Asymptotic Analysis - Volume 130, issue 1-2

Authors: Yang, Heng

Article Type: Research Article

Abstract: In this paper, we prove the existence of nontrivial solutions and ground state solutions for the following planar Schrödinger–Poisson system with zero mass − Δ u + ϕ u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where α ∈ ( 0 , 2 …) , I α : R 2 → R is the Riesz potential, f ∈ C ( R , R ) is of subcritical exponential growth in the sense of Trudinger–Moser. In particular, some new ideas and analytic technique are used to overcome the double difficulties caused by the zero mass case and logarithmic convolution potential. Show more

Keywords: Planar Schrödinger–Poisson system, Logarithmic convolution potential, zero mass

DOI: 10.3233/ASY-211741

Citation: Asymptotic Analysis, vol. 130, no. 1-2, pp. 1-21, 2022

Authors: Golovaty, Yuriy

Article Type: Research Article

Abstract: We study the asymptotic behaviour of eigenvalues of 2D vibrating systems with mass density perturbed in a vicinity of closed curves. The threshold case in which the resonance frequencies of the membrane and the frequencies of thin inclusion coincide is investigated. The perturbed eigenvalue problem can be realized as a family of self-adjoint operators acting on varying Hilbert spaces. However the so-called limit operator is non-self-adjoint and possesses the Jordan chains of length 2. Apart from the lack of self-adjointness, the operator has non-compact resolvent. As a consequence, its spectrum has a complicated structure, for instance, the spectrum contains a …countable set of eigenvalues with infinite multiplicity. The complete asymptotic analysis of eigenvalues has been carried out. Show more

Keywords: Asymptotics of eigenvalues, eigenvalue of infinite multiplicity, quasimode, non-self-adjoint operator, concentrated mass, singular perturbation

DOI: 10.3233/ASY-211743

Citation: Asymptotic Analysis, vol. 130, no. 1-2, pp. 23-51, 2022

Authors: Bizhanova, Galina

Article Type: Research Article

Abstract: There is studied the Hölder space solution u ε of the problem for parabolic equation with the time derivative ε ∂ t u ε | Σ in the boundary condition, where ε > 0 is a small parameter. The unique solvability of the perturbed problem and estimates of it’s solution are obtained. The convergence of u ε as ε → 0 to the solution of the unperturbed problem is proved. …Boundary layer is not appeared. Show more

Keywords: Parabolic equation, boundary–value problem, small parameter in the boundary condition, Hölder space, existence, uniqueness, coercive estimates, convergence of the solution

DOI: 10.3233/ASY-211744

Citation: Asymptotic Analysis, vol. 130, no. 1-2, pp. 53-87, 2022

Authors: Aouadi, Moncef | Mahfoudhi, Imed | Moulahi, Taoufik

Article Type: Research Article

Abstract: We study some spectral and numerical properties of the solutions to a thermoelastic problem with double porosity. The model includes Cattaneo-type evolution law for the heat flux to remove the physical paradox of infinite propagation speed of the classical Fourier’s law. Firstly, we prove that the operator determined by the considered problem has compact resolvent and generates a C 0 -semigroup in an appropriate Hilbert space. We also show that there is a sequence of generalized eigenfunctions of the linear operator that forms a Riesz basis. By a detailed spectral analysis, we obtain the expressions …of the spectrum and we deduce that the spectrum determined growth condition holds. Therefore we prove that the energy of the considered problem decays exponentially to a rate determined explicitly by the physical parameters. Finally, some numerical simulations based on Chebyshev spectral method for spatial discretization are given to confirm the exponential stability result and to show the distribution of the eigenvalues and the variables of the problem. Show more

Keywords: Thermoelasticity, double porosity, well-posedness, spectral analysis, numerical simulations

DOI: 10.3233/ASY-211745

Citation: Asymptotic Analysis, vol. 130, no. 1-2, pp. 89-126, 2022

Authors: Surnachev, Mikhail

Article Type: Research Article

Abstract: In this paper a weak Harnack inequality for the parabolic p ( x ) -Laplacian is established.

Keywords: Degenerate parabolic equations, p(x)-Laplacian, Harnack’s inequality, expansion of positivity

DOI: 10.3233/ASY-211746

Citation: Asymptotic Analysis, vol. 130, no. 1-2, pp. 127-165, 2022

Authors: Yoshida, Natsumi

Article Type: Research Article

Abstract: In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem with the far field condititon for the generalized Benjamin–Bona–Mahony–Burgers equation with a fourth-order dissipative term. When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, it is proved that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity. We can further obtain the same global asymptotic stability of the rarefaction wave to the generalized Korteweg–de Vries–Benjamin–Bona–Mahony–Burgers equation with a fourth-order dissipative term as the former one.

Keywords: Benjamin–Bona–Mahony–Burgers equation, Korteweg–de Vries–Benjamin–Bona–Mahony–Burgers equation, convex flux, asymptotic behavior, rarefaction wave

DOI: 10.3233/ASY-211747

Citation: Asymptotic Analysis, vol. 130, no. 1-2, pp. 167-185, 2022

Authors: Wang, Tianfang | Zhang, Wen | Zhang, Jian

Article Type: Research Article

Abstract: In this paper we study the Dirac equation with Coulomb potential − i α · ∇ u + a β u − μ | x | u = f ( x , | u | ) u , x ∈ R 3 where a is a positive constant, μ is a positive parameter, α = ( α 1 , α 2 , α 3 ) , α i …and β are 4 × 4 Pauli–Dirac matrices. The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. Under some suitable conditions, we prove that the problem possesses a ground state solution which is exponentially decay, and the least energy has continuous dependence about μ . Moreover, we are able to obtain the asymptotic property of ground state solution as μ → 0 + , this result can characterize some relationship of the above problem between μ > 0 and μ = 0 . Show more

Keywords: Dirac equation, Coulomb potential, asymptotic property, ground state solutions

DOI: 10.3233/ASY-211748

Citation: Asymptotic Analysis, vol. 130, no. 1-2, pp. 187-212, 2022

Authors: Cinelli, Ivan | Ferrari, Gianluca | Squassina, Marco

Article Type: Research Article

Abstract: We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity, in the theory of electrorheological fluids as well as in image processing for the regions where the variable exponent p ( x ) reaches the value 1.

Keywords: Anisotropic Sobolev spaces, nonlocal characterizations, image processing

DOI: 10.3233/ASY-211749

Citation: Asymptotic Analysis, vol. 130, no. 1-2, pp. 213-232, 2022

Authors: Michelangeli, Alessandro | Santamaria, Nicola

Article Type: Research Article

Abstract: For a mixture of interacting Bose gases initially prepared in a regime of condensation (uncorrelation), it is proved that in the course of the time evolution observables of disjoint sets of particles of each species have correlation functions that remain asymptotically small in the total number of particles and display a controlled growth in time. This is obtained by means of ad hoc estimates of Lieb–Robinson type on the propagation of the interaction, established here for the multi-component Bose mixture.

Keywords: Bose mixtures, composite BEC, many-body dynamics, correlation function, Lieb–Robinson bounds

DOI: 10.3233/ASY-211750

Citation: Asymptotic Analysis, vol. 130, no. 1-2, pp. 233-259, 2022

Authors: Hayashi, Nakao | Kaikina, Elena I. | Naumkin, Pavel I. | Ogawa, Takayoshi

Article Type: Research Article

Abstract: We study the nonlinear Neumann boundary value problem for semilinear heat equation ∂ t u − Δ u = λ | u | p , t > 0 , x ∈ R + n , u ( 0 , x ) = ε u 0 ( x ) , x ∈ R + n , − ∂ x u ( t , x ′ , 0 ) …= γ | u | q ( t , x ′ , 0 ) , t > 0 , x ′ ∈ R n − 1 where p = 1 + 2 n , q = 1 + 1 n and ε > 0 is small enough. We investigate the life span of solutions for λ , γ > 0 . Also we study the global in time existence and large time asymptotic behavior of solutions in the case of λ , γ < 0 and ∫ R + n u 0 ( x ) d x > 0 . Show more

Keywords: Nonlinear heat equation, large time asymptotics, initial-boundary value problem, half line

DOI: 10.3233/ASY-211751

Citation: Asymptotic Analysis, vol. 130, no. 1-2, pp. 261-295, 2022