# The case of middle intensity in spectral problems with the internal dissipation of an energy

We consider the following spectral problem:
$$\lambda^2u - \lambda\beta Ku - \Delta u = 0\;(in\;\Omega),\quad\frac{\partial u}{\partial n} + u = 0\;(on\;\Gamma), \quad K = K^\ast > 0.$$
Here $\Omega\subset\mathbb{R}^m$ is an domain with Lipschitz boundary $\Gamma = \partial\Omega$. The parameter $\beta > 0$ imitates the power of the internal dissipation of an energy.
The problem can be reduce to study another spectral problem seeing in sum of Hilbert spaces:
$$\begin{pmatrix} \beta K& iA^{{1}/{2}} \\ iA^{{1}/{2}}& 0 \end{pmatrix} \begin{pmatrix} u \\ \varsigma \end{pmatrix} = \lambda \begin{pmatrix} u \\ \varsigma \end{pmatrix}, \quad u \in \mathscr{D}(A) \cap \mathscr{D}(K), \quad \varsigma \in \mathscr{D}(A^{1/2}), \quad A \gg 0.$$
The methods of the spectral theory of the operator bundles and the theory of the self-adjoint operators in idefinite metric spaces are used.
Here we study the case, when $\beta K := 2\beta A^\delta,\;\beta > 0,\;\delta > 0$. Then we have
$$\begin{pmatrix} 2\beta A^\delta& iA^{{1}/{2}} \\ iA^{{1}/{2}}& 0 \end{pmatrix} \begin{pmatrix} u \\ \varsigma \end{pmatrix} = \lambda \begin{pmatrix} u \\ \varsigma \end{pmatrix}, \quad \delta \ge 0, \quad u \in \mathscr{D}(A) \cap \mathscr{D}(2\beta A^\delta), \quad \varsigma \in \mathscr{D}(A^{1/2}).$$
We consider here, that $0 < A^{-1} \in \mathscr{S}_\infty(E)$.
This problem contains two parameters: $\beta > 0$ и $\delta > 0$. The aim of consideration of this problem is a desire to trace, as spectrum mutates when $\delta$ grows at different positive $\beta$ and to obtain the statements about localization of the spectrum and the properties of own and joined elements.
It is found out that behavior of spectrum depends on intensity of internal dissipation in the system. It can be weak, middle and strong. The case of middle intensity of internal dissipation is studied, so case when $1/2 < \delta < 1$. The frontier case between weak and middle intensity, when $\delta = 1/2$, and the frontier case between middle and strong intensity, when $\delta = 1$, are considered.
We investigate that connection between operators $\beta K = 2\beta A^\delta$ both confirms the results got before and opens new effects in a spectral problem. The spectrum substantially depends on parameter $\delta$, so when $1/2 \le \delta < 1$ it locates near real semiaxis.
The detailed structure of spectrum depends on the parameter of internal dissipation $\beta$. In particular, in the frontier case $\delta = 1/2$ the spectrum structure substantially depends on $\beta$: when $0 < \beta < 1$ it is imaginary and when $\beta > 1$ — real and positive.
In the case of middle intensity of internal dissipation in all range of change $\delta$ and $\beta$ the systems of eigenfunctions form basis Rissa and $\mathscr{J}$-orthogonal basic in some Hilbert spaces. In the frontier case $\delta = 1$ there are not imaginary spectrum, it consists of real eigenvalues.

Keywords: Hilbert space, compact self-adjoint operator, classes of compact operators, characteristic equation, dynamics of the eigenvalues’ motion.

UDC:
517.9:532