Existence and energy decay of solutions for the Euler-Bernoulli viscoelastic equation with a delay. (English) Zbl 1326.35188
Summary: In this paper, we consider initial-boundary value problem of Euler-Bernoulli viscoelastic equation with a delay term in the internal feedbacks. Namely, we study the following equation
\[
u_{tt}(x,t)+ \Delta^2 u(x,t)-\int\limits_0^t g(t-s)\Delta^2 u(x,s)\mathrm{d}s+\mu_1u_t(x,t)+\mu_2 u_t(x,t-\tau)=0
\]
together with some suitable initial data and boundary conditions in \(\Omega\times (0,+\infty)\). For arbitrary real numbers \(\mu_1\) and \(\mu_2\), we prove that the above-mentioned model has a unique global solution under suitable assumptions on the relaxation function \(g\). Moreover, under some restrictions on \(\mu_1\) and \(\mu_2\), exponential decay results of the energy for the concerned problem are obtained via an appropriate Lyapunov function.
MSC:
| 35L35 | Initial-boundary value problems for higher-order hyperbolic equations |
| 93D15 | Stabilization of systems by feedback |
| 35R09 | Integro-partial differential equations |
| 74D05 | Linear constitutive equations for materials with memory |
| 35B40 | Asymptotic behavior of solutions to PDEs |
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