ABSTRACT
The ideas in Section 6.3 for Example 6 can be used to treat more general second-order systems. As we shall see in this chapter, the strength of the damping plays a direct role in the regularity properties for the associated semigroup and consequently, the solutions of related equations. Consider the general abstract second-order system
y¨(t) +A2y˙(t) +A1y(t) = f(t)
or, in variational form
〈y¨(t), ϕ〉V ∗,V + σ1(y(t), ϕ) + σ2(y˙(t), ϕ) = 〈f(t), ϕ〉V ∗,V (8.1)
where H is a given complex Hilbert space. As usual, we assume that σ1 and σ2 are sesquilinear forms on V where V ↪→ H ↪→ V ∗ is a Gelfand triple. We also assume that σ1 is continuous, V -elliptic, and symmetric (σ1(ϕ,ψ) = σ1(ψ,ϕ)). We assume that σ2 is continuous and satisfies a weakened ellipticity condition which we formally call H-semiellipticity.
Definition 8.1 A sesquilinear form σ on V is H-semielliptic if there is a constant b ≥ 0 such that
