As has been noted before, we have the conserved quantities , and given by Equations(25) – (27). We shall also use the notation , , and[75, Lemma 8.2, p. 681]. Note that in the spatially homogeneous case, [75, (6.9) – (6.10), p. 676] as well as the adjacent text. Consequently, it seems unlikely that it should be possible to approximate solutions such that is negative by spatially homogeneous solutions.
In practice, it is often convenient to apply an isometry to a solution so that the conserved quantities become as simple as possible. This is achieved in [75, Lemma 8.2, p. 681]:
Lemma 1 Consider a solution to Equations (11)–(12). If , there is an isometry such that if and are the constants of the transformed solution, then and . If , there is an isometry such that the constants of the transformed solution are and or .
Analyzing the asymptotic behavior of the transformed solution and then transforming back is often more convenient than analyzing the original solution.
Returning to the question of the asymptotics, we wish to interpret the conserved quantities as ODEs for and . Due to [75, Lemma 8.1, p. 680], we have the following result:
Naively, it would seem natural to consider the integral expressions to be error terms and to interpret what remains as ODEs for the averages. On the other hand, it would then seem that we have too many equations. In the end, the situation turns out to be somewhat more complicated; see below.
Among other things, Equations (61) – (63) imply the existence of a constant such that
is that it decays as . In other words, first taking the average and then taking the square leads to decay of the form . First taking the square and then taking the average leads to decay of the form . This behavior reflects the same sort of asymptotics as those characterized by Equation (52).
Naively estimating the integral on the right-hand side of Equation (61) leads to the conclusion that it is bounded but no more. Consequently, it seems unreasonable to think of this term as an error term. On the other hand, integrating with respect to time might lead to an improvement. In fact, since decays very quickly, see Equation (64), replacing with in Equation (61) leads to a term, which tends to zero. Consequently, we can replace by in (61) with a small error. Integrating Equation (61) and using such ideas leads to [75, Lemma 8.9, p. 685]:
Note that in the case of , this result gives detailed information concerning the asymptotics of ; see [75, Corollary 8.10, p. 685]:
for all .
Clearly, Lemma 3 yields important information concerning the asymptotics. Is it possible to apply similar ideas to Equations (62) and (63)? It turns out to be necessary to combine both equations in order to obtain a single equation for . The problem is the last term in Equation (62) and the second to last term in Equation (63). However, combining partial integrations, Taylor expansions in the last term in Equation (62) with various estimates, such as Equation (64), leads to [75, Lemma 8.8, p. 684]:
In the case of it is convenient to apply this result with . This leads to [75, Proposition 8.11, p. 685]:
In case of , we consequently have detailed information concerning the asymptotic behavior of as well. Furthermore, as was observed earlier, given a solution with the property that , there is an isometry of hyperbolic space such that the transformed solution is such that the corresponding equals zero. Thus, the only case that remains to be analyzed is . This case is more complicated (but more interesting). We shall therefore omit a description of the analysis.
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