Scattering for strictly hyperbolic systems with time‐dependent coefficients

Abstract

The present paper is devoted to finding conditions on the occurrence of scattering for strictly hyperbolic systems with time-dependent coefficients whose time-derivatives are in L 1 in time. More precisely, it will be shown that the solutions are asymptotically free if the coefficients are stable in the sense that their improper Riemann integrals converge as t → ±∞, while each nontrivial solution with radially symmetric data is never asymptotically free provided that the coefficients are not stable as t → ±∞. As a by-product, wave and scattering operators can be constructed. An important feature is that assumptions on only one derivative of the coefficients are made so that the results would be applicable to the asymptotic behaviour of Kirchhoff systems.

Global well-posedness of Kirchhoff systems

“…Let us first analyse certain basic properties of characteristic roots ϕ k (t, ξ). The next proposition is established in [17].…”

“…In this section we shall derive asymptotic integrations for linear hyperbolic systems with time-dependent coefficients, a kind of representation formula for their solutions. In fact, we have discussed such arguments in our recent paper [17] in the context of the scattering problems. To make the argument self-contained, we must give the proof completely, because the Fourier integral form of solutions U to Kirchhoff system (1.2) will be obtained by a careful observation of the construction of asymptotic integrations for linear systems.…”

“…The next proposition is known as Levinson's lemma in the theory of ordinary differential equations (see Coddington and Levinson [3], and also Hartman [9]); the new feature here is the additional dependence on ξ, which is crucial for our analysis (see also Proposition 2.3 from [17] and Theorem 3.1 from [16]). Proposition 3.3.…”

Asymptotically free property of the solutions of an abstract linear hyperbolic equation with time-dependent coefficients

“…(1.3) with time-dependent coefficients. There are a number of results concerning (1.2)-(1.3) (see, for example, [1,8,10,11,12,6], [14, Section 2] and references therein).…”

“…10) for every 0 ≤ s ≤ t. From (3.5), it follows thatlim k→∞ y k (t) − y(t) H×H = 0 for every t ∈ (0, ∞) \ N 1 , and therefore lim k→∞ (E(λ)y k )(t) − (E(λ)y)(t) H×H = 0…”