Abstract:

This paper discusses the shadowing properties of the semilinear nonautonomous evolution equation

on a Banach space $X$, where the linear operator $A(t) : D(A(t)) \subset X \rightarrow X$ may not be bounded and $u'(t)=A(t)u(t)$ admits exponential dichotomy. This paper first establishes shadowing properties under the classical Lipschitz condition and a weaker $BS^p $ type Lipschitz condition for $f$. Then we further introduce the concepts of $L^p$ pseudo orbits and $L^p$ shadowing property, establishing corresponding shadowing theorem. Finally, an example of a parabolic partial differential equation is provided as an application of the abstract results. Compared to existing literature, this paper not only weakens the Lipschitz condition for the nonlinear term and introduces and discusses the new $L^p$ shadowing property, but most importantly, it allows $A(t)$ to be an unbounded operator, thereby enabling the abstract results to be applied to partial differential equations.

Key words: abstract evolution equation, exponential dichotomy, shadowing property