Almost automorphic solutions for some partial functional differential equations (Q864695)

This interesting paper deals with the existence of almost automorphic solutions for the following partial functional differential equation \[ \begin{aligned} \frac{d}{dt} u(t)&=Au(t)+L(u_t)+f(t),\;t\geq 0,\\ u_0&=\varphi\in C([-r,0];X), \end{aligned}\tag{1} \] where \(A\) is a linear operator on a Banach space \(X\) not necessarily densely defined and satisfies the Hille-Yosida condition, \(L\) is a bounded linear operator from \(C([-r,0];X)\) to \(X\), \(f\) is an automorphic function from \(\mathbb R\) to \(X\), and \(u_t\in C([-r,0];X)\) is defined by \[ u_t(\theta)=u(t+\theta),\;\theta\in [-r,0]. \] The main result of the paper states that if (1) has a bounded solution on \(\mathbb R^+\) then it has an almost automorphic solution. The authors also give some applications to hyperbolic equations and Lotka-Volterra model describing the evolution of a single diffusive animal species.