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Parabolic singular limit of a wave equation with localized boundary damping

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  • We will consider the family of wave equations with boundary damping

    $\qquad\qquad \qquad\qquad \epsilon u_{t t} -\Delta u + \lambda u =f $ on $\Omega \times (0,T)$

    $(P_{\epsilon, \lambda, \Gamma_0})\qquad\qquad u_t + \frac{\partial u}{\partial \vec{n}} =g$ on $\Gamma_1 \times (0,T) $

    $\qquad\qquad u=0 $ on $\Gamma_0 \times (0,T)$

    where $0< \epsilon \leq \epsilon_0$, $\Omega \subset \mathbb R^N$ is a regular open connected set, $\lambda \geq 0$ and $\Gamma = \Gamma_0\cup \Gamma_1$ is a partition of the boundary of $\Omega$. We will also consider the case where $\Gamma_0$ is empty (see below for more precise assumptions on $\lambda$, $\Omega$ and $\Gamma_0$, $\Gamma_1$).
    For this problem the corresponding formal singular perturbation at $\epsilon =0$ is

    $\qquad\qquad \qquad\qquad -\Delta u + \lambda u =f$ on $\Omega \times (0,T) $

    $(P_{0, \lambda, \Gamma_0}) \qquad\qquad u_t + \frac{\partial u}{\partial \vec{n}} =g$ on $\Gamma_1 \times (0,T) $

    $\qquad\qquad u=0 $ on $ \Gamma_0 \times (0,T)$

    We are here concerned with the well possedness of both problems for the non--homogeneous case, i.e. $f=f(t,x)$, $g=g(t,x)$, and with the convergence, as $\epsilon$ approaches $0$, of the solutions of $(P_{\epsilon, \lambda, \Gamma_0})$ to solutions of $(P_{0, \lambda, \Gamma_0})$.

    Mathematics Subject Classification: 35L10, 35J05.


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