Spectra of Gurtin-Pipkin type of integro-differential equations and applications to waves in graded viscoelastic structures. (English) Zbl 1514.47116
The paper is concerned with the second-order Gurtin-Pipkin type of integro-differential equations \[u_{tt}(t)+T_au(t)+\displaystyle\int_0^t K_t(t-s)T_bu(s)\,ds=h,\tag{1}\] where \(T_a\) and \(T_b\) are unbounded self-adjoint operators with compact resolvents in a separable Hilbert space \(\mathcal{H}\), and the exponential kernel \(K\) is defined by \(K(t)=\sum_{j=1}^N a_j e^{-b_jt}\) with \(a_j>0\) and \(0<b_1<b_2<\dots <b_N\). The author investigates the spectral properties and spectral enclosures for equation \((1)\), and then he presents some applications to wave equations with Boltzmann damping.
Reviewer: Rodica Luca (Iaşi)
MSC:
| 47N20 | Applications of operator theory to differential and integral equations |
| 45K05 | Integro-partial differential equations |
| 35L05 | Wave equation |
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