Asymptotic Analysis - Volume 46, issue 1

Authors: Frénod, E.

Article Type: Research Article

Abstract: We show that the solution to an oscillatory–singularly perturbed ordinary differential equation may be asymptotically expanded into a sum of oscillating terms. Each of those terms writes as an oscillating operator acting on the solution to a nonoscillating ordinary differential equation with an oscillating correction added to it. The expression of the nonoscillating ordinary differential equations are defined by a recurrence relation. We then apply this result to problems where charged particles are submitted to large magnetic field.

Citation: Asymptotic Analysis, vol. 46, no. 1, pp. 1-28, 2006

Authors: Cuevas, Claudio | Vodev, Georgi

Article Type: Research Article

Abstract: We obtain large time decay of the Lp norms, 2<p<+∞, of solutions to the wave equation with a real-valued potential which decays slowly at infinity. This extends previous results by Beals and Strauss [Comm. Partial Differential Equations 18 (1993), 1365–1397] and Georgiev and Visciglia [Comm. Partial Differential Equations 28 (2003), 1325–1369].

Citation: Asymptotic Analysis, vol. 46, no. 1, pp. 29-42, 2006

Authors: François, Gilles

Article Type: Research Article

Abstract: This paper deals with a spectral problem for a second-order elliptic operator A stemming from a parabolic problem under a dynamical boundary condition. The discrete character and the convergence to infinity of the eigenvalue sequence of the problem \[\cases{Au=\lambda u& \mbox{in }\varOmega,\cr \noalign{\vskip3pt}\curpartial_{\nu_{A}}u=\lambda\sigma u& \mbox{on }\curpartial\varOmega,}\] are shown. By means of min–max formulae, a comparison of the eigenvalue sequence with the spectra of the Dirichlet and Neumann problem is obtained and yields an upper bound for λk . On the other hand, comparing with the Steklov problem leads to a lower bound. In the two-dimensional case, this …yields the exact growth order of the eigenvalue sequence and leads to inequalities about the constant of the leading term in its asymptotic behavior. For higher dimensions, the same arguments hold for a sufficiently smooth domain, while for a Lipschitz boundary, lower and upper bounds for the growth exponent are obtained. Finally, the variational characterization of the eigenvalues yields an upper bound for the number of nodal domains of the k-th eigenfunction. The one-dimensional case is also discussed. Show more

Keywords: elliptic operator, eigenvalue problems, asymptotic behavior of eigenvalues, dynamical boundary conditions for parabolic problems

Citation: Asymptotic Analysis, vol. 46, no. 1, pp. 43-52, 2006

Authors: Amadori, Debora

Article Type: Research Article

Abstract: We consider the scalar conservation law with oscillatory, periodic source term \[u^{\varepsilon}_{t}+f(u^{\varepsilon})_{x}=\dfrac{1}{\varepsilon}V'\big(\dfrac{x}{\varepsilon}\big),\quad x\in \mathbb{R},\ t>0,\ \varepsilon>0,\] and with initial data \[u^{\varepsilon}(x,0)=u_{o}\big(x,\dfrac{x}{\varepsilon}\big).\] For possibly resonant initial data, we prove a corrector-type result for this problem, extending a previous one by E and Serre [Asymptotic Anal. 5 (1992), 311–316]: an asymptotic representation \[$U(x,t,\tfrac{x}{\varepsilon})$ is identified for the sequence uε (x,t), and the strong convergence of the asymptotic expansion is shown.

Keywords: conservation laws, periodic source term, oscillations, homogenization

Citation: Asymptotic Analysis, vol. 46, no. 1, pp. 53-79, 2006

Authors: Dkhil, F. | Stevens, A.

Article Type: Research Article

Abstract: In this paper we consider a nonlocal integro-differential model as it was discussed by Bates and Chen [Electron. J. Differential Equations 1999(26) (1999), 1–19]. It is known that unique, stable traveling waves exist for the classical reaction–diffusion model as well as for the nonlocal model and for combinations of both for certain bistable nonlinearities. Here we are concerned with the traveling wave speed and how small perturbations with a nonlocal term affect the speed of the original reaction–diffusion problem. We show that an expansion for the wave speed of the perturbed problem exists and calculate the sign of the first-order …coefficient. Show more

Keywords: nonlocal integro-differential equation, reaction–diffusion equation, traveling wave speed

Citation: Asymptotic Analysis, vol. 46, no. 1, pp. 81-91, 2006