Asymptotic Analysis - Volume 128, issue 2

Authors: Buccheri, S. | da Silva, J.V. | de Miranda, L.H.

Article Type: Research Article

Abstract: In this work, given p ∈ ( 1 , ∞ ) , we prove the existence and simplicity of the first eigenvalue λ p and its corresponding eigenvector ( u p , v p ) , for the following local/nonlocal PDE system (0.1) − Δ p u + ( − Δ ) p r u = 2 α α + β λ | u | α − 2 …| v | β u in  Ω − Δ p v + ( − Δ ) p s v = 2 β α + β λ | u | α | v | β − 2 v in  Ω u = 0 on  R N ∖ Ω v = 0 on  R N ∖ Ω , where Ω ⊂ R N is a bounded open domain, 0 < r , s < 1 and α ( p ) + β ( p ) = p . Moreover, we address the asymptotic limit as p → ∞ , proving the explicit geometric characterization of the corresponding first ∞ -eigenvalue, namely λ ∞ , and the uniformly convergence of the pair ( u p , v p ) to the ∞ -eigenvector ( u ∞ , v ∞ ) . Finally, the triple ( u ∞ , v ∞ , λ ∞ ) verifies, in the viscosity sense, a limiting PDE system. Show more

Keywords: First eigenvalue problem, simplicity, local/nonlocal p-Laplacians, Hölder ∞-Laplacian and ∞-Laplacian

DOI: 10.3233/ASY-211702

Citation: Asymptotic Analysis, vol. 128, no. 2, pp. 149-181, 2022

Authors: Makki, Ahmad | Miranville, Alain | Petcu, Madalina

Article Type: Research Article

Abstract: In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

Keywords: Cahn–Hilliard/Allen–Cahn equations, dynamic boundary conditions, well-posedness, global attractors, fractal dimension

DOI: 10.3233/ASY-211703

Citation: Asymptotic Analysis, vol. 128, no. 2, pp. 183-209, 2022

Authors: Messaoudi, Salim A. | Talahmeh, Ala A. | Al-Gharabli, Mohammad M. | Alahyane, Mohamed

Article Type: Research Article

Abstract: Problems with variable exponents have attracted a great deal of attention lately and various existence, nonexistence and stability results have been established. The importance of such problems has manifested due to the recent advancement of science and technology and to the wide application in areas such as electrorheological fluids (smart fluids) which have the property that the viscosity changes drastically when exposed to heat or electrical fields. To tackle and understand these models, new sophisticated mathematical functional spaces have been introduced, such as the Lebesgue and Sobolev spaces with variable exponents. In this work, we are concerned with a system …of wave equations with variable-exponent nonlinearities. This system can be regarded as a model for interaction between two fields describing the motion of two “smart” materials. We, first, establish the existence of global solutions then show that solutions of enough regularities stabilize to the rest state ( 0 , 0 ) either exponentially or polynomially depending on the range of the variable exponents. We also present some numerical tests to illustrate our theoretical findings. Show more

Keywords: Variable-exponent nonlinearity, system wave equations, existence, decay

DOI: 10.3233/ASY-211704

Citation: Asymptotic Analysis, vol. 128, no. 2, pp. 211-238, 2022

Authors: Ambrosio, Vincenzo

Article Type: Research Article

Abstract: In this paper we consider singularly perturbed nonlinear Schrödinger equations with electromagnetic potentials and involving continuous nonlinearities with subcritical, critical or supercritical growth. By means of suitable variational techniques, truncation arguments and Lusternik–Schnirelman theory, we relate the number of nontrivial complex-valued solutions with the topology of the set where the electric potential attains its minimum value.

Keywords: Magnetic Laplacian, Variational methods, Lusternik–Schnirelman theory

DOI: 10.3233/ASY-211705

Citation: Asymptotic Analysis, vol. 128, no. 2, pp. 239-272, 2022

Authors: Cavalcanti, Marcelo M. | Gonzalez Martinez, Victor H.

Article Type: Research Article

Abstract: In the present paper, we are concerned with the semilinear viscoelastic wave equation in an inhomogeneous medium Ω subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space.

Keywords: Wave equation, Kelvin–Voigt damping, frictional damping, source term

DOI: 10.3233/ASY-211706

Citation: Asymptotic Analysis, vol. 128, no. 2, pp. 273-293, 2022