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General decay of solutions of a nonlinear Timoshenko system with a boundary control of memory type. (English) Zbl 1240.35338

The authors establish an asymptotic decay of the energy of the solution for the system \[ \rho _1(x)u_{tt}=\Delta u+\alpha \operatorname {div}v-\beta u-a(x)f_1(u,v), \]
\[ \rho _2(x)v_{tt}=\Delta v-\alpha \operatorname {div}u-a(x)f_2(u,v) \] in \(\Omega \times \mathbb {R}^{+}\) satisfying some initial conditions and the boundary conditions \(u(x,t)=v(x,t)=0\) on \(\Gamma _0\times \mathbb {R}^{+}\) and \[ u(x,t)=-\int _0^{t}g_1(t-s)\frac {\partial u}{\partial n}(s)\, ds,\; v(x,t)=-\int _0^{t}g_2(t-s)\frac {\partial v}{\partial n}(s)\, ds \] on \(\Gamma _1\times \mathbb {R}^{+},\) where \(\Gamma _0,\Gamma _1\) are closed and disjoint parts of \(\partial \Omega \), \(\Gamma _0\cup \Gamma _1=\partial \Omega \) and \(\Gamma _0\) is of a positive measure.

MSC:

35L51 Second-order hyperbolic systems
74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
35B40 Asymptotic behavior of solutions to PDEs
35L53 Initial-boundary value problems for second-order hyperbolic systems
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