Asymptotic Analysis - Volume 116, issue 3-4

Authors: Ekohela, Clesh Deseskel Elion | Bissanga, Gabriel | Batchi, Macaire

Article Type: Research Article

Abstract: The aim of this article is to study asymptotic behavior of higher-order (in space) anisotropic perturbed Caginalp phase-field systems with relaxations hyperbolic in term of attractors. The main difficulty comes from the fact that the phase spaces for the perturbed (ϵ ≠ 0 ) and unperturbed (ϵ = 0 ) equations are not the same; indeed, the limit problem is parabolic. Therefore, the previous approach employing for parabolic systems, cannot be applied and have to be adapted. In particular, this necessitates a study of the time boundary layer in order to estimate the difference of solutions …between the perturbed and unperturbed equations. Finally, we obtain the existence of the global attractor, as well as the existence of exponential attractors. Show more

Keywords: Anisotropy, boundary layer, Caginalp system, dissipativity, exponential attractors, hyperbolic relaxation, higher-order systems, global attractor, perturbed damped wave equation

DOI: 10.3233/ASY-191540

Citation: Asymptotic Analysis, vol. 116, no. 3-4, pp. 149-217, 2020

Authors: Sivaji Ganesh, Sista | Tewary, Vivek

Article Type: Research Article

Abstract: We consider the spectrum of a second-order elliptic operator in divergence form with periodic coefficients, which is known to be completely described by Bloch eigenvalues. We show that under small perturbations of the coefficients, a multiple Bloch eigenvalue can be made simple. The Bloch wave method of homogenization relies on the regularity of spectral edge. The spectral tools that we develop, allow us to obtain simplicity of an internal spectral edge through perturbation of the coefficients. As a consequence, we are able to establish Bloch wave homogenization at an internal edge in the presence of multiplicity by employing the perturbed …Bloch eigenvalues. We show that all the crossing Bloch modes contribute to the homogenization at the internal edge and that higher and lower modes do not contribute to the homogenization process. Show more

Keywords: Genericity, Bloch eigenvalues, periodic operators, homogenization

DOI: 10.3233/ASY-191542

Citation: Asymptotic Analysis, vol. 116, no. 3-4, pp. 219-248, 2020

Authors: Ambrosio, Vincenzo

Article Type: Research Article

Abstract: In this paper we consider the following class of fractional Kirchhoff equations with critical growth: ( ε 2 s a + ε 4 s − 3 b ∫ R 3 | ( − Δ ) s 2 u | 2 d x ) ( − Δ ) s u + V ( x ) u = f ( u ) + | u | 2 s ∗ …− 2 u in  R 3 , u ∈ H s ( R 3 ) , u > 0 in  R 3 , where ε > 0 is a small parameter, a , b > 0 are constants, s ∈ ( 3 4 , 1 ) , 2 s ∗ = 6 3 − 2 s is the fractional critical exponent, ( − Δ ) s is the fractional Laplacian operator, V is a positive continuous potential and f is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we prove the existence of a family of positive solutions u ε which concentrates around a local minimum of V as ε → 0 . Show more

Keywords: Fractional Kirchhoff equation, variational methods, critical growth

DOI: 10.3233/ASY-191543

Citation: Asymptotic Analysis, vol. 116, no. 3-4, pp. 249-278, 2020