# zbMATH — the first resource for mathematics

On the uniform decay of the local energy. (English) Zbl 0937.35118
Author’s summary: It is proved by P. D. Lax et al. [Commun. Pure Appl. Math. 16, 477-486 (1963; Zbl 0161.08001)] and C. S. Moravetz [Commun. Pure Appl. Math. 19, 439-444 (1966; Zbl 0161.08002)] that in odd dimensional spaces any uniform decay of the local energy implies that it must decay exponentially. We extend this to even dimensional spaces and to more general perturbations (including the transmission problem) showing that any uniform decay of the local energy implies that it must decay like $$O(t^{-2n})$$, $$t\gg 1$$ being the time and $$n$$ being the space dimension.

##### MSC:
 35P25 Scattering theory for PDEs 35J15 Second-order elliptic equations 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
##### Keywords:
cutoff resolvent; local energy decay
Full Text: