for . Differentiating with respect to , integrating by parts and using Equation (29) leads to the conclusion that the energies are decaying. Combining this observation with Sobolev embedding leads to the conclusion that, regardless of the choice of , is bounded to the future. Consequently, regardless of the choice of , does not grow faster than linearly to the future. Inserting this information into Equation (29) and integrating yields two smooth functions and such that[50, Theorem III.5, pp. 102–103].
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