWLINE\frac{d}{dt}u(t)=Au(t)+G(t,u(t)),\;t\geq s\geq 0,NEWLINE\]NEWLINE where \((A,D(A))\) is a Hille-Yosida operator on a Banach space \(X\) and \(G\) is a continuous function on \(\mathbb{R}_+ \times \overline{D(A)}\) with values in the extrapolated Favard class corresponding to \(A\). This kind of equation naturally occurs when dealing with boundary conditions that change in time. In their main result they present principles on linearized stability and instability for solutions to such an equation. The approach is based on the theory of extrapolation spaces. The results are applied to nonautonomous semilinear retarded differential equations.