Semigroups and Linear Partial Differential Equations with Delay

Abstract

We prove the equivalence of the well-posedness of a partial differential equation with delay and an associated abstract Cauchy problem. This is used to derive sufficient conditions for well-posedness, exponential stability and norm continuity of the solutions. Applications to a reaction-diffusion equation with delay are given.

Robustness with respect to small time-varied delay for linear dynamical systems on Banach spaces

“…Dropping the assumption that B is bounded, Liu ([Liu]) showed that if A generates a holomorphic semigroup and B is (−A) α -bounded, then the exponential stability of (1.1) (with τ (t) ≡ r) on the phase space C(−r, 0; D(A)) is robust. Bátkai et al ([Ba1,Ba2]) proved a similar result on the phase space X × L p (−r, 0; D(B)).…”

“…The robustness of delay equations has been studied by many authors (see cf. [Ba1,Ba2,Da,EN,Hu,FN,JGH,Liu]). …”

The Modulus Semigroup for Linear Delay Equations

“…It is shown in [2] that the delay equation (DE) is equivalent to an abstract Cauchy problem U (t) = AU(t) (t 0),…”

On strong convex compactness property of spaces of nonlinear operators

“…Almost at the same time as [15], Schliichtermann [14] proved that both the space of weakly compact operators and the space of conditionally weakly compact 412 X. Song and J. Peng [2] operators enjoy the property. Moreover, he remarked that the mentioned conclusions can be extended to the nonlinear case if the uniform boundedness in Voigt's definition of the strong convex compactness property is replaced with the uniform integrability property (see, [14,Remark 2.4] or (2.3) below).…”

“…If the perturbation operator K is linear, then it is easy to show that so is the perturbed semigroup {S t } t^ o whenever the unperturbed semigroup {T t }t% o is strongly continuous or weakly completely continuous. See [2,5,6,7,8,11] for details about property persistence of operator semigroups under linear perturbation. However, in the nonlinear case, we do not know if the perturbed {S t }t% o inherits the weak continuity, strong continuity or weakly complete continuity of the unperturbed semigroup…”