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Speed of convergence on the real axis of resonances. (Vitesse de convergence vers le réel des résonnances.) (French) Zbl 0885.35063

The author considers the boundary-value problem \[ (\partial^2_t-\Delta)u= 0\quad\text{in }\Theta^c\times\mathbb{R}_t,\quad u|_{t=0}= u_0\in H^1_0(\Theta^c),\quad \partial_tu|_{t=0}= u_1\in L^2(\Theta^c),\quad u|_{\partial\Theta}= 0, \] where \(\Theta\subset\mathbb{R}^d\) denotes a compact obstacle in \(\mathbb{R}^d\), \(u_0\) and \(u_1\) are given functions in \(\Theta^c= \mathbb{R}^d\backslash\Theta\) while \(\Delta= \sum_{i,j} \partial_ia_{ij}(x)\partial_j\) stands for the Laplacian operator (\(a_{ij}\) are also supposed to be given). The main aim of the author is to prove that under certain restrictions there can be found constants \(c_1\), \(c_2\), \(\varepsilon_0>0\) such that the outgoing resolvent \(R(\lambda)\) defined by \(R(\xi)f= \int^{+\infty}_0 \exp(-\text{it }\xi)v(t)dt\) has no pole in the region \(\{\lambda\in\mathbb{C},\text{ Im }\lambda<\varepsilon_0 \exp(-c_2|\text{Re }\lambda |)\}\cap\{|\lambda|> c_1\}\). The function \(v(t)\) appearing in the above expression of \(R(\xi)\) stands for the solution corresponding to the initial data \((0,f)\). As corollaries of the main result, he also shows that
i) there exist \(\chi_1\), \(\chi_2\), \(c\), and \(c_3>0\) such that in the region mentioned above one has \[ |\chi_1 R(\lambda) \chi_2|_{C(L^2,H^d_c)}\leq c\exp(c_3|\text{Re } \lambda|) \] and
(ii) for every \(R>0\) and every \(k>0\) there exists a \(C>0\) such that for every data \((u_0,u_1)\) with support in \((B(0, R_1)\cap\overline\Theta^c)\) one has \[ \Biggl( \int_{B(0,R)\cap\Theta^c} |\nabla u|^2(t)+ |\partial_tu|^2(t)\Biggr)^{1/2}\leq {C\over \log(2+ t)^k} |(u_0, u_1)|. \] The proof is given for the simplest case, where \(\Theta\) is of a ball.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
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