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Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping. (English) Zbl 1240.35330
The authors are concerned with the viscoelastic Kirchhoff equation with the Balakrishnan-Taylor damping. By integral inequalities and multiplier techniques they establish polynomial decay estimates for energy of the initial-boundary problem. \[ u_{tt}-(\xi _0+\xi _1\| \nabla u(t)\| ^2_2+\sigma (\nabla u(t),\nabla u_t(t))) \Delta u+\int _0^t h(t-s)\Delta u(s)\,ds=| u| ^p u \] in \(\Omega \times [0,+\infty )\), \(u(x,0)=u_0(x)\), \(u_t(x,0)=u_1(x)\) in \(\Omega \) and \(u(x,t)=0\) in \(\partial \Omega \times [0,+\infty ),\) where \(\Omega \) is a bounded domain in \(\mathbb {R}^n\) with smooth boundary.
The work extends authors’ previous result [Demonstr. Math. 44, No. 1, 67–90 (2011; Zbl 1227.35074)], where an exponential decay and a blow up result have been established.

35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
45K05 Integro-partial differential equations
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