On the asymptotics of the nonreal spectrum of the integro-differential Gurtin-Pipkin equation with relaxation kernels representable in the form of the Stielties integral (Q2135107)

The author studies the Gurtin-Pipkin integro-differential equation of the form \[ u''(t) + A^{2}u(t)-\int^{t}_{0}R(t-\tau)A^{2}u(\tau)d\tau=f(t),\tag{1} \] where \( A \) is an unbounded self-adjoint positive operator with compact inverse in a separable Hilbert space \( H \) and \( u(t) \) and \( f(t) \) are vector functions \( \mathbb{R}_{+}\rightarrow H. \) Equation (1) contains a relaxation kernel \( R(t) \) representable as the Stiletjes integral as follows: \[ R(t)=\int_{0}^{+\infty}e^{-tx}d\sigma(x),\quad t\in [0,\infty). \] Note that the Gurtin-Pipkin equation (1) is an abstract hyperbolic equation in a separable Hilbert space \( H \). The main goal of the present paper is to find the asymptotics of the non-real spectrum of the symbol of the integro-differential equation (1). It is shown that this asymptotics can be used in the study of high-frequency oscillations in determining wave propagation velocity and also in compiling the Riesz basis in the problems under consideration. Further, the application of the results obtained in this paper to the kernels most widely used in practice is demonstrated.