Characterization of Well-Posedness for Second-Order Abstract Differential Equations on Continuous Function Spaces

We give a characterization of closed and densely defined linear operators A on a Banach space X such that the Dirichlet boundary value problem for the homogeneous linear second order differential equation u″(t)+Au(t)=0,t∈[0,π], is uniformly well-posed. When −A is a sectorial operator, we show that the problem is uniformly well-posed. In the case where −A is the infinitesimal generator of a uniformly bounded cosine function with 1∈ρ(Cos(2π)), we prove that the problem is uniformly well-posed for A and −A. Some concrete examples are given. © 2025 Elsevier Inc.