Energy decay of the solution for a weak viscoelastic equation with a time-varying delay. (English) Zbl 1400.35031
Summary: In this paper, we consider a weak viscoelastic equation with internal time-varying delay
\[
u_{tt}(x, t)-\Delta u(x, t)+\alpha (t)\int_0^t g(t-s) \Delta u(x,s)ds+ \mu u_t \left(x, t-\tau(t)\right)=0
\]
in a bounded domain. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we establish a general decay result for the energy. This work generalizes and improves earlier results in the literature.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 74Dxx | Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) |
| 93D20 | Asymptotic stability in control theory |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
| 35R09 | Integro-partial differential equations |
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