Asymptotic Analysis - Volume 134, issue 3-4

Authors: Amirat, Youcef | Münch, Arnaud

Article Type: Research Article

Abstract: We perform an asymptotic analysis with respect to the parameter ε > 0 of the solution of the scalar advection–diffusion equation y t ε + M ( x , t ) y x ε − ε y x x ε = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T ) , supplemented with Dirichlet boundary conditions. For small values of ε , the solution y ε …exhibits a boundary layer of size O ( ε ) in the neighborhood of x = 1 (assuming M > 0 ) and an internal layer of size O ( ε 1 / 2 ) in the neighborhood of the characteristic starting from the point ( 0 , 0 ) . Assuming that these layers interact each other after a finite time T > 0 and using the method of matched asymptotic expansions, we construct an explicit approximation P ε satisfying ‖ y ε − P ε ‖ L ∞ ( 0 , T ; L 2 ( 0 , 1 ) ) = O ( ε 1 / 2 ) . We emphasize the additional difficulties with respect to the case M constant considered recently by the authors. Show more

Keywords: Asymptotic analysis, singular perturbation, internal and boundary layers, Sobolev estimates

DOI: 10.3233/ASY-231836

Citation: Asymptotic Analysis, vol. 134, no. 3-4, pp. 297-343, 2023

Authors: Petkov, Vesselin

Article Type: Research Article

Abstract: Let Ω = R 3 ∖ K ¯ , where K is an open bounded domain with smooth boundary Γ. Let V ( t ) = e t G b , t ⩾ 0 , be the semigroup related to Maxwell’s equations in Ω with dissipative boundary condition ν ∧ ( ν ∧ E ) + γ ( x ) ( ν ∧ H ) = 0 , γ ( x ) > 0 , ∀ x ∈ Γ …. We study the case when γ ( x ) ≠ 1 , ∀ x ∈ Γ , and we establish a Weyl formula for the counting function of the eigenvalues of G b in a polynomial neighbourhood of the negative real axis. Show more

Keywords: Dissipative boundary conditions, dissipative eigenvalues, Weyl formula

DOI: 10.3233/ASY-231837

Citation: Asymptotic Analysis, vol. 134, no. 3-4, pp. 345-367, 2023

Authors: Benoit, Antoine | Loyer, Romain

Article Type: Research Article

Abstract: This article aims to finalize the classification of weakly well-posed hyperbolic boundary value problems in the half-space. Such problems with loss of derivatives are rather classical in the literature and appear for example in (Arch. Rational Mech. Anal. 101 (1988 ) 261–292) or (In Analyse Mathématique et Applications (1988 ) 319–356 Gauthier-Villars). It is known that depending on the kind of the area of the boundary of the frequency space on which the uniform Kreiss–Lopatinskii condition degenerates then the energy estimate can include different losses. The three first possible areas of degeneracy have been studied in (Annales …de l’Institut Fourier 60 (2010 ) 2183–2233) and (Differential Integral Equations 27 (2014 ) 531–562) by the use of geometric optics expansions. In this article we use the same kind of tools in order to deal with the last remaining case, namely a degeneracy in the glancing area. In comparison to the first cases studied we will see that the equation giving the amplitude of the leading order term in the expansion, and thus initializing the whole construction of the expansion, is not a transport equation anymore but it is given by some Fourier multiplier. This multiplier needs to be invert in order to recover the first amplitude. As an application we discuss the existing estimates of (Discrete Contin. Dyn. Syst., Ser. B 23 (2018 ) 1347–1361; SIAM J. Math. Anal. 44 (2012 ) 1925–1949) for the wave equation with Neumann boundary condition. Show more

Keywords: Hyperbolic boundary value problems, geometric optics expansions, weak well-posedness, glancing modes

DOI: 10.3233/ASY-231838

Citation: Asymptotic Analysis, vol. 134, no. 3-4, pp. 369-411, 2023

Authors: Li, Chunqiu | Peng, Zhen

Article Type: Research Article

Abstract: This article is concerned with the bifurcation from infinity of the following elliptic system arising from biology − κ Δ u = λ u + f ( x , u ) − v , − Δ v = u − v , in a bounded domain Ω ⊂ R N . We regard this problem as a stationary problem of some reaction-diffusion system. By using a method of a pure dynamical nature, we will establish some multiplicity results on bifurcations from …infinity for this system under an appropriate Landesman-Lazer type condition. Show more

Keywords: Conley index, bifurcation from infinity, elliptic equation, parabolic equation

DOI: 10.3233/ASY-231839

Citation: Asymptotic Analysis, vol. 134, no. 3-4, pp. 413-436, 2023

Authors: Duerinckx, Mitia | Gloria, Antoine

Article Type: Research Article

Abstract: In this note, we provide a short and robust proof of the Clausius–Mossotti formula for the effective conductivity in the dilute regime, together with an optimal error estimate. The proof makes no assumption on the underlying point process besides stationarity and ergodicity, and it can be applied to dilute systems in many other contexts.

Keywords: Dilute expansion, homogenization, effective coefficients

DOI: 10.3233/ASY-231840

Citation: Asymptotic Analysis, vol. 134, no. 3-4, pp. 437-453, 2023

Authors: Yu, Min | Li, Weijia | Yan, Weiping

Article Type: Research Article

Abstract: This paper considers nonlinear Kirchhoff equation with Kelvin–Voigt damping. This model is used to describe the transversal motion of a stretched string. The existence of smooth stationary solutions of nonlinear Kirchhoff equation has been studied widely. In the present contribution, we prove that a class of stationary solutions is asymptotic stable by overcoming the “loss of derivative” phenomenon causing from the Kirchhoff operator. The key point is to find a suitable weighted function when we carry out the energy estimate for the linearized equation.

Keywords: Kirchhoff equation, Global existence, Asymptotic stability, Ground state, Kelvin–Voigt damping

DOI: 10.3233/ASY-231841

Citation: Asymptotic Analysis, vol. 134, no. 3-4, pp. 455-484, 2023

Authors: Dasgupta, Aparajita | Kumar, Vishvesh | Mondal, Shyam Swarup

Article Type: Research Article

Abstract: In this paper, we deal with the initial value fractional damped wave equation on G , a compact Lie group, with power-type nonlinearity. The aim of this manuscript is twofold. First, using the Fourier analysis on compact Lie groups, we prove a local in-time existence result in the energy space for the fractional damped wave equation on G . Moreover, a finite time blow-up result is established under certain conditions on the initial data. In the next part of the paper, we consider fractional wave equation with lower order terms, i.e., damping and mass with the same power type nonlinearity …on compact Lie groups, and prove the global in-time existence of small data solutions in the energy evolution space. Show more

Keywords: Fractional damped wave equation, well-posedness, compact Lie groups, L2–L2-estimates, finite blow-up

DOI: 10.3233/ASY-231842

Citation: Asymptotic Analysis, vol. 134, no. 3-4, pp. 485-511, 2023

Authors: Su, Keqin | Yang, Xin-Guang | Miranville, Alain | Yang, He

Article Type: Research Article

Abstract: This paper is concerned with the dynamics of the two-dimensional Navier–Stokes equations with multi-delays in a Lipschitz-like domain, subject to inhomogeneous Dirichlet boundary conditions. The regularity of global solutions and of pullback attractors, based on tempered universes, is established, extending the results of Yang, Wang, Yan and Miranville (Discrete Contin. Dyn. Syst. 41 (2021 ) 3343–3366). Furthermore, the robustness of pullback attractors when the delays, considered as small perturbations, disappear is also derived. The key technique in the proofs is the application of a retarded Gronwall inequality and a variable index for the tempered pullback dynamics, allowing to …obtain uniform estimates and the compactness of the process. Show more

Keywords: Navier–Stokes equations, multi-delays, Lipschitz-like domain, robustness

DOI: 10.3233/ASY-231845

Citation: Asymptotic Analysis, vol. 134, no. 3-4, pp. 513-552, 2023

Authors: Ibrahim, H. | Younes, R.

Article Type: Research Article

Abstract: We consider quasilinear elliptic equations Δ p u + f ( u ) = 0 in the quarter-plane Ω, with zero Dirichlet data. For some general nonlinearities f , we prove the existence of a positive solution with a prescribed limiting profile. The question is motivated by the result in (Adv. Nonlinear Stud. 13 (1) (2013) 115–136), where the authors establish that for solutions u ( x 1 , x 2 ) of the preceding Dirichlet problem, lim x 1 …→ ∞ u ( x 1 , x 2 ) = V ( x 2 ) , where V is a solution of the corresponding one-dimensional problem with V ( + ∞ ) = z and z is a root of f . Starting with such a profile V and a carefully selected z , the authors of this paper apply Perron’s method in order to prove the existence of a solution u with limiting profile V . The work in this paper is similar in spirit to that in (Math. Methods Appl. Sci. 39 (14) (2016) 4129–4138), where the authors compare the sub and the super solutions by using arguments based on the strong maximum principle for semilinear equations. However, for the quasilinear case, such a maximum principle is lacking. This difficulty is overcome by employing a less classical weak sweeping principle that requires a careful boundary analysis. Show more

Keywords: p-Laplace equation, asymptotic behavior, positive solution

DOI: 10.3233/ASY-231846

Citation: Asymptotic Analysis, vol. 134, no. 3-4, pp. 553-574, 2023

Authors: Dai, Guowei | Duan, Ben | Liu, Fang

Article Type: Research Article

Abstract: In this paper, we investigate the Laplace’s equation for the electrical potential of charge drops on exterior domain, and overdetermined boundary conditions are prescribed. We determine the local bifurcation structure with respect to the surface tension coefficient as bifurcation parameter. Furthermore, we establish the stability and the instability near the bifurcation point.

Keywords: Bifurcation, stability, charged drops

DOI: 10.3233/ASY-231853

Citation: Asymptotic Analysis, vol. 134, no. 3-4, pp. 575-588, 2023