Shadowing properties of evolution equations with exponential trichotomy on Banach spaces
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...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2025-07-01
|
| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2025/73/abstr.html |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849235492853252096 |
|---|---|
| author | Kun Tu Hui-Sheng Ding |
| author_facet | Kun Tu Hui-Sheng Ding |
| author_sort | Kun Tu |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-dcf8f0dd01e840cc9ade276f1fd8749c |
| institution | Kabale University |
| issn | 1072-6691 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj-art-dcf8f0dd01e840cc9ade276f1fd8749c2025-08-20T04:02:45ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912025-07-01202573,113Shadowing properties of evolution equations with exponential trichotomy on Banach spacesKun Tu0Hui-Sheng Ding1 Jiangxi Normal Univ., Nanchang, Jiangxi, China Jiangxi Normal Univ., Nanchang, Jiangxi, China 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.http://ejde.math.txstate.edu/Volumes/2025/73/abstr.htmlabstract evolution equation exponential trichotomyshadowing property |
| spellingShingle | Kun Tu Hui-Sheng Ding Shadowing properties of evolution equations with exponential trichotomy on Banach spaces Electronic Journal of Differential Equations abstract evolution equation exponential trichotomy shadowing property |
| title | Shadowing properties of evolution equations with exponential trichotomy on Banach spaces |
| title_full | Shadowing properties of evolution equations with exponential trichotomy on Banach spaces |
| title_fullStr | Shadowing properties of evolution equations with exponential trichotomy on Banach spaces |
| title_full_unstemmed | Shadowing properties of evolution equations with exponential trichotomy on Banach spaces |
| title_short | Shadowing properties of evolution equations with exponential trichotomy on Banach spaces |
| title_sort | shadowing properties of evolution equations with exponential trichotomy on banach spaces |
| topic | abstract evolution equation exponential trichotomy shadowing property |
| url | http://ejde.math.txstate.edu/Volumes/2025/73/abstr.html |
| work_keys_str_mv | AT kuntu shadowingpropertiesofevolutionequationswithexponentialtrichotomyonbanachspaces AT huishengding shadowingpropertiesofevolutionequationswithexponentialtrichotomyonbanachspaces |