Asymptotic Analysis - Volume 119, issue 1-2

Authors: Jihui, Wu | Shu, Wang

Article Type: Research Article

Abstract: In this paper, we consider a class of the degenerate Cahn–Hilliard equation with a smooth double-well potential. By applying a semi-discretization technique and asymptotic analysis method on non-degenerate Cahn–Hilliard equation, we obtain the existence and regularity of weak solutions to the Cahn–Hilliard equation with degenerate mobility. Moreover, we define a new entropy and obtain the global entropy estimates.

Keywords: Degenerate, non-degenerate, semi-discretization, entropy

DOI: 10.3233/ASY-191563

Citation: Asymptotic Analysis, vol. 119, no. 1-2, pp. 1-38, 2020

Authors: Kurseeva, Valeria | Moskaleva, Marina | Valovik, Dmitry

Article Type: Research Article

Abstract: The paper focuses on a nonlinear eigenvalue problem of Sturm–Liouville type with real spectral parameter under first type boundary conditions and additional local condition. The nonlinear term is an arbitrary monotonically increasing function. It is shown that for small nonlinearity the negative eigenvalues can be considered as perturbations of solutions to the corresponding linear eigenvalue problem, whereas big positive eigenvalues cannot be considered in this way. Solvability results are found, asymptotics of negative as well as positive eigenvalues are derived, distribution of zeros of the eigenfunctions is presented. As a by-product, a comparison theorem between eigenvalues of two problems with …different data is derived. Applications of the found results in electromagnetic theory are given. Show more

Keywords: Nonlinear Sturm–Liouville problem, asymptotical analysis, distribution of eigenvalues, comparison theorem, periodicity of eigenfunctions

DOI: 10.3233/ASY-191565

Citation: Asymptotic Analysis, vol. 119, no. 1-2, pp. 39-59, 2020

Authors: Vargas Junior, Edson Cilos | da Luz, Cleverson Roberto

Article Type: Research Article

Abstract: In this work we study decay rates for a σ -evolution equation in R n under effects of a damping term represented by the action of a fractional Laplacian operator and a time-dependent coefficient, b ( t ) ( − Δ ) θ u t ( t , x ) . We consider that b is ‘confined’ in the curve g ( t ) = ( 1 + t ) α ln γ ( 1 + t ) …for large t ⩾ t 0 and without any control on d d t b ( t ) . Show more

Keywords: Wave equation, plate equation, fractional damping, sharp decay rates, non-effective damping, multiplier method

DOI: 10.3233/ASY-191566

Citation: Asymptotic Analysis, vol. 119, no. 1-2, pp. 61-86, 2020

Authors: Said, Mona Ben | Nier, Francis | Viola, Joe

Article Type: Research Article

Abstract: The present article is concerned with global subelliptic estimates for Kramers–Fokker–Planck operators with polynomials of degree less than or equal to two. The constants appearing in those estimates are accurately formulated in terms of the coefficients, especially when those are large.

Keywords: Subelliptic estimates, compact resolvent, Kramers–Fokker–Planck operator, quaternions, Bargmann transform

DOI: 10.3233/ASY-191569

Citation: Asymptotic Analysis, vol. 119, no. 1-2, pp. 87-116, 2020

Authors: Delourme, Bérangère | Fliss, Sonia | Joly, Patrick | Vasilevskaya, Elizaveta

Article Type: Research Article

Abstract: We are interested in a 2D propagation medium obtained from a localized perturbation of a reference homogeneous periodic medium. This reference medium is a “thick graph”, namely a thin structure (the thinness being characterized by a small parameter ε > 0 ) whose limit (when ε tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. In the first part of this work, we proved that such a geometrical perturbation is able to produce localized eigenmodes …(the propagation model under consideration is the scalar Helmholtz equation with Neumann boundary conditions). This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We used a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough. The objective of the present work is to complement the previous one by constructing and justifying a high order asymptotic expansion of these eigenvalues (with respect to the small parameter ε ) using the method of matched asymptotic expansions. In particular, the obtained expansion can be used to compute a numerical approximation of the eigenvalues and of their associated eigenvectors. An algorithm to compute each term of the asymptotic expansion is proposed. Numerical experiments validate the theoretical results. Show more

Keywords: Spectral theory, periodic media, quantum graphs, matched asymptotic expansion

DOI: 10.3233/ASY-191573

Citation: Asymptotic Analysis, vol. 119, no. 1-2, pp. 117-152, 2020