Asymptotic Analysis - Volume 118, issue 1-2

Authors: Arora, R. | Giacomoni, J. | Goel, D. | Sreenadh, K.

Article Type: Research Article

Abstract: In this article, we study the existence, multiplicity, regularity and asymptotic behavior of the positive solutions to the problem of half-Laplacian with singular and exponential growth nonlinearity in one dimension (see below ( P λ ) ). We prove two results regarding the existence and multiplicity of solutions to the problem ( P λ ) . In the first result, existence and multiplicity have been proved for classical solutions via bifurcation theory while in the latter result multiplicity has been proved for critical exponential nonlinearity by variational methods. An independent …question of symmetry and monotonicity properties of classical solution has been answered in the paper. To characterize the behavior of large solutions, we further study isolated singularities for the singular semi linear elliptic equation in Ω ⊂ R n , n ⩾ 2 s involving exponential growth nonlinearities in the more general framework of ( − Δ ) s operator and for all 0 < s < 1 (see below ( P s ) ). Show more

Keywords: Singular problem, exponential nonlinearity, half-Laplacian, asymptotic behavior, moving plane method

DOI: 10.3233/ASY-191557

Citation: Asymptotic Analysis, vol. 118, no. 1-2, pp. 1-34, 2020

Authors: Jleli, Mohamed | Samet, Bessem

Article Type: Research Article

Abstract: We are concerned with a semilinear parabolic differential inequality posed in an exterior domain of R N , N ⩾ 3 . Some blow-up and existence results are established for the considered problem. The novetly of this paper lies in considering a nontrivial Dirichlet boundary condition, which depends both on time and space. Indeed, to the best of our knowledge, in all articles which deal with the blow-up of solutions in exterior domains, the considered boundary condition is trivial or depends only on space.

Keywords: Blow-up, exterior domain, nontrivial Dirichlet boundary condition

DOI: 10.3233/ASY-191555

Citation: Asymptotic Analysis, vol. 118, no. 1-2, pp. 35-47, 2020

Authors: Lastra, A. | Malek, S.

Article Type: Research Article

Abstract: The asymptotic behavior of a family of singularly perturbed PDEs in two time variables in the complex domain is studied. The appearance of a multilevel Gevrey asymptotics phenomenon in the perturbation parameter is observed. We construct a family of analytic sectorial solutions in ϵ which share a common asymptotic expansion at the origin, in different Gevrey levels. Such orders are produced by the action of the two independent time variables.

Keywords: Asymptotic expansion, Borel–Laplace transform, Fourier transform, initial value problem, formal power series, nonlinear integro-differential equation, nonlinear partial differential equation, singular perturbation

DOI: 10.3233/ASY-191568

Citation: Asymptotic Analysis, vol. 118, no. 1-2, pp. 49-79, 2020

Authors: Chesnel, Lucas | Nazarov, Sergei A. | Taskinen, Jari

Article Type: Research Article

Abstract: We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for …perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given. Show more

Keywords: Linear water-wave problem, asymptotic analysis, invisibility, scattering matrix, weighted spaces with detached asymptotics

DOI: 10.3233/ASY-191556

Citation: Asymptotic Analysis, vol. 118, no. 1-2, pp. 81-122, 2020

Authors: Oquendo, Higidio Portillo | da Luz, Cleverson Roberto

Article Type: Research Article

Abstract: This article deals with the asymptotic behavior of the solutions of a Timoshenko beam with a fractional damping. The damping acts only in one of the equations and depends on a parameter θ ∈ [ 0 , 1 ] . Timoshenko systems with frictional or Kelvin–Voigt dampings are particular cases of this model. We prove that, for regular initial data, the semigroup of this system decays polynomially with rates that depend on θ and some relations between the structural parameters of the system. We also show that the decay rates obtained are optimal and the only possibility …to obtain exponential decay is when θ = 0 and the wave propagation speeds of the equations coincide. Show more

Keywords: Timoshenko beam, frictional damping, Kelvin–Voigt damping, polynomial decay, exponential decay

DOI: 10.3233/ASY-191552

Citation: Asymptotic Analysis, vol. 118, no. 1-2, pp. 123-142, 2020