On t-dependent hyperbolic systems. Part 2

Abstract

We consider hyperbolic equations with time-dependent coefficients and develop an abstract framework to derive the asymptotic behaviour of the representation of solutions for large times. We are dealing with generic situations where the large-time asymptotics is of hyperbolic type.Our approach is based on diagonalisation procedures combined with asymptotic integration arguments.

Small Data Wave Maps in Cyclic Spacetime

“…Consider now the equation y tt (t) + (λα(t) − q(t)) y(t) = 0 (21) with the periodic coefficients α(t) = b 2 (t) and…”

“…Thus, X λ (t, t 0 ) gives a fundamental solution to the equation (21). In what follows we often omit subindex λ of X λ (t, t 0 ).…”

“…Due to Theorem 2.3.1 [3] there exist the open instability intervals. The Assumption ISIN states that there exists the nonempty open instability interval Λ ⊂ (0, ∞) for equation (21).…”

“…One can find in [3,12] the detailed description of functions α = α(t) and q = q(t) satisfying this condition. For instance, in Theorem 4.4.1 [3] one can find asymptotic formula, which allows to estimate the length of the instability intervals of the equation obtained from (21) by Liouville transformation. Then, according to the next lemma one can find in the instability interval Λ a number λ such that a non-diagonal element of the monodromy matrix does not vanish.…”

“…[23]) Let b(t) be defined on R non-constant, positive, smooth function, which is 1-periodic. Then there exists an open subset Λ 0 ⊂ Λ such that b 21 = 0 for all λ ∈ Λ 0 .Next we use the periodicity of b = b(t) and the eigenvalues µ 0 > 1, µ −1 0 < 1 of the matrix X λ (1, 0) to construct solutions of(21) with prescribed values on a discrete set of time. The eigenvalues of matrix X λ (1, 0) are µ 0 and µ −10 with b 11 + b 22 = µ 0 + µ −1 0 .…”