Semigroups and linear partial differential equations with delay (Q5956498)

The authors study the problem NEWLINE\[NEWLINEu'(t)= Au(t)+ \phi u_t,\quad t\geq 0,\quad u(0)= x,\quad u_0= f,\tag{DE}NEWLINE\]NEWLINE with \(x\in X\), \(X\) is a Banach space: \(A: D(A)\subseteq X\to X\) is a closed and densely defined linear operator, \(f\in L^p([- 1,0], X)\), \(1\leq p<\infty\), \(\phi: W^{1,p}([-1,0], X)\to X\) is a bounded linear operator (the delay operator), \(u: [-1,\infty)\to X\), \(u_t: [-1,0]\to X\), \(u_t(\sigma)= u(t+\sigma)\) for \(\sigma\in [-1,0]\).NEWLINENEWLINENEWLINEThe existence, uniqueness and some properties of the classical solutions to (DE) are investigated. For this purpose, the Banach space \({\mathcal E}:= X\times L^p([-1,0], X)\), the operator \({\mathcal A}:= \left(\begin{smallmatrix} A &\phi\\ 0 &{d\over d\sigma}\end{smallmatrix}\right)\) with domain \(D({\mathcal A})= \{{x\choose f}\in D(A)\times W^{1,p}([- 1,0], X): f(0)= x\}\) and the problem NEWLINE\[NEWLINE\dot{\mathcal U}(t)= {\mathcal A}{\mathcal U}(t),\quad t\geq 0,\quad{\mathcal U}(0)= {x\choose f},\tag{ACP}NEWLINE\]NEWLINE are considered. It is proved that the problems (DE) and (ACP) are equivalent in a certain sense. This equivalence is used to study existence, uniqueness, the continuity on the initial values, exponential stability and norm continuity of the solutions. Applications to a reaction-diffusion equation with delay are given.