Let us simplify the above equation further and consider

where and are constants and and . Clearly, this equation is of no interest in itself; there is no need to develop methods for analyzing an equation, which can be solved explicitly. However, considering this equation from a different perspective might lead to the development of methods that can be used in a more general situation. Since energies have turned out to be very useful in the analysis of systems of nonlinear wave equations, let us consider

We know that this quantity decreases exponentially as . Let us try to prove this statement without explicitly solving the equation. A natural first step is to differentiate:

Clearly, decreases. However, this is only a qualitative statement. In order to obtain a quantitative statement, let us introduce

We shall refer to this quantity as a correction term. It is important to note that this object has the following two properties. First

As a consequence, there are constants such that

recall that . Second,

Combining these two properties, we obtain . This estimate is optimal.

In the case of the polarized Gowdy equation (51) it is possible to carry out a similar argument. In fact, the correction

can be used to prove that [75, Section 4, pp. 668–670] for the details.

Ideas similar to the above can be used in the general Gowdy case, though there are additional complications. Due to the nonlinear character of the problem, it is necessary to divide the proof of the decay of the energy into two parts. The first part involves proving that if the energy is small initially, then the energy decays as . The second step consists of proving that the energy converges to zero. The small data result is, just as above, based on the introduction of a certain correction with the properties that

and that

Combining these two inequalities, it is possible to conclude that , assuming to be small enough initially.One way to take the step from small data to large data is to prove that converges to zero. What is known a priori is that

Just as before, this implies that decays, but not that converges to zero. However, it does prove that the right-hand side of Equation (57) is integrable. This information might not seem so useful. However, if it were possible to prove the integrability of to the future, we would be allowed to conclude that converges to zero (recall that is monotonically decreasing). This would then finish the result. In order to take the step from integrability of the right-hand side of (57) to the integrability of , we need to prove thatare bounded to the future. Note that these expressions are far from arbitrary. It is natural to integrate by parts twice and to use the equations (it is of some interest to note that the order in which one considers the two expressions is very important). Doing so leads to the desired conclusion; see [75] for the details.

It is of interest to note that the results concerning the decay rate can be generalized to a larger class of spacetimes [81].

http://www.livingreviews.org/lrr-2010-2 |
This work is licensed under a Creative Commons License. Problems/comments to |