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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.)
##### Keywords:
Galerkin method; exponential decay of energy
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##### References:
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