Asymptotic Analysis - Volume 125, issue 1-2

Authors: Ghader, Mouhammad | Nasser, Rayan | Wehbe, Ali

Article Type: Research Article

Abstract: We investigate the stability of a one-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin–Voigt type and with boundary or localized internal delay feedback. The main novelty in this paper is that the Kelvin–Voigt and the delay damping are both localized via non smooth coefficients. Under sufficient assumptions, in the case that the Kelvin–Voigt damping is localized faraway from the tip and the wave is subjected to a boundary delay feedback, we prove that the energy of the system decays polynomially of type t − 4 . However, an exponential decay of …the energy of the system is established provided that the Kelvin–Voigt damping is localized near a part of the boundary and a time delay damping acts on the second boundary. While, when the Kelvin–Voigt and the internal delay damping are both localized via non smooth coefficients near the boundary, under sufficient assumptions, using frequency domain arguments combined with piecewise multiplier techniques, we prove that the energy of the system decays polynomially of type t − 4 . Otherwise, if the above assumptions are not true, we establish instability results. Show more

Keywords: Wave equation, Kelvin–Voigt damping, time delay, semigroup, stability

DOI: 10.3233/ASY-201649

Citation: Asymptotic Analysis, vol. 125, no. 1-2, pp. 1-57, 2021

Authors: Biswas, Tania | Dharmatti, Sheetal | Mohan, Manil T. | Perisetti, Lakshmi Naga Mahendranath

Article Type: Research Article

Abstract: The Cahn–Hilliard–Navier–Stokes system describes the evolution of two isothermal, incompressible, immiscible fluids in a bounded domain. In this work, we consider the stationary nonlocal Cahn–Hilliard–Navier–Stokes system in two and three dimensions with singular potential. We prove the existence of a weak solution for the system using pseudo-monotonicity arguments and Browder’s theorem. Further, we establish the uniqueness and regularity results for the weak solution of the stationary nonlocal Cahn–Hilliard–Navier–Stokes system for constant mobility parameter and viscosity. Finally, in two dimensions, we establish that the stationary solution is exponentially stable (for convex singular potentials) under suitable conditions on mobility parameter and viscosity.

Keywords: Steady state nonlocal Cahn–Hilliard–Navier–Stokes systems, maximal monotone operator, pseudo-monotonicity, exponential stability

DOI: 10.3233/ASY-201650

Citation: Asymptotic Analysis, vol. 125, no. 1-2, pp. 59-99, 2021

Authors: Kang, Hyeonbae | Li, Xiaofei | Sakaguchi, Shigeru

Article Type: Research Article

Abstract: The polarization tensor is a geometric quantity associated with a domain. It is a signature of the small inclusion’s existence inside a domain and used in the small volume expansion method to reconstruct small inclusions by boundary measurements. In this paper, we consider the question of the polarization tensor vanishing structure of general shape. The only known examples of the polarization tensor vanishing structure are concentric disks and balls. We prove, by the implicit function theorem on Banach spaces, that a small perturbation of a ball can be enclosed by a domain so that the resulting inclusion of the core-shell …structure becomes polarization tensor vanishing. The boundary of the enclosing domain is given by a sphere perturbed by spherical harmonics of degree zero and two. This is a continuation of the earlier work (Kang, Li, Sakaguchi) for two dimensions. Show more

Keywords: Polarization tensor, polarization tensor vanishing structure, weakly neutral inclusion, neutral inclusion, existence, perturbation of balls, implicit function theorem, invisibility cloaking

DOI: 10.3233/ASY-201651

Citation: Asymptotic Analysis, vol. 125, no. 1-2, pp. 101-132, 2021

Authors: Qin, Yuming | Sheng, Ye

Article Type: Research Article

Abstract: In this paper, we investigate one-dimensional thermoelastic system of Timoshenko type III with double memory dampings. At first we give the global existence of solutions by using semigroup theory. Then we can prove the energy decay of solutions by constructing a series of Lyapunov functionals and obtain the existence of absorbing ball. Finally, we prove the asymptotic compactness by using uniform contractive functions and obtain the existence of uniform attractor.

Keywords: Timoshenko system, global existence, asymptotic behavior, uniform attractor

DOI: 10.3233/ASY-201653

Citation: Asymptotic Analysis, vol. 125, no. 1-2, pp. 133-157, 2021

Authors: Mavoungou, Urbain Cyriaque | Batangouna, Narcisse | Langa, Franck Davhys Reval | Moukoko, Daniel | Batchi, Macaire

Article Type: Research Article

Abstract: In this paper, we study of the dissipativity, global attractor and exponential attractor for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms, with initial and homogenous Dirichlet boundary condition.

Keywords: Caginalp hyperbolic phase field, singular potential, Dirichlet boundary, bounded absorbing set, dissipativity, global attractor, exponential attractor

DOI: 10.3233/ASY-201655

Citation: Asymptotic Analysis, vol. 125, no. 1-2, pp. 159-186, 2021

Authors: Mi, Ling | Chen, Chuan

Article Type: Research Article

Abstract: In this paper, we consider the m -Hessian equation S m [ D 2 u ] = b ( x ) f ( u ) > 0 in Ω, subject to the singular boundary condition u = ∞ on ∂ Ω . We give estimates of the asymptotic behavior of such solutions near ∂ Ω when the nonlinear term f satisfies a new structure condition.

Keywords: Hessian equations, singular boundary condition, asymptotic behavior

DOI: 10.3233/ASY-201656

Citation: Asymptotic Analysis, vol. 125, no. 1-2, pp. 187-202, 2021