On a Bohr-Neugebauer property for some almost automorphic abstract delay equations
The authors consider and investigate the Bohr-Neugebauer property for the partial functional differential equation \[ \frac{d}{dt}u(t)=Au(t)+L(u_{t})+f(t), \quad t\in\mathbb{R}. \] Here, \(A\) is a linear operator on a Banach space \(X\), \(L(u_{t})\) represents delays in the equation, the domain \(D(A)\) is not necessarily dense, and \(A\) satisfies the classical Hille-Yosida condition. Also, for \(r>0\) and \(C\left([-r,0], X\right)\) the space of continuous functions \(f: [-r,0]\rightarrow X\) endowed with the uniform norm topology, the delay \(u_{t}\in C\left([-r,0], X\right)\) is defined by \(u_{t}(s):=u(t+s)\), for \(s\in[-r,0]\), and \(L\) is a bounded linear operator from \(C\left([-r,0], X\right)\) to \(X\). Furthermore, \[ \frac{d}{dt}u(t)=Au(t)+L(u_{t})+f(t)\text{ for }t\geq \tau \] and \(u_{\tau}=\phi\in C\left([-r,0], X\right)\) is the associate Cauchy problem for the above equation. The authors, after quoting a brief and comprehensive report on the problem of almost periodicity and almost automorphic functions, start with the study of relative compactness of bounded solutions of the above equation, and they present a Bohr-Neugebauer type theorem for some almost automorphic finite dimensional ordinary differential equations. Then using spectral decomposition as a consequence of their compactness assumption they extend the Bohr-Neugebauer theorem for the above equation. Also, they present two new results of integration of almost automorphic functions. The authors complete their article by proposing an application to a reaction-diffusion equation with continuous delay in a bounded open subset of \(\mathbb{R}^{n}\) with a smooth boundary. In the opinion of the reviewer, the authors have written a very interesting article with significant results that contribute to the further understanding and research of the almost periodic and almost automorphic effects on differential equations in Banach spaces.
partial functional differential equations
scientific article; zbMATH DE number 6979943
scientific article