Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 39, pp. 1-17.

Title: Reduction principle for partial functional differential equation without compactness
       
Authors: Meryem El Attaouy (Cadi Ayyad Univ., Marrakech, Morocco)
         Khalil Ezzinbi (Cadi Ayyad Univ., Marrakech, Morocco)
         Gaston Mandata N'Guerekata (Morgan State Univ., Baltimore, MD, USA )     
         
Abstract:
This article establishes a reduction principle for partial functional differential
equation without compactness of the semigroup generated by the linear part.
Under conditions more general than the compactness of the C0-semigroup generated
by the linear part, we establish the quasi-compactness of the C0-semigroup
associated to the linear part of the partial functional differential equation.
This result allows as to construct a reduced system that is posed by an ordinary
differential equation posed in a finite dimensional space. Through this result
we study the existence of almost automorphic and almost periodic solutions
for partial functional differential equations.
For illustration, we study a transport model.

Submitted May 23, 2022. Published June 20, 2023.
Math Subject Classifications: 35G15, 35G20, 35G25, 35G30.
Key Words: Functional differential equations; quasi-compact semigroup;
 variation of constants formula;  Stepanov-almost automorphic function;
 almost automorphic solution; almost periodic solution.