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Exponential stability in linear viscoelasticity. (English) Zbl 1117.35052
The paper concerns with the exponential decay of solutions to the following problem:
\[ \partial_{tt}u(t)+\alpha Au(t)+\beta \partial_t u(t)-\int_0^{\infty} \mu(s)Au(t-s)\,ds=0,t>0, \]
\[ u(t)=w_0(t), \,t\leq 0, \]
\[ u_t(0)=v_0, \]
where \(\alpha>0,\,\beta\geq 0, \,A \) is a strictly positive selfadjoint linear operator and \(\mu(s)\) is a summable decreasing function. Under condition \[ \mu(t+s)\leq C\exp(-\delta t) \mu(s), \,t\geq 0 \text{ and a.e. } s>0\tag{1} \] with \(\delta >0\) the authors prove the exponential stability of the corresponding semigroup \(S(t)\) for \(\beta=0\) provided smallness of a set where \(\mu\) is flat. For \(\beta>0\), (1) is sufficient for the exponential stability.

MSC:
35L90 Abstract hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
45K05 Integro-partial differential equations
47D06 One-parameter semigroups and linear evolution equations
74D05 Linear constitutive equations for materials with memory
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