Generalised energy conservation law for wave equations with variable propagation speed

Abstract

We investigate the long time behaviour of the L 2 -energy of solutions to wave equations with variable speed of propagation. The novelty of the approach is the combination of estimates for higher order derivatives of the coefficient with a stabilisation property.

The threshold of effective damping for semilinear wave equations

“…We notice that (33) reduces to λ ′ (t) ≥ 0 if b ≡ 0 (see [8]). Dealing with (20), thanks to the special structure of b(t) given by (32) we see that (33) is satisfied for any µ ≥ 0. there exists an anti-derivative Λ(t) of λ(t) and a constant α ∈ R such that…”

Waves in the Witten Bubble of Nothing and the Hawking Wormhole

“…Parenthetically, we mention that the De Sitter propagator in flat coordinates has been studied in [31], and some energy estimates for general wave equations with time dependent coefficients are investigated in [35].…”

The influence of oscillations on global existence for a class of semi-linear wave equations

“…The oscillating function b = b(t) is required to be smooth, positive, bounded and has to satisfy (here we follow [12]) (H 1 ) b(t) is -stabilizing towards b ∞ , i.e. we assume that there exists a strictly increasing continuous function (t), (0) = 0, (t) (t) for t>0 and (t) = o( (t)),t →∞, such that…”

“…The construction of the function b(t) was done in [12], but for convenience of the reader we give some details.…”