Let $\mathcal{M}\subset\mathbb{R}^{3}$ be an oriented compact surface on which we consider the system: $$\left \{ \begin{array}{l@{\quad}l} u_{tt} - \Delta_{\mathcal{M}} u + a(x)g_{0}(u_{t})=0 & \text{in } \mathcal{M}\times\mathopen{)} 0,\infty( ,\\ \partial_{\nu_{co}}u +u + b(x)g(u_t)=0 & \text{on } \partial \mathcal {M}\times\mathopen{)}0,+\infty(. \end{array} \right . $$ If $\mathcal{M}$ along with the localizers a , b and the nonlinear feedbacks g , g 0 satisfy certain conditions then uniform (but not necessarily exponential) decay rates of the finite energy of solutions can be established. We present a unified approach that bridges and extends a number of earlier results on stabilization of 2nd-order hyperbolic equations on manifolds. The methodology captures geometric requirements for damping acting simultaneously on subsets of the interior and of the boundary, and shows how placements of these feedbacks can complement each other depending on the underlying surface. In addition, the results conveniently incorporate the existing theory that allows elimination of geometric conditions from the controlled boundary (in absence of nearby interior damping), and elimination of damping entirely from certain boundary neighborhoods. The model also admits feedbacks that grow sub- or super-linearly not only at the origin, but also at infinity and demonstrates an interplay between the regularity of solutions and asymptotic energy decay rates.
Applied Mathematics and Optimization – Springer Journals
Published: Feb 1, 2014
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