College of Mathematics and Statistics
Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: tukun@jxnu.edu.cn
College of Mathematics and Statistics
Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: dinghs@mail.ustc.edu.cn
Kun Tu, Hui-Sheng Ding
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.
DOI: 10.58997/ejde.2025.73
Show me the PDF file (369 KB), TEX file for this article.
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Kun Tu College of Mathematics and Statistics Jiangxi Normal University Nanchang, Jiangxi 330022, China email: tukun@jxnu.edu.cn |
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Hui-Sheng Ding College of Mathematics and Statistics Jiangxi Normal University Nanchang, Jiangxi 330022, China email: dinghs@mail.ustc.edu.cn |
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