Asymptotic Analysis - Volume 123, issue 1-2

Authors: D’Abbicco, Marcello | Ebert, Marcelo Rempel

Article Type: Research Article

Abstract: In this paper we study the asymptotic profile (as t → ∞ ) of the solution to the Cauchy problem for the linear plate equation u t t + Δ 2 u − λ ( t ) Δ u + u t = 0 when λ = λ ( t ) is a decreasing function, assuming initial data in the energy space and verifying a moment condition. For sufficiently small data, we find the critical exponent …for global solutions to the corresponding problem with power nonlinearity u t t + Δ 2 u − λ ( t ) Δ u + u t = | u | p . In order to do that, we assume small data in the energy space and, possibly, in L 1 . In this latter case, we also determinate the asymptotic profile of the solution to the semilinear problem for supercritical power nonlinearities. Show more

Keywords: Asymptotic profile, diffusion phenomena, critical exponents, plate equation

DOI: 10.3233/ASY-201624

Citation: Asymptotic Analysis, vol. 123, no. 1-2, pp. 1-40, 2021

Authors: Deugoué, G. | Tachim Medjo, T.

Article Type: Research Article

Abstract: In this article, we derive a large deviation principle for a 2D Allen–Cahn–Navier–Stokes model under random influences. The model consists of the Navier–Stokes equations for the velocity, coupled with an Allen–Cahn equation for the order (phase) parameter. The proof relies on the weak convergence method that was introduced in (Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011 ) 725–747) and based on a variational representation on infinite-dimensional Brownian motion.

Keywords: Allen–Cahn, Navier–Stokes, strong solutions, Gaussian noise, large deviations

DOI: 10.3233/ASY-201625

Citation: Asymptotic Analysis, vol. 123, no. 1-2, pp. 41-78, 2021

Authors: Esposito, Luca | Roy, Prosenjit | Sk, Firoj

Article Type: Research Article

Abstract: We analyze the asymptotic behavior of the eigenvalues of nonlinear elliptic problems under Dirichlet boundary conditions and mixed (Dirichlet, Neumann) boundary conditions on domains becoming unbounded. We make intensive use of Picone identity to overcome nonlinearity complications. Altogether the use of Picone identity makes the proof easier with respect to the known proof in the linear case. Surprisingly the asymptotic behavior under mixed boundary conditions critically differs from the case of pure Dirichlet boundary conditions for some class of problems.

Keywords: Eigenvalues problems, ℓ goes to plus infinity, dimension reduction

DOI: 10.3233/ASY-201626

Citation: Asymptotic Analysis, vol. 123, no. 1-2, pp. 79-94, 2021

Authors: Xu, Yao | Niu, Weisheng

Article Type: Research Article

Abstract: This paper concentrates on the quantitative homogenization of higher-order elliptic systems with almost-periodic coefficients in bounded Lipschitz domains. For almost-periodic coefficients in the sense of H. Weyl, we establish uniform local L 2 estimates for the approximate correctors. Under an additional assumption (1.8 ) on the frequencies of the coefficients, we derive the existence of true correctors as well as the O ( ε ) convergence rate in H m − 1 . As a byproduct, the large-scale Hölder estimate and a Liouville theorem are obtained …for higher-order elliptic systems with almost-periodic coefficients in the sense of Besicovitch. Since (1.8 ) is not well-defined for equivalence classes of almost-periodic functions in the sense of H. Weyl or Besicovitch, we provide another condition yielding the O ( ε ) convergence rate under perturbations of the coefficients. Show more

Keywords: Almost periodic homogenization, higher-order elliptic systems, quantitative estimates

DOI: 10.3233/ASY-201627

Citation: Asymptotic Analysis, vol. 123, no. 1-2, pp. 95-137, 2021

Authors: Bahrouni, Anouar | Ho, Ky

Article Type: Research Article

Abstract: In this paper, we give some properties of the new fractional Sobolev spaces with variable exponents and apply them to study a class of eigenvalue problems involving the fractional p ( · ) -Laplace operator. We obtain sequences of eigenvalues going asymptotically to infinity and we also establish sufficient conditions to get zero value for the principal eigenvalue, which is a striking difference between the variable exponent case and the constant exponent case. As an application, we obtain several existence and nonexistence results for the eigenvalue problem according to the asymptotic growth of the nonlinearity and the range …of the spectral parameter. Show more

Keywords: Fractional Sobolev spaces, variable exponents, eigenvalue problems, variational methods

DOI: 10.3233/ASY-201628

Citation: Asymptotic Analysis, vol. 123, no. 1-2, pp. 139-156, 2021

Authors: Melo, Wilberclay G. | Rocha, Natã F. | Zingano, Paulo R.

Article Type: Research Article

Abstract: This work guarantees the existence of a positive instant t = T and a unique solution ( u , w ) ∈ [ C ( [ 0 , T ] ; H a , σ s ( R 2 ) ) ] 3 (with a > 0 , σ > 1 , s > 0 and s ≠ 1 ) for the micropolar equations. Furthermore, we consider the global existence in time of this solution …in order to prove the following decay rates: lim t → ∞ t s 2 ‖ ( u , w ) ( t ) ‖ H ˙ a , σ s ( R 2 ) 2 = lim t → ∞ t s + 1 2 ‖ w ( t ) ‖ H ˙ a , σ s ( R 2 ) 2 = lim t → ∞ ‖ ( u , w ) ( t ) ‖ H a , σ λ ( R 2 ) = 0 , ∀ λ ⩽ s . These limits are established by applying the estimate ‖ F − 1 ( e T | · | ( u ˆ , w ˆ ) ( t ) ) ‖ H s ( R 2 ) ⩽ [ 1 + 2 M 2 ] 1 2 , ∀ t ⩾ T , where T relies only on s , μ , ν and M (the inequality above is also demonstrated in this paper). Here M is a bound for ‖ ( u , w ) ( t ) ‖ H s ( R 2 ) (for all t ⩾ 0 ) which results from the limits lim t → ∞ t s 2 ‖ ( u , w ) ( t ) ‖ H ˙ s ( R 2 ) = lim t → ∞ ‖ ( u , w ) ( t ) ‖ L 2 ( R 2 ) = 0 . Show more

Keywords: Micropolar equations, decay rates, Sobolev–Gevrey spaces

DOI: 10.3233/ASY-201630

Citation: Asymptotic Analysis, vol. 123, no. 1-2, pp. 157-179, 2021

Authors: Wang, Danhua | Liu, Wenjun

Article Type: Research Article

Abstract: In this paper, we consider the Cauchy problem related to the standard linear solid model with Gurtin–Pipkin thermal law in the whole space. Under some assumptions on the relaxation function g , we establish the well-posedness result by using semigroup theory. Besides, by using the energy method in the Fourier space, we prove the decay estimate result under the non-critical case. Our result indicates that the decay property is of the regularity-loss type, which is in line with the decay property of Cattaneo system.

Keywords: Well-posedness, the standard linear solid model, Gurtin–Pipkin thermal law, decay estimate

DOI: 10.3233/ASY-201631

Citation: Asymptotic Analysis, vol. 123, no. 1-2, pp. 181-201, 2021