*Preliminary estimates on components.*

Let us consider the notations introduced in Section 3, where we reduced ourselves to proving (3.19) through (3.21).
In this first paragraph we derive some *k*-independent estimates on ${\rho}_{k}(t)$ and ${\theta}_{k}(t)$ that are needed several times in the sequel.
The constants ${M}_{8},\mathrm{\dots},{M}_{23}$, we introduce hereafter, depend on the solution (as the constants ${M}_{1},\mathrm{\dots},{M}_{7}$ of Section 3), but they do not depend on *k*.
First of all, from (3.17) and (3.18), it follows that

$\sum _{k=0}^{\mathrm{\infty}}{\rho}_{k}^{2}(t)\le {M}_{4}$

and, in particular, we find

${\rho}_{k}(t)\le {M}_{8}\mathit{\hspace{1em}}\text{for all}t\ge 0,\text{and all}k\in \mathbb{N},$(7.1)

and

$\sum _{k=0}^{\mathrm{\infty}}{\rho}_{k}^{2}(t){\mathrm{sin}}^{2}{\theta}_{k}(t)\le {M}_{4}.$(7.2)

From this estimate and (3.16), it follows that

$|{\rho}_{k}^{\prime}(t)|\le {M}_{9}{\rho}_{k}(t)\mathit{\hspace{1em}}\text{for all}t\ge 0,\text{and all}k\in \mathbb{N}.$(7.3)

This implies, in particular, that

$\sum _{k=0}^{\mathrm{\infty}}{[{\rho}_{k}^{\prime}(t)]}^{2}\le {M}_{10}\mathit{\hspace{1em}}\text{for all}t\ge 0$

and

$|{\rho}_{k}^{\prime}(t)|\le {M}_{11}\mathit{\hspace{1em}}\text{for all}t\ge 0,\text{and all}k\in \mathbb{N}.$(7.4)

Moreover, from (7.3), it follows that

${\rho}_{k}(t)\le {\rho}_{k}(0){e}^{{M}_{9}t}\mathit{\hspace{1em}}\text{for all}t\ge 0,\text{and all}k\in \mathbb{N}.$(7.5)

Let us consider now the series

$\sum _{k=0}^{\mathrm{\infty}}{\rho}_{k}^{m}(t),\sum _{k=0}^{\mathrm{\infty}}{[{\rho}_{k}^{m}(t)]}^{\prime},$

where $m\ge 2$ is a fixed exponent (in the sequel we need only the cases $m=2$ and $m=4$).
From the previous estimates, we have

$\sum _{k=0}^{\mathrm{\infty}}{\rho}_{k}^{m}(t)\le {M}_{12},\sum _{k=0}^{\mathrm{\infty}}|{[{\rho}_{k}^{m}(t)]}^{\prime}|\le {M}_{12},$(7.6)

where of course the constant ${M}_{12}$ depends also on *m*.
Moreover, from (7.5) and the square-integrability of the sequence ${\rho}_{k}(0)$, it follows that both series are normally convergent on compact subsets of $[0,+\mathrm{\infty})$.

We stress that we can not hope that these series are normally convergent in $[0,+\mathrm{\infty})$, even when $m=2$.
Indeed, normal convergence would imply uniform convergence, and hence the possibility to exchange the series and the limit as $t\to +\mathrm{\infty}$, while the conclusion of Theorem 2.1 says that this is not the case, at least when *J* is an infinite set.

Finally, plugging (3.16) and (7.2) into (3.15), after integration, we obtain that

${\theta}_{k}(t)=-{\lambda}_{k}{e}^{t}-{\psi}_{k}(t),$(7.7)

for a suitable function ${\psi}_{k}:[0,+\mathrm{\infty})\to \mathbb{R}$ of class ${C}^{1}$ satisfying

$|{\psi}_{k}^{\prime}(t)|\le {M}_{13}\mathit{\hspace{1em}}\text{for all}t\ge 0,\text{and all}k\in \mathbb{N}.$(7.8)

*Estimates on trigonometric coefficients.*
For every $k\in \mathbb{N}$, we set

${a}_{k}(t):={\mathrm{sin}}^{2}{\theta}_{k}(t)-\frac{1}{2},{b}_{k}(t):={\mathrm{sin}}^{4}{\theta}_{k}(t)-\frac{3}{8}$

and, for every $k\ne h$,

${c}_{h,k}(t):={\mathrm{sin}}^{2}{\theta}_{h}(t){\mathrm{sin}}^{2}{\theta}_{k}(t)-\frac{1}{4}.$

These functions represent the corrections we have to take into account when we approximate the trigonometric functions with their time-average, as we did at the beginning of Section 4.

It is easy to see that

$sup\{|{a}_{k}(t)|,|{b}_{k}(t)|,|{c}_{h,k}(t)|\}\le 1\mathit{\hspace{1em}}\text{for all}t\ge 0,$(7.9)

where the supremum is taken over all admissible indices or pairs of indices.
Now we claim that

$\left|{\displaystyle {\int}_{t}^{s}}{a}_{k}(\tau )\mathit{d}\tau \right|\le {M}_{14}{e}^{-t}$$\text{for all}s\ge t\ge 0,\text{and all}k\in \mathbb{N},$(7.10)$\left|{\displaystyle {\int}_{t}^{s}}{b}_{k}(\tau )\mathit{d}\tau \right|\le {M}_{15}{e}^{-t}$$\text{for all}s\ge t\ge 0,\text{and all}k\in \mathbb{N},$(7.11)

and

$\left|{\int}_{t}^{s}{c}_{h,k}(\tau )\mathit{d}\tau \right|\le {M}_{16}\left(1+\frac{1}{|{\lambda}_{k}-{\lambda}_{h}|}\right){e}^{-t}\mathit{\hspace{1em}}\text{for all}s\ge t\ge 0,\text{and all}h\ne k.$(7.12)

In order to prove (7.10), we just observe that

${a}_{k}(t)=-\frac{1}{2}\mathrm{cos}(2{\theta}_{k}(t)),$

and hence, by (7.7),

${a}_{k}(t)=-\frac{1}{2}\mathrm{cos}(-2{\lambda}_{k}{e}^{t}-2{\psi}_{k}(t))=-\frac{1}{2}\mathrm{cos}(2{\lambda}_{k}{e}^{t}+2{\psi}_{k}(t)).$

Thanks to (7.8), the assumptions of Lemma 6.1 are satisfied with $\alpha :=2{\lambda}_{k}$, ${L}_{3}:=2{M}_{13}$ and $\psi (t):={\psi}_{k}(t)$.
Thus, we obtain that

$\left|{\int}_{t}^{s}{a}_{k}(\tau )\mathit{d}\tau \right|\le \frac{3+2{M}_{13}}{2{\lambda}_{k}}{e}^{-t}\le {M}_{17}{e}^{-t},$

where in the last inequality, we exploited that all eigenvalues are larger than a fixed positive constant.

The proof of (7.11) is analogous, just starting from the trigonometric identity

${b}_{k}(t)=-\frac{1}{2}\mathrm{cos}(2{\theta}_{k}(t))+\frac{1}{8}\mathrm{cos}(4{\theta}_{k}(t)).$

Also the proof of (7.12) is analogous, but in this case the trigonometric identity is

${c}_{h,k}=-\frac{1}{4}\mathrm{cos}(2{\theta}_{h})-\frac{1}{4}\mathrm{cos}(2{\theta}_{k})+\frac{1}{8}\mathrm{cos}(2{\theta}_{h}+2{\theta}_{k})+\frac{1}{8}\mathrm{cos}(2{\theta}_{h}-2{\theta}_{k}).$

All the four terms can be treated through Lemma 6.1, but now in the last term the differences between eigenvalues are involved.
As a consequence, for the last term, we obtain an estimate of the form

$\left|{\int}_{t}^{s}\mathrm{cos}(2{\theta}_{h}(\tau )-2{\theta}_{k}(\tau ))\mathit{d}\tau \right|\le \frac{3+4{M}_{13}}{2|{\lambda}_{k}-{\lambda}_{h}|}{e}^{-t}.$

If we want this estimate to be uniform for $k\ne h$, we have to assume that the differences between eigenvalues are bounded away from 0, and this is exactly the point where assumption (2.7) comes into play in the proof of Theorem 2.5.
*Equation for the energy.*
Let $R(t)$ be the total energy as defined in (3.17).
We claim that $R(t)$ solves a differential equation of the form

${R}^{\prime}(t)=R(t)-\frac{1}{2}{R}^{2}(t)-\frac{1}{4}\sum _{k=0}^{\mathrm{\infty}}{\rho}_{k}^{4}(t)+{\mu}_{1}(t)+{\mu}_{2}(t),$(7.13)

where (for the sake of shortness, we do not write the explicit dependence on *t* in the right-hand sides)

${\mu}_{1}(t):=2\sum _{k=0}^{\mathrm{\infty}}({\mathrm{\Gamma}}_{1,k}{\rho}_{k}^{2}+{a}_{k}{\rho}_{k}^{2}-{b}_{k}{\rho}_{k}^{4}),{\mu}_{2}(t):=-2\sum _{k=0}^{\mathrm{\infty}}\left({\rho}_{k}^{2}\sum _{i\ne k}{c}_{i,k}{\rho}_{i}^{2}\right).$(7.14)

We also claim that ${\mu}_{1}(t)$ satisfies

$\left|{\int}_{t}^{s}{\mu}_{1}(\tau )\mathit{d}\tau \right|\le {M}_{18}{e}^{-t}\mathit{\hspace{1em}}\text{for all}s\ge t\ge 0.$(7.15)

The verification of (7.13) is a lengthy but elementary calculation, which starts by writing

${R}^{\prime}(t)=2\sum _{k=0}^{\mathrm{\infty}}{\rho}_{k}(t){\rho}_{k}^{\prime}(t),$

and by replacing ${\rho}_{k}^{\prime}(t)$ with the right-hand side of (3.14).
The crucial point is that when computing the product

${\rho}_{k}^{2}{\mathrm{sin}}^{2}{\theta}_{k}\cdot \sum _{i=0}^{\mathrm{\infty}}{\rho}_{i}^{2}{\mathrm{sin}}^{2}{\theta}_{i},$

one has to isolate the term of the series with $i=k$.
In this way, the product becomes

${\rho}_{k}^{4}{\mathrm{sin}}^{4}{\theta}_{k}+{\rho}_{k}^{2}\sum _{i\ne k}{\rho}_{i}^{2}{\mathrm{sin}}^{2}{\theta}_{i}{\mathrm{sin}}^{2}{\theta}_{k},$

and now one can express ${\mathrm{sin}}^{4}{\theta}_{k}$ in terms of ${b}_{k}$, and ${\mathrm{sin}}^{2}{\theta}_{i}{\mathrm{sin}}^{2}{\theta}_{k}$ in terms of ${c}_{i.k}$.
The rest is straightforward algebra.

The proof of (7.15) follows from several applications of Lemma 6.3 with different choices of ${f}_{k}(t)$ and ${g}_{k}(t)$.

•

For the term ${\mathrm{\Gamma}}_{1,k}{\rho}_{k}^{2}$, we choose ${f}_{k}(t):={\rho}_{k}^{2}(t)$ and ${g}_{k}(t):={\mathrm{\Gamma}}_{1,k}(t)$.
Indeed, the assumptions on ${f}_{k}(t)$ follow from (7.6) with $m=2$ and from the normal convergence of the same series on compact subsets of $[0,+\mathrm{\infty})$, while the assumptions on ${g}_{k}(t)$ follow from (3.16).

•

For the term ${a}_{k}{\rho}_{k}^{2}$, we choose ${f}_{k}(t):={\rho}_{k}^{2}(t)$ and ${g}_{k}(t):={a}_{k}(t)$.
The assumptions on ${f}_{k}(t)$ are satisfied as before, while those on ${g}_{k}(t)$ follow from (7.9) and (7.10).

•

For the term ${b}_{k}{\rho}_{k}^{4}$, we choose ${f}_{k}(t):={\rho}_{k}^{4}(t)$ and ${g}_{k}(t):={b}_{k}(t)$.
Now we need the estimates for the series (7.6) with $m=4$ in order to verify the assumptions on ${f}_{k}(t)$, and (7.9)
and (7.11) in order to provide the requires estimates on ${g}_{k}(t)$.

*Equation for quotients.*
For every pair of indices *h* and *k* in *J*, we consider the ratio ${Q}_{h,k}(t)$ introduced in (4.8).
We remind that components with indices in *J* never vanish, and therefore the quotient is well defined and positive for every $t\ge 0$.
After some lengthy calculations, we obtain

${Q}_{h,k}^{\prime}(t)={\alpha}_{h}(t){Q}_{h,k}(t)(1-{Q}_{h,k}^{2}(t))+{\alpha}_{h}(t){\beta}_{h,k}(t){Q}_{h,k}^{3}(t)+{\gamma}_{h,k}(t){Q}_{h,k}(t),$(7.16)

where

${\alpha}_{h}(t):=\frac{1}{8}{\rho}_{h}^{2}(t),{\beta}_{h,k}(t):=8({c}_{h,k}(t)-{b}_{k}(t)),$${\gamma}_{h,k}(t):={a}_{k}-{a}_{h}+{\mathrm{\Gamma}}_{1,k}-{\mathrm{\Gamma}}_{1,h}+{\rho}_{h}^{2}({b}_{h}-{c}_{h,k})+\sum _{i\notin \{h,k\}}{\rho}_{i}^{2}({c}_{i,h}-{c}_{i,k}).$

We observe that the first term of equation (7.16) is the same as in equation (4.10), which was derived by neglecting all the rest.

We claim that

$sup\{|{\alpha}_{h}(t)|,|{\alpha}_{h}^{\prime}(t)|,|{\beta}_{h,k}(t)|,|{\gamma}_{h,k}(t)|\}\le {M}_{19}\mathit{\hspace{1em}}\text{for all}t\ge 0,$(7.17)

where the supremum is taken over all admissible indices or pairs of indices, and that

$\left|{\displaystyle {\int}_{t}^{s}}{\beta}_{h,k}(\tau )\mathit{d}\tau \right|\le {M}_{20}\left(1+{\displaystyle \frac{1}{|{\lambda}_{k}-{\lambda}_{h}|}}\right){e}^{-t},$(7.18)$\left|{\displaystyle {\int}_{t}^{s}}{\gamma}_{h,k}(\tau )\mathit{d}\tau \right|\le {M}_{21}\left(1+{\displaystyle \frac{1}{|{\lambda}_{k}-{\lambda}_{h}|}}+\underset{i\notin \{h,k\}}{sup}\left({\displaystyle \frac{1}{|{\lambda}_{i}-{\lambda}_{k}|}}+{\displaystyle \frac{1}{|{\lambda}_{i}-{\lambda}_{h}|}}\right)\right){e}^{-t}$(7.19)

for every pair of admissible indices and every $s>t\ge 0$.
We point out that the supremum in (7.19) is finite because the sequence of eigenvalues is increasing.

Estimate (7.17) follows from (7.1) and
(7.4) in the case of ${\alpha}_{h}(t)$ and ${\alpha}_{h}^{\prime}(t)$, from (7.9) in the case of ${\beta}_{h,k}(t)$, and from
(7.9), (3.16) and (3.18) in the case of ${\gamma}_{h,k}(t)$.

Estimate (7.18) follows from (7.11) and (7.12).

Finally, in order to verify (7.19), we consider the expression for ${\gamma}_{h,k}$, and we apply

•

inequality (7.10) to the term ${a}_{k}-{a}_{h}$,

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inequality (3.16) to the term ${\mathrm{\Gamma}}_{1,k}-{\mathrm{\Gamma}}_{1,h}$,

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Lemma 6.2, (7.11) and (7.12) to the term ${\rho}_{h}^{2}({c}_{h,k}-{b}_{h})$,

•

Lemma 6.3 and (7.12) to the last term (the series).

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