Asymptotic Analysis - Volume 120, issue 3-4

Authors: Chen, Jianhua | Huang, Xianjiu | Qin, Dongdong | Cheng, Bitao

Article Type: Research Article

Abstract: In this paper, we study the following generalized quasilinear Schrödinger equation − div ( ε 2 g 2 ( u ) ∇ u ) + ε 2 g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = K ( x ) | u | p − 2 u + | u | 22 ∗ − 2 u , x ∈ R …N , where N ⩾ 3 , ε > 0 , 4 < p < 22 ∗ , g ∈ C 1 ( R , R + ) , V ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global minimum, and K ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem with critical growth, and establish a phenomenon of exponential decay. Moreover, by Ljusternik–Schnirelmann theory, we also prove the existence of multiple solutions. Show more

Keywords: Generalized quasilinear Schrödinger equation, existence, asymptotic behavior, ground state solutions, critical exponents

DOI: 10.3233/ASY-191586

Citation: Asymptotic Analysis, vol. 120, no. 3-4, pp. 199-248, 2020

Authors: Ghezaiel, Emna | Abdelwahed, Mohamed | Chorfi, Nejmeddine | Hassine, Maatoug

Article Type: Research Article

Abstract: This work focuses on the topological sensitivity analysis of a three-dimensional parabolic type problem. The considered application model is described by the heat equation. We derive a new topological asymptotic expansion valid for various shape functions and geometric perturbations of arbitrary form. The used approach is based on a rigorous mathematical framework describing and analyzing the asymptotic behavior of the perturbed temperature field.

Keywords: Geometric inverse problem, topological sensitivity analysis, asymptotic expansion, parabolic operator

DOI: 10.3233/ASY-191587

Citation: Asymptotic Analysis, vol. 120, no. 3-4, pp. 249-272, 2020

Authors: Fang, Fei | Ji, Chao

Article Type: Research Article

Abstract: In this paper, we first study the cone Moser–Trudinger inequalities and their best exponents on both bounded and unbounded domains R + 2 . Then, using the cone Moser–Trudinger inequalities, we study the asymptotic behavior of Cerami sequences and the existence of weak solutions to the nonlinear equation − Δ B u = f ( x , u ) , in  x ∈ int ( B ) , u = 0 , on  ∂ B , …where Δ B is an elliptic operator with conical degeneration on the boundary x 1 = 0 , and the nonlinear term f has the subcritical exponential growth or the critical exponential growth. Show more

Keywords: Cone Moser–Trudinger inequalities, best exponent, asymptotic estimate, rearrangement, mountain pass theorem

DOI: 10.3233/ASY-191588

Citation: Asymptotic Analysis, vol. 120, no. 3-4, pp. 273-299, 2020

Authors: Feng, Yuehong | Li, Xin | Wang, Shu

Article Type: Research Article

Abstract: This paper is concerned with smooth solutions of the non-isentropic Euler–Poisson system for ion dynamics. The system arises in the modeling of semi-conductor, in which appear one small parameter, the momentum relaxation time. When the initial data are near constant equilibrium states, with the help of uniform energy estimates and compactness arguments, we rigorously prove the convergence of the system for all time, as the relaxation time goes to zero. The limit system is the drift-diffusion system.

Keywords: Non-isentropic Euler–Poisson system, global smooth solution, global zero-relaxation limit

DOI: 10.3233/ASY-191589

Citation: Asymptotic Analysis, vol. 120, no. 3-4, pp. 301-318, 2020

Authors: Li, Xintao | Huang, Shoujun | Yan, Weiping

Article Type: Research Article

Abstract: This paper studies the wave-breaking mechanism and dynamical behavior of solutions near the explicit self-similar singularity for the two component Camassa–Holm equations, which can be regarded as a model for shallow water dynamics and arising from the approximation of the Hamiltonian for Euler’s equation in the shallow water regime.

Keywords: Camassa–Holm equation, blow-up solution, asymptotic analysis, stability

DOI: 10.3233/ASY-191590

Citation: Asymptotic Analysis, vol. 120, no. 3-4, pp. 319-336, 2020

Authors: Cárdenas, Esteban | Raikov, Georgi | Tejeda, Ignacio

Article Type: Research Article

Abstract: We consider the Landau Hamiltonian H 0 , self-adjoint in L 2 ( R 2 ) , whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues Λ q , q ∈ Z + . We perturb H 0 by a non-local potential written as a bounded pseudo-differential operator Op w ( V ) with real-valued Weyl symbol V , such …that Op w ( V ) H 0 − 1 is compact. We study the spectral properties of the perturbed operator H V = H 0 + Op w ( V ) . First, we construct symbols V , possessing a suitable symmetry, such that the operator H V admits an explicit eigenbasis in L 2 ( R 2 ) , and calculate the corresponding eigenvalues. Moreover, for V which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of H V adjoining any given Λ q . We find that the effective Hamiltonian in this context is the Toeplitz operator T q ( V ) = p q Op w ( V ) p q , where p q is the orthogonal projection onto Ker ( H 0 − Λ q I ) , and investigate its spectral asymptotics. Show more

Keywords: Landau Hamiltonian, non-local potentials, Weyl pseudo-differential operators, eigenvalue asymptotics, logarithmic capacity

DOI: 10.3233/ASY-191591

Citation: Asymptotic Analysis, vol. 120, no. 3-4, pp. 337-371, 2020

Authors: Chen, Dongxiang | Ren, Siqi | Wang, Yuxi | Zhang, Zhifei

Article Type: Research Article

Abstract: In this paper, we prove the global well-posedness of the 2-D magnetic Prandtl model in the mixed Prandtl/Hartmann regime when the initial data is a small perturbation of the Hartmann layer in Sobolev space.

Keywords: Magnetic Prandtl equation, Hartmann layer, MHD equations, global stability

DOI: 10.3233/ASY-191593

Citation: Asymptotic Analysis, vol. 120, no. 3-4, pp. 373-393, 2020