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Nonlinear boundary stabilization for Timoshenko beam system. (English) Zbl 1316.35177
Summary: This paper is concerned with the existence and decay of solutions of the following Timoshenko system: \[ \left\| \begin{matrix} u'' - \mu(t)\Delta u+\alpha_1 \sum_{i = 1}^n \frac{\partial v}{\partial x_i} = 0, \text{ in }\Omega\times(0, \infty),\\ v''-\Delta v - \alpha_2 \sum_{i = 1}^n \frac{\partial u}{\partial x_i} = 0, \text{ in }\Omega \times(0, \infty),\end{matrix}\right. \] subject to the nonlinear boundary conditions: \[ \left\|\begin{matrix} u = v = 0 \text{ in }\Gamma_0 \times(0, \infty), \\ \frac{\partial u}{\partial \nu} + h_1(x, u') = 0 \text{ in }\Gamma_1 \times(0, \infty), \\ \frac{\partial v}{\partial \nu} + h_2(x, v') + \sigma(x) u = 0 \text{ in }\Gamma_1 \times(0, \infty), \end{matrix}\right. \] and the respective initial conditions at \(t=0\). Here \(\Omega\) is a bounded open set of \(\mathbb R^n\) with boundary \(\Gamma\) constituted by two disjoint parts \(\Gamma_0\) and \(\Gamma_1\) and \(\nu(x)\) denotes the exterior unit normal vector at \(x \in \Gamma_1\). The functions \(h_i(x, s)\, (i = 1, 2)\) are continuous and strongly monotone in \(s \in \mathbb R\). The existence of solutions of the above problem is obtained by applying the Galerkin method with a special basis, the compactness method and a result of approximation of continuous functions by Lipschitz continuous functions due to Strauss. The exponential decay of energy follows by using the multiplier and the Zuazua method.

35L52 Initial value problems for second-order hyperbolic systems
35B40 Asymptotic behavior of solutions to PDEs
35Q74 PDEs in connection with mechanics of deformable solids
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Full Text: DOI arXiv
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