Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 73, pp. 1-13.

Title: Shadowing properties of evolution equations with exponential trichotomy on Banach spaces

Authors: Kun Tu (Jiangxi Normal Univ., Nanchang, Jiangxi, China)
         Hui-Sheng Ding (Jiangxi Normal Univ., Nanchang, Jiangxi, China)

Abstract:
In this article we investigate the shadowing properties of the semilinear
non-autonomous evolution equation
$$
u'(t) = A(t)u(t) + f(t, u(t)), \quad t\geq 0
$$
on a Banach space $X$. Here the linear operator
$A(t) : D(A(t)) \subset X \to X$ may not be bounded, and the homogeneous equation
$u'(t)=A(t)u(t)$ admits a general exponential trichotomy.
We obtain two shadowing properties under $BS^p $ type and $L^2$ type Lipschitz
conditions on $f$, respectively. Moreover, a concrete example of parabolic partial
differential equation is provided to illustrate the applicability of our
abstract results. Compared with known results, the main feature of this paper lies 
in relaxing the Lipschitz conditions on $f$, considering the shadowing properties under 
the framework of general exponential trichotomies, and most importantly, allowing  
$A(t)$  to be unbounded, which enables the abstract results to be directly applied 
to partial differential equations.

Submitted May 1, 2025. Published July 15, 2025.
Math Subject Classifications: 34G20, 37C50, 47J35.
Key Words: Abstract evolution equation;  exponential trichotomy; shadowing property.