Flow invariance for nonautonomous nonlinear partial differential delay equations (Q1951382)

In this paper, the following nonautonomous partial differential delay equation \[ \left\{\begin{aligned} &\dot u(t)+B(t)u(t)\ni F(t,u_t), \quad 0\leq s\leq t,\\ &u_0=\phi, \end{aligned}\right. \] is considered, where the operators \(B(t)\subset X\times X\) are nonlinear and multivalued \(\omega\)-accretive operators in a Banach space \(X,\) \(\omega\in \mathbb R,\) \(\phi: I\to X,\) \(I=[-R,0]\) or \((-\infty,0],\) \(u_t(\theta)=u(t+\theta),\) \(\theta\in I.\) Several fundamental results on existence and flow-invariance of solutions are proved. The abstract results are applied to models from population dynamics.