Asymptotic Analysis - Volume 84, issue 3-4

Authors: Huang, Aimin | Pham, Du

Article Type: Research Article

Abstract: In this article, our goal is to prove the existence and uniqueness of solution for 1D and 2D semi-linear hyperbolic equations in a bounded domain with a monotone nonlinear term. We use elliptic regularization and a finite difference scheme in time to build the approximate solutions for the semi-linear hyperbolic equations, and we utilize the regularization method together with the monotonicity and convexity of the nonlinear term to show the existence of the resulting stationary problems. Finally, the existence of the solution for the evolution problem is done by studying the convergence of the approximate solutions and by using the …standard Minty method, and the uniqueness is achieved. Show more

Keywords: semi-linear hyperbolic, nonlinear monotonicity, existence and uniqueness, integration by parts, polynomial growth

DOI: 10.3233/ASY-131168

Citation: Asymptotic Analysis, vol. 84, no. 3-4, pp. 123-146, 2013

Authors: Israel, Haydi | Miranville, Alain | Petcu, Madalina

Article Type: Research Article

Abstract: The aim of this paper is to study the well-posedness and long time behavior, in terms of finite-dimensional attractors, of a perturbed Cahn–Hilliard equation. This equation differs from the usual Cahn–Hilliard by the presence of the term ε(−Δu+f(u)). In particular, we prove the existence of a robust family of exponential attractors as ε goes to zero.

Keywords: Cahn–Hilliard equation, regular potential, global attractor, exponential attractor

DOI: 10.3233/ASY-131172

Citation: Asymptotic Analysis, vol. 84, no. 3-4, pp. 147-179, 2013

Authors: De Cave, Linda Maria

Article Type: Research Article

Abstract: In this paper we study nonlinear elliptic boundary value problems with singular nonlinearities whose simplest example is −div (|∇u|p−2 ∇u)=f/uγ in Ω, u=0 on ∂Ω, where Ω is a bounded open set in RN (N≥2), γ>0, 1<p<N, 0≤f∈Lm (Ω), m≥1. The main difficulty is due to the right hand side f(x)/uγ , since u=0 on the boundary. In order to overcome this “obstacle”, we approach our above model problem thanks to the smooth Dirichlet problems un ∈W1,p 0 (Ω): −div (|∇un |p−2 ∇un )=min(f(x),n)/(|un |+1/n)γ and we prove that there exists a …solution u as limit (in a suitable topology) of the sequence {un }. To be more precise, we prove that the above model problem has a suitable solution u for every f in L1 (Ω) and for every γ>0 and how the regularity of u depends on the summability of f, on p and on γ. Show more

Keywords: nonlinear elliptic equations, singular elliptic equations, quasilinear elliptic equations with p-Laplacian

DOI: 10.3233/ASY-131173

Citation: Asymptotic Analysis, vol. 84, no. 3-4, pp. 181-195, 2013

Authors: Bonnaillie-Noël, V. | Dambrine, M.

Article Type: Research Article

Abstract: The presence of small inclusions or of a surface defect modifies the solution of the Laplace equation posed in a reference domain Ω0 . If the characteristic size of the perturbation is small, then one can expect that the solution of the problem posed on the perturbed geometry is close to the solution of the reference shape. Asymptotic expansion with respect to that small parameter – the characteristic size of the perturbation – can then be performed. We consider in the present work the case of two circular defects with homogeneous Dirichlet boundary conditions in a bidimensional domain, we distinguish …the cases where the distance between the object is of order 1 and the case where it is larger than the characteristic size of the defects but small with respect to the size of the domain. In both cases, we derive the complete expansion and provide some numerical illustrations. Show more

Keywords: perforated domain, Dirichlet boundary conditions, asymptotic expansion

DOI: 10.3233/ASY-131174

Citation: Asymptotic Analysis, vol. 84, no. 3-4, pp. 197-227, 2013

Authors: Mouzaoui, Lounès

Article Type: Research Article

Abstract: We study the asymptotic behavior of the Schrödinger equation in the presence of a nonlinearity of Hartree type in the semi-classical regime. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading order amplitude of the solution without altering the rapid oscillations. We show the validity of the WKB-analysis when the potential in the nonlinearity is singular around the origin. No new resonant wave is created in our model, this phenomenon is inhibited due to the nonlinearity. The nonlocal nature of this latter leads us to build our result on a high-frequency averaging effects. In the …proof we make use of the Wiener algebra and the space of square-integrable functions. Show more

Keywords: Schrödinger equation, Hartree nonlinearity, semi-classical regime, WKB-method, Wiener algebra

DOI: 10.3233/ASY-131175

Citation: Asymptotic Analysis, vol. 84, no. 3-4, pp. 229-245, 2013

Article Type: Other

Citation: Asymptotic Analysis, vol. 84, no. 3-4, pp. 247-247, 2013