Asymptotic Analysis - Volume 135, issue 3-4

Authors: Li, Weijia | Shangguan, Yuqi | Yan, Weiping

Article Type: Research Article

Abstract: This paper deals with global stability dynamics for the Klein–Gordon–Zakharov system in R 2 . We first establish that this system admits a family of linear mode unstable explicit quasi-periodic wave solutions. Next, we prove that the Kelvin–Voigt damping can help to stabilize those linear mode unstable explicit quasi-periodic wave solutions for the Klein–Gordon–Zakharov system in the Sobolev space H s + 1 ( R 2 ) × H s + 1 ( R 2 ) × H s + 1 …( R 2 ) for any s ⩾ 1 . Moreover, the Kelvin–Voigt damped Klein–Gordon–Zakharov system admits a unique Sobolev regular solution exponentially convergent to some special solutions (including quasi-periodic wave solutions) of it. Our result can be extended to the n -dimension dissipative Klein–Gordon–Zakharov system for any n ⩾ 1 . Show more

Keywords: Stabilizability, quasi-periodic wave solutions, Klein–Gordon–Zakharov system, Kelvin–Voigt damping

DOI: 10.3233/ASY-231856

Citation: Asymptotic Analysis, vol. 135, no. 3-4, pp. 305-348, 2023

Authors: Babadjian, Jean-François | Millot, Vincent | Rodiac, Rémy

Article Type: Research Article

Abstract: This note addresses the question of convergence of critical points of the Ambrosio–Tortorelli functional in the one-dimensional case under pure Dirichlet boundary conditions. An asymptotic analysis argument shows the convergence to two possible limits points: either a globally affine function or a step function with a single jump at the middle point of the space interval, which are both critical points of the one-dimensional Mumford–Shah functional under a Dirichlet boundary condition. As a byproduct, non minimizing critical points of the Ambrosio–Tortorelli functional satisfying the energy convergence assumption as in (Babadjian, Millot and Rodiac (2022 )) are proved to exist.

Keywords: Mumford–Shah functional, Ambrosio–Tortorelli functional, Γ-convergence, critical points, brittle fracture

DOI: 10.3233/ASY-231857

Citation: Asymptotic Analysis, vol. 135, no. 3-4, pp. 349-362, 2023

Authors: Antonia, Railane | Molica Bisci, Giovanni | de Lima, Henrique F. | Santos, Márcio S.

Article Type: Research Article

Abstract: We investigate complete hypersurfaces with some positive higher order mean curvature in a semi-Riemannian warped product space. Under standard curvature conditions on the ambient space and appropriate constraints on the higher order mean curvatures, we establish rigidity and nonexistence results via Liouville type results and suitable maximum principles related to the divergence of smooth vector fields on a complete noncompact Riemannian manifold. Applications to standard warped product models, like the Schwarzschild, Reissner-Nordström and pseudo-hyperbolic spaces, as well as steady state type spacetimes, are given and a particular study of entire graphs is also presented.

Keywords: Semi-Riemannian warped products, complete hypersurfaces, higher order mean curvatures, entire graphs, maximum principles

DOI: 10.3233/ASY-231858

Citation: Asymptotic Analysis, vol. 135, no. 3-4, pp. 363-398, 2023

Authors: Kakumani, Bhargav Kumar | Yadav, Suman Prabha

Article Type: Research Article

Abstract: In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Here we prove the existence of the solution to the problem using the Faedo–Galerkin method. Also, we prove few general decay rate results.

Keywords: Viscoelasticity, global existence, decay estimates, convexity, logarithmic nonlinearity

DOI: 10.3233/ASY-231859

Citation: Asymptotic Analysis, vol. 135, no. 3-4, pp. 399-419, 2023

Authors: Brandolini, Barbara | de Bonis, Ida | Ferone, Vincenzo | Volzone, Bruno

Article Type: Research Article

Abstract: We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are presented, depending on the summability of the right-hand side of the equation. The maximum principle arguments employed in the core of the proofs of the main results offer a nonstandard, flexible alternative to the ones described in (Arch. Ration. Mech. Anal. 239 (2021 ) 1733–1770, Theorem 31). Some interesting consequences are L p regularity results and nonlocal energy estimates for solutions.

Keywords: Symmetrization, fractional Laplacian, nonlocal elliptic equations, singular elliptic equations

DOI: 10.3233/ASY-231860

Citation: Asymptotic Analysis, vol. 135, no. 3-4, pp. 421-444, 2023

Authors: Feng, Weixun | Chen, Zhi | Qin, Dongdong | Tang, Xianhua

Article Type: Research Article

Abstract: In this paper, we consider the d-dimensional (d ⩾ 2 ) Oldroyd-B model with only dissipation in the equation of stress tensor, and establish a small data global well-posedness result in critical L p framework. Particularly, we give a positive answer to the problem proposed recently by Wu-Zhao (J. Differ. Equ. 316 (2022 )) involving the upper bound for the time integral of the lower frequency piece of the stress tensor, and show that it is indeed independent of the time. Moreover, we improve the results in (J. Math. Fluid …Mech. 24 (2022 )) by relaxing the space dimension d = 2 , 3 to any d ⩾ 2 . Show more

Keywords: Oldroyd-B model, tensor dissipation, global solutions, time independent bound

DOI: 10.3233/ASY-231861

Citation: Asymptotic Analysis, vol. 135, no. 3-4, pp. 445-461, 2023

Authors: Giga, Yoshikazu | Łasica, Michał | Rybka, Piotr

Article Type: Research Article

Abstract: We derive the dynamic boundary condition for the heat equation as a limit of boundary layer problems. We study convergence of their weak and strong solutions as the width of the layer tends to zero. We also discuss Γ-convergence of the functionals generating these flows. Our analysis of strong solutions depends on a new version of the Reilly identity.

Keywords: Boundary layer, dynamic boundary conditions, convergence of gradient flows, Reilly identity, Γ-convergence

DOI: 10.3233/ASY-231862

Citation: Asymptotic Analysis, vol. 135, no. 3-4, pp. 463-508, 2023

Authors: Almousa, Nouf | Bucur, Claudia | Cornale, Roberta | Squassina, Marco

Article Type: Research Article

Abstract: In the study of concavity properties of positive solutions to nonlinear elliptic partial differential equations the diffusion and the nonlinearity are typically independent of the space variable. In this paper we obtain new results aiming to get almost concavity results for a relevant class of anisotropic semilinear elliptic problems with spatially dependent source and diffusion.

Keywords: Approximate convexity principles, anisotropic problems, semilinear elliptic problems

DOI: 10.3233/ASY-231863

Citation: Asymptotic Analysis, vol. 135, no. 3-4, pp. 509-524, 2023

Authors: Möller, Jakob | Mauser, Norbert J.

Article Type: Research Article

Abstract: In this paper we introduce the (unipolar) pressureless Euler–Poisswell equation for electrons as the O ( 1 / c ) semi-relativistic approximation and the (unipolar) pressureless Euler–Darwin equation as the O ( 1 / c 2 ) semi-relativistic approximation of the (unipolar) pressureless Euler–Maxwell equation. In the “infinity-ion-mass” limit, the unipolar Euler–Maxwell equation arises from the bipolar Euler–Maxwell equation, describing a coupled system for a plasma of electrons and ions. The non-relativistic limit c → ∞ of the Euler–Maxwell equation is the repulsive Euler–Poisson equation with electric …force. We propose that the Euler–Poisswell equation, where the Euler equation with electric force is coupled to the magnetostatic O ( 1 / c ) approximation of Maxwell’s equations, is the correct semi-relativistic O ( 1 / c ) approximation of the Euler–Maxwell equation. In the Euler–Poisswell equation the fluid dynamics are decoupled from the magnetic field since the Lorentz force reduces to the electric force. The first non-trivial interaction with the magnetic field is at the O ( 1 / c 2 ) level in the Euler–Darwin equation. This is a consistent and self-consistent model of order O ( 1 / c 2 ) and includes the full Lorentz force, which is relativistic at O ( 1 / c 2 ) . The Euler–Poisswell equation also arises as the semiclassical limit of the quantum Pauli–Poisswell equation, which is the O ( 1 / c ) approximation of the relativistic Dirac–Maxwell equation. We also present a local wellposedness theory for the Euler–Poisswell equation. The Euler–Maxwell system couples the non-relativistic Euler equation and the relativistic Maxwell equations and thus it is not fully consistent in 1 / c . The consistent fully relativistic model is the relativistic Euler–Maxwell equation where Maxwell’s equations are coupled to the relativistic Euler equation – a model that is neglected in the mathematics community. We also present the relativistic Euler–Darwin equation resulting as a O ( 1 / c 2 ) approximation of this model. Show more

Keywords: Quantum physics, mathematical modeling, Euler equation, non-relativistic limit, semi-relativistic approximation

DOI: 10.3233/ASY-231864

Citation: Asymptotic Analysis, vol. 135, no. 3-4, pp. 525-543, 2023

Authors: Li, Tatsien | Rao, Bopeng

Article Type: Research Article

Abstract: We first show that under a suitable balanced repartition of the mixed controls within the system, Kalman’s rank condition is still necessary and sufficient for the uniqueness of solution to the adjoint system associated with incomplete internal and boundary observations, therefore for the approximate controllability of the primary system by means of mixed controls. Then we study the stability of the approximately synchronizable state by groups with respect to applied controls.

Keywords: Mixed controls, stability of synchronizable state, wave equations

DOI: 10.3233/ASY-231865

Citation: Asymptotic Analysis, vol. 135, no. 3-4, pp. 545-579, 2023