A spectral mapping theorem for semigroups solving PDEs with nonautonomous past
This paper builds a spectral mapping theorem for semigroups solving the partial differential equation of the form \[ \dot{u}(t)=Bu(t)+\Phi \tilde{u}_t,\quad t\geq 0\,, \] \[ u(0)=x\in X,\qquad u_0=f\in L^p(\mathbb{R}_{-},\,X) \eqno(1) \] in a Banach space \(X\), where \(B: D(B)\subset X\to X\) is the generator of a strongly continuous semigroup on X, \(\Phi: D(\Phi)\subset L^p(\mathbb{R}_{-},\,X)\to X\) is a linear delay operator which can be expressed by the Riemann-Stieltjes integral \( \Phi\,g=\int_{-\infty}^0 g\,d\eta \) with a bounded variation function \(\eta: \mathbb R\to \mathcal{L}(X)\), and \(\tilde{u}_t: \mathbb{R}_{-}\to X\) is the modified history function \[ \tilde{u}_t(\tau)=\begin{cases} \dot{u}(t)=U(\tau,\,t+\tau)\,f(t+\tau),\quad & t\leq t+\tau\leq 0,\\ U(\tau,\,0)\,u(t+\tau),\quad & \tau\leq 0\leq t+\tau, \end{cases} \] for an evolution family \(U(t,\,s)\,(t\leq s\leq 0)\). The author rewrites equation (1) to the abstract Cauchy problem \[ \dot{\mathcal{U}}(t)=\mathcal{C}\,\mathcal{U}(t),\quad t\geq 0,\qquad \mathcal{U}(0)= \Big({x \atop f}\Big) \] in the product space \(\mathcal{E}=X\times L^p(\mathbb{R}_{-},\,X)\), where \(\mathcal{C}: D(\mathcal{C})\subset \mathcal{E}\to \mathcal{E}\) is an operator relating to \(B\), \(\Phi\) and the evolution family \(U(t,\,s)\,(t\leq s\leq 0)\) and generates a strongly continuous semigroup \(\mathcal{T}(t)(t\geq 0)\) on \(\mathcal{E}\). The author obtains a spectral mapping theorem for the semigroup \(\mathcal{T}(t)(t\geq 0)\) and then uses this theorem to get spectral conditions for the exponential stability of equation (1). This result is particularly interesting.
scientific article; zbMATH DE number 2163624