Dichotomy of linear partial differential equations of neutral type in Banach spaces (Q2665518)

This paper investigates the neutral equation \[ \begin{cases} \frac{d}{dt}(z(t) - Dz_t ) = A(z(t)- Dz_t ) + Lz_t , t\geq 0,\\ \lim\limits_{t\to 0}(z(t)- Dz_t ) = x, z_0 = \varphi, \end{cases} \] where the state function \(z : [-r ,+\infty)\to X\) and its history \( z_t (\cdot) = z(t +\cdot) : [-r , 0] \to X, A : D(A)\subset X\to X \) generates a \(C_0\)-semigroup \((T (t))_{t\geq0} \) and \(L, D : W^{1,p}([-r , 0], X)\to X \) are Riemann-Stieltjes integrals defined by \[L f =\int_{-r}^0 d\mu(\theta) f (\theta), Dg = \int_{-r}^0 dν(\theta)g(\theta), \] where \(\mu, ν : [-r , 0] \to L(X)\) are functions of bounded variations continuous and vanish at zero. The equation above is equivalent to the following Cauchy problem \[\dot w (t) = \mathcal A^{D,L}w(t), t \geq 0, w(0) = \binom{x}{\varphi},\] where the operator \(\mathcal A^{D,L}: D(\mathcal A^{D,L})\subset X\to X \) is given by \[ \mathcal A^{D,L}:=\binom{A \quad L}{0 \quad \frac{d}{d\theta}},\] \[ D(\mathcal A^{D,L}) =\left\{ \binom{x}{\varphi}\in D(A)\times W^{1,p}([-r , 0], X); \varphi(0) = x + D\varphi\right\}. \] Then, the operator \((\mathcal A^{D,L}, D(\mathcal A^{D,L}))\) generates a strongly continuous semigroup \((\mathcal T^{ D,L} (t))_{t\geq 0}\) on \(\mathcal X = X\times L_p([-r , 0], X)\). Based on regular linear systems and feedback theory, they prove the following result: Assume that \((T (t))_{t\geq 0}\) has exponential dichotomy. Then, the neutral semigroup \((\mathcal T^{ D,L} (t))_{t\geq 0}\) has exponential dichotomy and the dimension of its unstable subspace coincides with the dimension of the unstable space of \((T (t))_{t\geq 0}\). If in addition \((T (t))_{t\geq 0}\) is exponentially stable, then \((\mathcal T^{ D,L} (t))_{t\geq 0}\) is exponentially stable. This result can write as follows: Assume that \((T (t))_{t\geq 0}\) is an immediately norm continuous semigroup and \(\sigma(A+Le_\lambda)\cap i\mathbb R = \emptyset\), where \((e_\lambda x)(\theta)=e^{\lambda \theta }x, \, x\in X, \lambda \in \mathbb C, \theta\in [-r,0]\). Then, \((\mathcal T^{ D,L} (t))_{t\geq 0}\) has exponential dichotomy and the dimension of its unstable subspace coincides with the dimension of the unstable space of \((T (t))_{t\geq 0}\).