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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
\[ E_{\chi}(t)\leq C\left(\frac{\log(2+t)}{\langle t\rangle} \right)^{2s/(3n+ \varepsilon)}\left(\|u_0\|^2_{H^{1+s}(X)}+ \|u_1\|^2_{H^{s}(X)}\right). \]

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