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Asymptotic stability for anisotropic Kirchhoff systems. (English) Zbl 1175.35013

Summary: We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative anisotropic Kirchhoff systems, involving the \(p(x)\)-Laplacian operator, governed by time-dependent nonlinear damping forces and strongly nonlinear power-like variable potential energies. This problem had been considered earlier for potential energies which arise from restoring forces, whereas here we allow also the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. The results are further extended to the more delicate problem involving higher order damping terms.

MSC:

35B35 Stability in context of PDEs
35L70 Second-order nonlinear hyperbolic equations
45K05 Integro-partial differential equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
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