On the Periodic Solutions for a class of Partial Differential Equations with unbounded Delay

Résumé

Trough this work we investigate the existence of periodic solutions for the following partial differential equations with infinite delay of the form $\dot{\textit{w}}(t) = \mathcal{L}\textit{w}(t) + \mathcal{D}(\textit{w}_{t}) + \mathcal{H}(t)$. We assume that the operator $(\mathcal{L},\mathscr{D}(\mathcal{L}))$ is generally nondensely defined operator and verifies the Hille-Yosida condition. Using the theory of perturbation of semi-Fredholm operators, we propose some sufficient conditions on the linear operators $\mathcal{L}$, $\mathcal{D}$ and the phase space $\mathscr{B}$ to guarantee the existence of periodic solutions for this class of partial differential equations from bounded ones on the positive real half-line without considering the compactness of the semigroup generated by the part of $\mathcal{L}$ on the closure of it's domain. At the end, an application with numerical simulations, is given to confirm the applicability of the obtained theoretical results.