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Existence and decay of solutions of a damped Kirchhoff equation. (English) Zbl 1162.35054
An abstract formulation of an initial-boundary value problem for the generalized Kirchhoff equation with a damping was studied \[ u^{\prime\prime}(t)+M(\mid A^{\frac{1}{2}}u(t)\mid^2)Au(t)+\delta u^{\prime}(t)=0,\,t>0, \] \[ u(0)=u^0,\,u^{\prime}(0)=u^1, \] where \(A\) is a linear nonnegative self-adjoint operator, \(\delta\) is a positive constant, \[ M(s)\in C^1(\mathbb R^+),M(s)\geq m_0>0. \] Existence and uniqueness of global small solutions are proved. To show the exponential decay of the energy, the authors additionally imposes the condition \(A\geq\beta ,\,\beta>0.\)

35L70 Second-order nonlinear hyperbolic equations
35B35 Stability in context of PDEs
45K05 Integro-partial differential equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs