# zbMATH — the first resource for mathematics

Applications of cutoff resolvent estimates to the wave equation. (English) Zbl 1189.58012
The solutions $$u(x,t)$$ to the following wave equation on $$X\times\mathbb R_t$$
$\begin{gathered} (-D_t^2-\Delta)u(x,t)=0,\,(x,t)\in X\times[0,\infty);\\ u(x,0)=u_0\in H^1(X)\cap C_c^\infty(X),\quad D_tu(x,0)=u_1\in L^2(X)\cap C_c^\infty(X), \end{gathered}\tag $$*$$$
where $$D_t=-i\partial_t$$ are considered. For $$u$$ satisfying $$(*)$$ and $$\chi\in C_c^\infty(X)$$, the local energy is defined as $$E_\chi(t)=1/2(\|\chi\partial_tu\|^2_{L^2(X)}+ \|\chi u\|^2_{H^1(X)})$$.
The main result is given in Theorem 1 and could be described as: if there is a hyperbolic trapped set which is sufficiently “thin” ($$P_E(1/2)<0$$, for definitions see [S. Nonnenmacher and M. Zworski, Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math., Palaiseau 2005–2006, XXII1–XXII6 (2006; Zbl 1122.35090)]), then the local energy decays at least polynomially, with an exponent depending on the dimension.
More precisely: Suppose $$(X,g)$$ is a Riemannian manifold of odd dimension $$n\geq3$$ without boundary, asymptotically Euclidian, and $$(X,g)$$ admits a compact hyperbolic fractal trapped set $$K_E$$ in the energy level $$E>0$$ with topological pressure $$P_E(1/2)<0$$. Assume there is no other trapping and $$(-\Delta-\lambda^2)^{-1}$$ admits a holomorphic continuation to a neighbourhood around $$\mathbb R\subset\mathbb C$$. Then for each $$\varepsilon>0$$ and $$s>0$$, there is a constant $$C>0$$, depending on $$\varepsilon$$, $$s$$, and support of $$\chi$$, $$u_0$$, and $$u_1$$, such that
$E_{\chi}(t)\leq C\left(\frac{\log(2+t)}{\langle t\rangle} \right)^{2s/(3n+ \varepsilon)}\left(\|u_0\|^2_{H^{1+s}(X)}+ \|u_1\|^2_{H^{s}(X)}\right).$

##### MSC:
 58J45 Hyperbolic equations on manifolds 35L15 Initial value problems for second-order hyperbolic equations 58J05 Elliptic equations on manifolds, general theory
Full Text: