PFDE with nonautonomous past (Q1848986)

The authors of this very interesting paper study the Cauchy problem associated to partial differential equations (PDEs) with infinite delay where the history function is modified by an evolution family. By the theory of evolution semigroups and using a delay semigroup with translation property and extrapolation spaces, well-posedness is shown. The asymptotic behavior of the solution semigroup is characterized by an operator-valued characteristic equation. The authors consider the Cauchy problem \[ \dot x(t)=Bx(t)+ \Phi x_t\tag{1} \] \(t\geq 0\), \(x_t(s)= x(s+t)\), \(s\leq 0\) with infinite delay and initial data \(x_0=f\in C^1_0\). This leads to the PDE \[ \partial u(t,s)/ \partial t=\partial u(t,s)/ \partial s+A(s)u (t,s) \tag{2} \] for the linear operator \(A(s)= -\partial U(s,r)/ \partial r|_{r=0}\) on the Banach space \(X\) and therefore \[ \partial/\partial s(t,0)= Bu(t,0)+ \Phi u(t,\cdot).\tag{3} \] Here \(\Phi\) is the delay operator, \(B\) is a linear operator. It is shown that under some assumptions, a strongly continuous semigroup \((T_{B,\Phi}(t))_{t\geq 0}\) exists on some Banach space \(E\) such that \(u(t,s)=(T_{B,\Phi} (t)f)(s)\) solves equation (2). In addition, the authors estimate the critical growth bound of \((T_{B,\Phi} (t))_{t\geq 0}\) and obtain as a consequence that the spectrum of the generator of \((T_{B,\Phi} (t))_{ t\geq 0}\) determines the growth bound of \((T_{B,\Phi} (t))_{t\geq 0}\), hence of the solutions of (2) and (3).