Singular differential equations with delay. (English) Zbl 1022.35080

The authors study the initial value problem \[ d/dt(Mu(t))=-Lu(t)+L_1u(t-1),\quad t\geq 0,\qquad u(t)=\phi (t),\quad t\in [- 1,0] \] where \(M,L,L_1\) are closed linear operators in a Banach space \(X\), \(L\) is invertible and \(T=ML^{-1}\in L(X)\). They show existence of strict solutions provided \(z=0\) is a simple pole of \((z+T)^{-1}\) and \(\phi \) satisfies a compatibility condition. In the case of a reflexive Banach space, the pole hypotheses is relaxed. The theorems extend a class of operators for which existence results hold analogous to those for regular equation with \(M=I\). A variant of the variation of parameters formula is given as well as applications to concrete equations.


35R10 Partial functional-differential equations
35K65 Degenerate parabolic equations
35G10 Initial value problems for linear higher-order PDEs
34G10 Linear differential equations in abstract spaces