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!
|
| Summary: | 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. |
|---|---|
| ISSN: | 1072-6691 |