Universal methods for nonlinear spectral problems

This Chapter deals with a spectral geometry of non-linear spectral problems (NLSP). Known also as polynomial operator pencils, such problems appear in many physical applications and, in particular, in equations which determine spectra of single-particle modes. A NLSP is defined here as a linear combination of different powers of an eigenvalue multiplied by operator coefficients which do not commute to each other in general. After describing a method how spectra of NLSP can be found one defines a relevant spectral function, a ‘pseudo-trace’. For a rather wide a class of physically motivated NLSP the pseudo-trace has an asymptotic expansion of the same structure as standard heat kernel asymptotics. Much space of this Chapter is devoted to calculations of the expansion coefficients for NLSP which are expressed as local invariant functionals of background fields.

In this paper, we consider the Klein–Gordon equation u″+−αΔu+β|u′|pu′+γ|u|qu=0 with damping term and acoustic boundary conditions. This work is devoted to prove the existence of global solutions and uniform decay rates of solutions of the Klein–Gordon equation with damping term and acoustic boundary conditions.

The nonlinear wave equation with acoustic and fractional boundary conditions, coupled with logarithmic source and delay terms, is significant for its ability to model complex systems, its contribution to the advancement of mathematical theory, and its wide-ranging applicability to real-world problems. This paper examines the global existence and general decay of solutions to a wave equation characterized by coupling with logarithmic source and delay terms, and governed by both fractional and acoustic boundary conditions. The global existence of solutions is analyzed under a range of hypotheses, and the general decay behavior is established through the construction and application of an appropriate Lyapunov function.

Results on a nonlinear wave equation with acoustic and fractional boundary conditions coupling by logarithmic source and delay terms: Global existence and asymptotic behavior of solutions

Under the acoustic boundary conditions, the initial boundary value problem of a wave equation with multiple nonlinear source terms is considered. This paper gives the energy functional of regular solutions for the wave equation and proves the decreasing property of the energy functional. Firstly, the existence of a global solution for the wave equation is proved by the Faedo–Galerkin method. Then, in order to obtain the nonexistence of global solutions for the wave equation, a new functional is defined. When the initial energy is less than zero, the special properties of the new functional are proved by the method of contraction. Finally, the conditions for the nonexistence of global solutions of the wave equation with acoustic boundary conditions are analyzed by using these special properties.

In this article, we consider the mixed problem for the wave equation with acoustic boundary conditions. This work is devoted to proving the existence of global solutions of the wave equation with acoustic boundary conditions.

In this paper, we deal with the wave equation with acoustic boundary conditions. The exponential stabilization is obtained by Lyapunov approach and Riemannian geometry method. We then apply our main theorem to the wave equations with memory‐type acoustic boundary conditions, which is not available in the literature and give an example in the end.

CHAPTER 3 - Multidimensional non-linear spectral boundary value problems and soliton superposition of their asymptotic solutions

Numerical method for solving the nonlinear three-parametric spectral problems arising, in particular, in the nonlinear antenna synthesis theory with a flat radiating aperture is considered. Numerical algorithms for finding the connected components of the spectrum of linear homogeneous two-dimensional integral equations describing the bifurcation surface of solutions of synthesis problems of flat antenna according to the prescribed requirements to energy directivity pattern are constructed and justified.

The theory and methods of solving the nonlinear multi-parametric spectral problems and their application in investigations of spectral properties of linear differential equations, which comprise the spectral parameters in equation coefficients and boundary conditions nonlinearly, are considered. Investigation of the problem of non-uniqueness of solutions to the non-linear integral equations of the Hammerstein type, which are used in the theory of the synthesis of antennas with a plane radiating aperture, is presented. A number of numerical examples of solving specific spectral problems are given.

Numerical method for solving the nonlinear two-parameter spectral problems arising in nonlinear synthesis theory of antennas with a flat radiating aperture is considered. The numerical algorithms to find the connected components of the spectrum of linear homogeneous two-dimensional integral equations which describe the branching lines of solutions of synthesis problems of the flat apertures and antenna arrays according to the prescribed requirements to amplitude directivity pattern are constructed and justified. For the case of rectangular aperture the analysis of branching of the solutions of basic equations of synthesis with using the proposed method is presented. Results of numerical experiments are given.

In this paper, we are concerned with the energy decay rate of the nonlinear viscoelastic problem with dynamic and acoustic boundary conditions.

Polynomial decay of a variable coefficient wave equation with an acoustic undamped boundary condition

1. Nonlinear spectral problems arise in various areas of analysis and mathematical physics. Most developed are the theory andmethods of solving such problemswith one-dimensional spectral parameter (see [1–9]). In this paper, we consider a nonlinear spectral problemwith two-dimensional spectral parameter. The use of implicit function methods allows us to study qualitative characteristics of the existing spectrum of holomorphic operator-functions and to construct comparatively simple algorithms for numerical finding of spectral lines or connected components of spectral domains.

Linear independence, equivalence and minimality of root vectors for certain nonlinear spectral problems