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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
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