Well-posedness of second-order degenerate differential equations with finite delay in vector-valued function spaces

Shangquan Bu and Gang Cai

Vol. 288 (2017), No. 1, 27–46

DOI: 10.2140/pjm.2017.288.27

Abstract

We give necessary and sufficient conditions of the $L^p$-well-posedness (respectively, $B_{p,q}^s$-well-posedness) for the second-order degenerate differential equation with finite delay: ${\left(M{u}^{\prime }\right)}^{\prime }\left(t\right)+\alpha u^{\prime }\left(t\right)=Au\left(t\right)+G{u}_t^{\prime }+F{u}_t+f\left(t\right)$, $\left(t\in \left[0,2\pi \right]\right)$ with periodic boundary conditions $u\left(0\right)=u\left(2\pi \right)$, $\left(M{u}^{\prime }\right)\left(0\right)=\left(M{u}^{\prime }\right)\left(2\pi \right)$, where $A$ and $M$ are closed linear operators on a Banach space $X$ satisfying $D\left(A\right)\subset D\left(M\right)$, and $F$ and $G$ are bounded linear operators from $L^p\left(\left[-2\pi ,0\right];X\right)$ (respectively, $B_{p,q}^s\left(\left[-2\pi ,0\right];X\right)$) into $X$.

Keywords

Degenerate differential equations, delay equations, well-posedness, Lebesgue–Bochner spaces, Besov spaces, Fourier multipliers

Mathematical Subject Classification 2010

Primary: 34G10, 34K30, 43A15, 47D06

Milestones

Received: 12 April 2016

Accepted: 25 August 2016

Published: 8 April 2017