Now, for the second term in the right, considering the previous result together with Eq. (22), we have

•> , il — iJ-4c7r2fc2(t-t0)\ bk(e~ f*2nikv - l)e-^^dr\ < #fc|U -

and taking into account that ^-4e?r2fc2-- < 1, we get Eq. (21). □

Proposition 3.1. i) We assume that I = YlkeKnJ bkC-k = 0, with K fl J finite set, and that a solution of Eq. (16) satisfies /Q°° |u(s)|g?s < oo. Then the system reaches the rest stationary solution, that:

(v(t) —> 0, as t —> oo afc(i) —► &fe, as i oo dfc(i) -> ¿bfc, ast-> oo ii) Conversely, if I = YlkexnJ &fcc-k ^ 0 then for every solution

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