Asymptotic Analysis - Volume 123, issue 3-4

Authors: Zhang, Youpei | Tang, Xianhua

Article Type: Research Article

Abstract: We are concerned with the mathematical and asymptotic analysis of solutions to the following nonlinear problem − Δ A u = λ β ( x ) | u | q u + f ( | u | ) u in  Ω , u = 0 on  ∂ Ω , where Δ A u is the magnetic Laplace operator, Ω ⊂ R N …is a smooth bounded domain, A : Ω ↦ R N is the magnetic potential, u : Ω ↦ C , λ is a real parameter, β ∈ L ∞ ( Ω , R ) is an indefinite potential, q is nonnegative, and f : [ 0 , + ∞ ) ↦ R is a reaction that oscillates either in a neighborhood of the origin or at infinity. We analyze two distinct cases, in close relationship with the oscillatory growth of the reaction. Additionally, we give asymptotic estimates for the norm of the solutions in related function spaces. Show more

Keywords: Magnetic Laplace operator, blow-up, low- and high-energy solutions, oscillation, variational methods

DOI: 10.3233/ASY-201629

Citation: Asymptotic Analysis, vol. 123, no. 3-4, pp. 203-236, 2021

Authors: Ercole, Grey | Medeiros, Aldo H.S. | Pereira, Gilberto A.

Article Type: Research Article

Abstract: We study the behavior as p → ∞ of u p , a positive least energy solution of the problem [ ( − Δ p ) α + ( − Δ q ( p ) ) β ] u = μ p | u ( x u ) | p − 2 u ( x u ) δ x u …in  Ω u = 0 in  R N ∖ Ω | u ( x u ) | = ‖ u ‖ ∞ , where Ω ⊂ R N is a bounded, smooth domain, δ x u is the Dirac delta distribution supported at x u , lim p → ∞ q ( p ) p = Q ∈ ( 0 , 1 ) if  0 < β < α < 1 ( 1 , ∞ ) if  0 < α < β < 1 and lim p → ∞ μ p p > R − α , with R denoting the inradius of Ω. Show more

Keywords: Asymptotic behavior, Dirac delta distribution, fractional Sobolev spaces, viscosity solutions

DOI: 10.3233/ASY-201632

Citation: Asymptotic Analysis, vol. 123, no. 3-4, pp. 237-262, 2021

Authors: Zhu, Kaixuan | Xie, Yongqin | Zhou, Feng | Li, Xin

Article Type: Research Article

Abstract: In this paper, we consider a non-autonomous reaction-diffusion equation with hereditary effects and the nonlinearity f satisfying the polynomial growth of arbitrary p − 1 (p ⩾ 2 ) order. We employ the asymptotic a priori estimate method (see (J. Differential Equations 223 (2006 ) 367–399)) to our problem and establish an existence criterion for the ( C L 2 ( Ω ) , C L p ( Ω ) ) -uniform (w.r.t σ ∈ Σ …) attractors (see Theorem 2.18 ). Then, we obtain the ( C L 2 ( Ω ) , C L p ( Ω ) ) and ( C L 2 ( Ω ) , C H 0 1 ( Ω ) ) -uniform (w.r.t σ ∈ Σ ) attractors by applying the existence criterion and the uniform (w.r.t σ ∈ Σ ) Condition (C) respectively. Show more

Keywords: Reaction-diffusion equations, delays, uniform attractors

DOI: 10.3233/ASY-201633

Citation: Asymptotic Analysis, vol. 123, no. 3-4, pp. 263-288, 2021

Authors: Webler, Claudete M. | Zanchetta, Janaina P.

Article Type: Research Article

Abstract: The following coupled damped Klein–Gordon–Schrödinger equations are considered i ψ t + Δ ψ + i α b ( x ) ( | ψ | 2 + 1 ) ψ = ϕ ψ χ Ω R in  R n × ( 0 , ∞ ) ( α > 0 ) , ϕ t t − Δ ϕ + ϕ + a ( x ) ϕ t = | ψ | 2 …χ Ω R in  R n × ( 0 , ∞ ) , where Ω R = R n ∖ B R = { | x | ⩾ R } and χ Ω R represents the characteristic function of Ω R . Assuming that a , b ∈ W 1 , ∞ ( R n ) are nonnegative functions such that a ( x ) ⩾ a 0 > 0 in Ω R and b ( x ) ⩾ b 0 > 0 in Ω R , the exponential decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous ones given by Cavalcanti et al. in the references (NoDEA, Nonlinear Differ. Equ. Appl. 15 (2008 ) 91–113), (NoDEA, Nonlinear Differ. Equ. Appl. 7 (2000 ) 285–307), (Communications on Pure and Applied Analysis 17 (2018 ) 2039–2061) and (Evolution Equations and Control Theory 8 (2019 ) 847–865). Show more

Keywords: Klein–Gordon–Schrödinger, localized damping, unbounded domains

DOI: 10.3233/ASY-201634

Citation: Asymptotic Analysis, vol. 123, no. 3-4, pp. 289-315, 2021

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

Article Type: Research Article

Abstract: This work is concerned with the problem of identifying the shape, size and location of a small embedded tumor from measured temperature on the skin surface. The temperature distribution in the biological tissue is governed by the Pennes model equation. The proposed approach is based on the Kohn–Vogelius formulation and the topological sensitivity analysis method. The ill-posed geometric inverse problem is reformulated as a topology optimization. The temperature field perturbation, caused by the presence of a small anomaly, is analyzed and estimated. A topological asymptotic formula, describing the variation of the considered Kohn–Vogelius type functional with respect to the presence of …a small anomaly is derived. Show more

Keywords: Asymptotic analysis, topological sensitivity, inverse problem, Kohn–Vogelius formulation

DOI: 10.3233/ASY-201635

Citation: Asymptotic Analysis, vol. 123, no. 3-4, pp. 317-333, 2021

Authors: Tuan, Nguyen Huy | Caraballo, Tomás | Thach, Tran Ngoc

Article Type: Research Article

Abstract: In this paper, we study two terminal value problems (TVPs) for stochastic bi-parabolic equations perturbed by standard Brownian motion and fractional Brownian motion with Hurst parameter h ∈ ( 1 2 , 1 ) separately. For each problem, we provide a representation for the mild solution and find the space where the existence of the solution is guaranteed. Additionally, we show clearly that the solution of each problem is not stable, which leads to the ill-posedness of each problem. Finally, we propose two regularization results for both considered problems by using the filter …regularization method. Show more

Keywords: Bi-parabolic equation, standard Brownian motion, fractional Brownian motion, terminal value problem, ill-posedness

DOI: 10.3233/ASY-201637

Citation: Asymptotic Analysis, vol. 123, no. 3-4, pp. 335-366, 2021

Authors: Pierre, Olivier | Coulombel, Jean-François

Article Type: Research Article

Abstract: This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the system of ideal incompressible magnetohydrodynamics. Current vortex sheets are piecewise smooth solutions that satisfy suitable jump conditions on the (free) discontinuity surface. In this article, we complete an earlier work by Alì and Hunter (Quart. Appl. Math. 61 (3) (2003 ) 451–474) and construct approximate solutions at any arbitrarily large order of accuracy to the three-dimensional free boundary problem when the initial discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the current vortex sheet give …rise to ‘surface waves’ on either side of the sheet. Such waves decay exponentially in the normal direction to the current vortex sheet and, in the weakly nonlinear regime which we consider here, their leading amplitude is governed by a nonlocal Hamilton–Jacobi type equation known as the ‘HIZ equation’ (standing for Hamilton–Il’insky–Zabolotskaya (J. Acoust. Soc. Amer. 97 (2) (1995 ) 891–897)) in the context of Rayleigh waves in elastodynamics. The main achievement of our work is to develop a systematic approach for constructing arbitrarily many correctors to the leading amplitude. We exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. The verification of these solvability conditions is based on mere algebra and arguments of combinatorial analysis, namely a Leibniz type formula which we have not been able to find in the literature. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the current vortex sheet equations. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves (C. R. Math. Acad. Sci. Paris 349 (23–24) (2011 ) 1239–1244) does not arise in the same way for the current vortex sheet problem. Show more

Keywords: Magnetohydrodynamics, current-vortex sheet, weakly nonlinear analysis, surface waves, WKB expansion, stability

DOI: 10.3233/ASY-201638

Citation: Asymptotic Analysis, vol. 123, no. 3-4, pp. 367-401, 2021