Persistence of a normally hyperbolic manifold for a system of non densely defined Cauchy problems (Q2420506)

Invariant manifolds theory plays an important role in the understanding of the asymptotic behavior of dynamical systems. In this article the authors study the persistence of normally hyperbolic manifolds for a class of nondensely defined Cauchy problems using the notion exponential dichotomy and a generalized Lyapunov-Perron approach. The considered system of differential equations is: \[ \begin{aligned} \dot{u}(t) &= F(u(t)) + K (u(t),v(t)), \;\; t> t_0 \;\; \text{ and } \;\; u(t_0) = x \in X, \\ \dot{v}(t) &=[A + B(u(t))]v(t) + G(u(t),v(t)), \;\; t>0 \;\; \text{ and } \;\; v(t_0) = y \in \overline{D(A)}. \end{aligned} \tag{1} \] Here \( A: D(A) \subset Y \to Y \) is an unbounded linear operator with a possibly nondense domain, while \( B(\cdot): X \to \mathcal{L}(\overline{D(A)},Y). \) The functions \( K: X \times \overline{D(A)} \to X\) and \( G: X\times \overline{D(A)} \to Y\) are nonlinear bounded and Lipschitz continuous maps while \( F: X \to X \) can be a bounded linear operator or a Lipschitz continuous nonlinear map. The main difficulty arises from the fact that in general one has \(\overline{D(A)} \neq Y\). Some assumptions on the operator \(A\) in system (1) allows to take into account several classes of differential equations, namely backward functional-differential equations, parabolic differential equations with local or nonlocal boundary conditions, as well as hyperbolic differential equations with linear or nonlinear boundary conditions. The authors illustrate the results with two examples. The first example models the interaction between antimicrobial and bacteria, the second example is a Ross-MacDonald epidemic system. In both examples, it is shown that the results of the paper allow to reduce the infinite-dimensional system to a scalar ordinary differential equation.