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Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian

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  • We consider an Euler-Bernoulli equation in a bounded domain with a local dissipation of viscoelastic type involving the $p$-Laplacian. The dissipation is effective in a suitable nonvoid subset of the domain under consideration. This equation corresponds to the plate equation with a localized structural damping when both the parameter $p$ and the space dimension equal two. First we prove existence, uniqueness, and smoothness results. Then, using an appropriate perturbed energy coupled with multiplier technique, we provide a constructive proof for the exponential and polynomial decay estimates of the underlying energy. It seems to us that this is the first time that a dissipation involving the $p$-Laplacian is used in the framework of stabilization of second order evolution equations with locally distributed damping.
    Mathematics Subject Classification: Primary: 93D15; Secondary: 35D30, 35D35, 35Q74, 37L15, 74K20.


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