Asymptotic Analysis - Volume 140, issue 3-4

Authors: Casado-Díaz, Juan

Article Type: Research Article

Abstract: The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in L N ( Ω ) , with N the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space H 0 1 ( Ω ) . However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates …in (J. Differential Equations 258 (2015) 2290–2314), and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems. Show more

Keywords: Asymptotic behavior, elliptic problem, drift term, varying coefficients

DOI: 10.3233/ASY-241914

Citation: Asymptotic Analysis, vol. 140, no. 3-4, pp. 147-158, 2024

Authors: Guan, Minlan | Lai, Lizhen | Liu, Boxue | Qin, Dongdong

Article Type: Research Article

Abstract: In this paper, we study the following Hamilton–Choquard type elliptic system: − Δ u + u = ( I α ∗ F ( v ) ) f ( v ) , x ∈ R 2 , − Δ v + v = ( I β ∗ F ( u ) ) f ( u ) , x ∈ R 2 , where I α …and I β are Riesz potentials, f : R → R possessing critical exponential growth at infinity and F ( t ) = ∫ 0 t f ( s ) d s . Without the classic Ambrosetti–Rabinowitz condition and strictly monotonic condition on f , we will investigate the existence of ground state solution for the above system. The strongly indefinite characteristic of the system, combined with the convolution terms and critical exponential growth, makes such problem interesting and challenging to study. With the help of a proper auxiliary system, we employ an approximation scheme and the non-Nehari manifold method to control the minimax levels by a fine threshold, and succeed in restoring the compactness for the critical problem. Existence of a ground state solution is finally established by the concentration compactness argument and some detailed estimates. Show more

Keywords: Hamilton–Choquard elliptic system, Critical exponential growth, Ground state solution, Trudinger–Moser inequality

DOI: 10.3233/ASY-241916

Citation: Asymptotic Analysis, vol. 140, no. 3-4, pp. 159-189, 2024

Authors: Moutinho, Abdon

Article Type: Research Article

Abstract: In this paper, we consider the problem of elasticity and stability of the collision of two kinks with low speed v for the nonlinear wave equation known as the ϕ 6 model in dimension 1 + 1 . We construct a sequence of approximate solutions ( ϕ k ( v , t , x ) ) k ∈ N ⩾ 2 for this model to understand the effects of the collision in the movement of each soliton during a …large time interval. The construction uses a new asymptotic method which is not only restricted to the ϕ 6 model. Show more

Keywords: Kink, soliton, ϕ6 model, non-integrable model, scalar field, partial differential equation, ordinary differential equation, collision

DOI: 10.3233/ASY-241917

Citation: Asymptotic Analysis, vol. 140, no. 3-4, pp. 191-280, 2024

Authors: Disconzi, Marcelo M. | Shao, Yuanzhen

Article Type: Research Article

Abstract: We revisit the theory of first-order quasilinear systems with diagonalizable principal part and only real eigenvalues, what is commonly referred to as strongly hyperbolic systems. We provide a self-contained and simple proof of local well-posedness, in the Hadamard sense, of the Cauchy problem. Our regularity assumptions are very minimal. As an application, we apply our results to systems of ideal and viscous relativistic fluids, where the theory of strongly hyperbolic equations has been systematically used to study several systems of physical interest.

Keywords: Strong hyperbolicity, first-order quasilinear systems, relativistic fluids

DOI: 10.3233/ASY-241919

Citation: Asymptotic Analysis, vol. 140, no. 3-4, pp. 281-302, 2024