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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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References:
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