## General decay rate estimate for the energy of a weak viscoelastic equation with an internal time-varying delay term.(English)Zbl 1286.35040

The author investigates a viscoelastic initial-boundary value problem with a linear damping and a time-varying delay term:
\begin{aligned} &u_{tt}-\Delta u(x,t)+\alpha(t)\int_0^t g(t-s)\Delta u(x,s)\,ds+a_0u_t(x,t) +a_1u_t(x,t-\tau(t))=0,\, x\in \Omega;\\ &u(x,t)=0,\, x\in \partial\Omega\, u(x,0)=u_0(x),\;u_t(x,0)=u_1(x),\;x\in \Omega;\\ &u_t(x,t)=f_0(x,t),\;(x,t)\in \Omega\times [-\tau(0),0), \end{aligned}
where $$\Omega$$ is bounded in $$\mathbb{R}^n,(n\geq 2),\;\alpha, g$$ are positive non-increasing functions, $$a_0>0$$ and $$\tau(t)>0$$ represents the time-varying delay. By introducing suitable energy and Lyapunov functionals, under suitable assumptions a general decay rate estimate for the energy depending on both $$\alpha, g$$ is established.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35L05 Wave equation 35L15 Initial value problems for second-order hyperbolic equations 35L70 Second-order nonlinear hyperbolic equations 93D15 Stabilization of systems by feedback 74D05 Linear constitutive equations for materials with memory

### Keywords:

internal feedback; linear damping; Lyapunov functionals
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