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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.

MSC:
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.)
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