Shadowing properties of evolution equations with exponential trichotomy on Banach spaces - Kun Tu; Hui-Sheng Ding (Electronic Journal of Differential Equations)
In this article we investigate the shadowing properties of the semilinear non-autonomous evolution equation $$ u'(t) = A(t)u(t) + f(t, u(t)), \quad t≥ 0 $$ on a Banach space $X$. Here the linear operator $A(t) : D(A(t)) \subset X → X$ 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.