\(\mu \)-Pseudo almost periodic solutions to some semilinear boundary equations on networks

Abstract

This work deals with the existence and uniqueness of \(\mu \)-pseudo almost periodic solutions to some transport processes along the edges of a finite network with inhomogeneous conditions in the vertices. For that, the strategy consists of seeing these systems as a particular case of the semilinear boundary evolution equations

$$\begin{aligned} (SHBE)\;{\left\{ \begin{array}{ll} \displaystyle {\frac{du}{dt}} &{}= A_{m} u(t)+f(t,u(t)),\quad t\in {\mathbb {R}}, \\ L u(t)&{} = g(t,u(t)) ,\quad t \in {\mathbb {R}},\\ \end{array}\right. } \end{aligned}$$

where \(A:= A_m|ker L\) generates a C\(_0\)-semigroup admitting an exponential dichotomy on a Banach space. Assuming that the forcing terms taking values in a state space and in a boundary space respectively are only \(\mu \)-pseudo almost periodic in the sense of Stepanov, we show that (SHBE) has a unique \(\mu \)-pseudo almost periodic solution which satisfies a variation of constant formula. Then we apply the previous result to obtain the existence and uniqueness of \(\mu \)-pseudo almost periodic solution to our model of network.