Asymptotic Analysis - Volume 91, issue 3-4

Authors: Leseduarte, M.C. | Quintanilla, R.

Article Type: Research Article

Abstract: In this paper we investigate the asymptotic spatial behavior of the solutions for several models for the nerve fibers. First, our analysis deals with the coupling of two parabolic equations. We prove that, under suitable assumptions on the coefficients and the nonlinear function, the decay is similar to the one corresponding to the heat equation. A limit case of this system corresponds to the coupling of a parabolic equation with an ordinary differential equation. In this situation, we see that for suitable boundary conditions the solution ceases to exist for a finite value of the spatial variable. Next two sections …correspond to the coupling of a hyperbolic/parabolic and hyperbolic/ordinary differential problems. For the first one we obtain that the decay is like an exponential of a second degree polynomial in the spatial variable. In the second one, we prove a similar behaviour to the one corresponding to the wave equation. In these two sections we use in a relevant way an exponentially weighted Poincaré inequality which has been revealed very useful in several thermal and mechanical problems. This kind of results have relevance to understand the propagation of perturbations for nerve models. Show more

Keywords: nerve equation, spatial decay, estimates, spatial nonexistence, FitzHugh–Nagumo

DOI: 10.3233/ASY-141258

Citation: Asymptotic Analysis, vol. 91, no. 3-4, pp. 185-203, 2015

Authors: Knessl, Charles | Yao, Haishen

Article Type: Research Article

Abstract: We consider some singularly perturbed ODEs and PDEs that correspond to the mean first passage time T until a diffusion process exits a domain Ω in Rn . We analyze the limit of small diffusion relative to convection, and assume that in a part of Ω the convection field takes the process toward the exit boundary. In the remaining part the flow does not hit the exit boundary, instead taking the process toward a stable equilibrium point inside Ω. Thus Ω is divided into a part where the diffusion is with the flow and a complementary part where the diffusion …is against the flow. We study such first passage problems asymptotically and, in particular, determine how T changes as we go between the two parts of the domain. We shall show that the mean first passage time may be exponentially large even in the part of Ω that is with the flow, and where a typical sample path of the process hits the exit boundary on much shorter time scales. Show more

Keywords: first passage time, diffusion process, convection field

DOI: 10.3233/ASY-141259

Citation: Asymptotic Analysis, vol. 91, no. 3-4, pp. 205-231, 2015

Authors: Attouchi, Amal

Article Type: Research Article

Abstract: We investigate the boundedness and large time behavior of solutions of the Cauchy–Dirichlet problem for the one-dimensional degenerate parabolic equation with gradient nonlinearity: ut =(|ux |p−2 ux )x +|ux |q  in (0,∞)×(0,1),q>p>2. We prove that: either ux blows up in finite time, or u is global and converges in W1,∞ (0,1) to the unique steady state. This in particular eliminates the possibility of global solutions with unbounded gradient. For that purpose a Lyapunov functional is constructed by the approach of Zelenyak.

Keywords: asymptotic behavior, p-Laplacian, gradient term, boundedness, Lyapunov functional

DOI: 10.3233/ASY-141263

Citation: Asymptotic Analysis, vol. 91, no. 3-4, pp. 233-251, 2015

Authors: Cacciapuoti, Claudio | Figari, Rodolfo | Posilicano, Andrea

Article Type: Research Article

Abstract: We consider a one-dimensional evolution problem modeling the dynamics of an acoustic field coupled with a set of mechanical oscillators. We analyze solutions of the system of ordinary and partial differential equations with time-dependent boundary conditions describing the evolution in the limit of a continuous distribution of oscillators.

Keywords: homogenization, point interactions, field–sources interaction

DOI: 10.3233/ASY-141264

Citation: Asymptotic Analysis, vol. 91, no. 3-4, pp. 253-264, 2015

Authors: Nemes, Gergő

Article Type: Research Article

Abstract: The aim of this paper is to investigate the known large argument asymptotic series of the Lommel function by Stieltjes transform representations. We obtain a number of properties of this asymptotic expansion, including explicit and realistic error bounds, and an exponentially improved asymptotic expansion. An interesting consequence related to the large argument asymptotic series of the Struve function is also proved.

Keywords: asymptotic expansions, Lommel function, error bounds, Stokes phenomenon

DOI: 10.3233/ASY-141266

Citation: Asymptotic Analysis, vol. 91, no. 3-4, pp. 265-281, 2015

Authors: Abels, Helmut | Schaubeck, Stefan

Article Type: Research Article

Abstract: We prove rigorously the convergence of the Cahn–Larché system, which is a Cahn–Hilliard system coupled with the system of linearized elasticity, to a modified Hele–Shaw problem as long as a classical solution of the latter system exists. By matched asymptotic expansion we construct approximate solutions and estimate the difference between the true and the approximate solutions.

Keywords: sharp interface limit, diffuse interface model, Cahn–Hilliard equation, linearized elasticity, Hele–Shaw system, matched asymptotic expansion

DOI: 10.3233/ASY-141268

Citation: Asymptotic Analysis, vol. 91, no. 3-4, pp. 283-340, 2015

Authors: Mohammed, Mogtaba | Sango, Mamadou

Article Type: Research Article

Abstract: In this paper we establish new homogenization results for stochastic linear hyperbolic equations with periodically oscillating coefficients. We first use the multiple expansion method to drive the homogenized problem. Next we use the two scale convergence method and Prokhorov's and Skorokhod's probabilistic compactness results. We prove that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized stochastic hyperbolic problem with constant coefficients. We also prove a corrector result.

Keywords: homogenization, two-scale convergence, hyperbolic stochastic PDE, corrector result, Prokhorov and Skorokhod compactness results

DOI: 10.3233/ASY-141269

Citation: Asymptotic Analysis, vol. 91, no. 3-4, pp. 341-371, 2015

Article Type: Other

Citation: Asymptotic Analysis, vol. 91, no. 3-4, pp. 373-374, 2015