Linear neutral partial differential equations: a semigroup approach

Abstract

We study linear neutral PDEs of the form (∂/∂t)F u t = BFu t + Φu t , t ≥ 0; u 0 (t) = ϕ(t), t ≤ 0, where the function u(·) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator Φ, we construct a C 0 -semigroup on C 0 (R − ,X) yielding the solutions of the equation.

An age‐dependent population equation with diffusion and delayed birth process

“…Moreover, as in [9], one can prove that, for all λ ∈ C, 19) where the bounded linear operators λ : E → Ᏺ are defined as…”

An age‐dependent population equation with delayed birth process

“…Only particular results, e.g. for neutral di erential equations [17] or analytic semigroups [18], are known while a general perturbation result is still missing.…”

Hyperbolicity of solution semigroups for linear neutral differential equations

“…In this section, we briefly recall the results obtained in [11] on the well-posedness of (1.1) as well as the representation of the resolvent of the semigroup solving (1.1). Under the assumptions from Sect.…”

“…2.3], Wu and Xia [16], Adimy and Ezzinbi [1], Nagel and Huy [11]. Especially, in [11] we have proposed an abstract treatment of these equations by choosing a Banach space X and considering the solution u(·) as a function from [−r, ∞) to X for some positive constant r. Moreover, B is a linear operator on X (representing the partial differential operator), while F and are linear operators from an X-valued function space, e.g., C([−r, 0], X) into X. More precisely, we make the following assumption throughout the paper.…”

“…In [11], under these assumptions we have solved (1.1) by constructing an appropriate strongly continuous semigroup on the space E. This semigroup was obtained by proving that a certain operator (see Definition 2.3) satisfies the Hille-Yosida conditions as long as we can write the difference operator as F = δ 0 − with being "small" (see (2.5)). It was also shown that the smallness of can be replaced by the condition "having no mass in 0" (see Definition 2.5).…”