Reduction principle and dynamic behaviors for a class of partial functional differential equations
The paper is concerned with the nonhomogeneous linear functional differential problem with delay \[ \begin{cases}\frac{du}{dt}(t)=-Au(t)+L(u_t)+f(t)\,\,\,\text{for}\,\,\,t\geq \sigma,\\ u_{\sigma}=\varphi\in {\mathcal C}_{\alpha}:={\mathcal C}([-r,0];X_{\alpha}),\end{cases}\tag{P} \] where the operator \(-A:D(A)\to X\) is the infinitesimal generator of an analytic semigroup \((T(t))_{t\geq 0}\) on a Banach space \((X,|\cdot|)\), \(\alpha\in (0,1]\) and \(X_{\alpha}\) is the Banach space \(D(A^{\alpha})\) endowed with the norm \(|x|_{\alpha}=|A^{\alpha}x|\) for \(x\in D(A^{\alpha})\). Here, \(A^{\alpha}\) is the fractional power of \(A\), \({\mathcal C}_{\alpha}\) is the space of continuous functions from \([-r,0]\) to \(X_{\alpha}\) with the uniform norm topology, \(u_t\in {\mathcal C}_{\alpha}\) is defined by \(u_t(\theta)=u(t+\theta)\) for \(\theta\in [-r,0],\,\,t\geq \sigma,\) \(L\) is a bounded linear operator from \({\mathcal C}_{\alpha}\) into \(X\) and \(f\) is a continuous function from \([\sigma,+\infty)\) into \(X\). The authors present a new variation of constants formula for the mild solutions of \((P)\) and they prove a reduction principle that shows that the dynamic of bounded solutions on \(\mathbb{R}\) is governed by an ordinary differential equation in a finite dimensional space. The existence (and uniqueness -- in the hyperbolic case) of almost automorphic, almost periodic and periodic solutions of \((P)\) are then established. In the last part of the paper the authors apply the obtained results to a partial differential equation with several delays.
scientific article; zbMATH DE number 5573694
scientific article