Asymptotic Analysis - Volume 139, issue 3-4

Authors: Liu, Chungen | Zhong, Yuyou | Zuo, Jiabin

Article Type: Research Article

Abstract: In this paper, we study a fractional Schrödinger–Poisson system with p -Laplacian. By using some scaling transformation and cut-off technique, the boundedness of the Palais–Smale sequences at the mountain pass level is gotten. As a result, the existence of non-trivial solutions for the system is obtained.

Keywords: Fractional Schrödinger–Poisson system, p-Laplacian, mountain pass lemma

DOI: 10.3233/ASY-241903

Citation: Asymptotic Analysis, vol. 139, no. 3-4, pp. 139-155, 2024

Authors: Poblete, Felipe | Silva, Clessius | Viana, Arlúcio

Article Type: Research Article

Abstract: This paper studies the existence of local and global self-similar solutions for a Boussinesq system with fractional memory and fractional diffusions u t + u · ∇ u + ∇ p + ν ( − Δ ) β u = θ f , x ∈ R n , t > 0 , θ t + u · ∇ θ + g α ∗ ( − Δ ) γ θ …= 0 , x ∈ R n , t > 0 , div u = 0 , x ∈ R n , t > 0 , u ( x , 0 ) = u 0 , θ ( x , 0 ) = θ 0 , x ∈ R n , where g α ( t ) = t α − 1 Γ ( α ) . The existence results are obtained in the framework of pseudo-measure spaces. We find that the existence and self-similarity of global solutions is strongly influenced by the relationship among the memory and the fractional diffusions. Indeed, we obtain the existence and self-similarity of global solutions only when γ = ( α + 1 ) β . Moreover, we prove a stability result for the global solutions and the existence of asymptotically self-similar solutions. Show more

Keywords: Nonlocal Navier–Stokes, Boussinesq system, PDEs in connection with fluid mechanics, fractional memory, self-similarity

DOI: 10.3233/ASY-241904

Citation: Asymptotic Analysis, vol. 139, no. 3-4, pp. 157-181, 2024

Authors: Kundu, A. | Mahato, H.S.

Article Type: Research Article

Abstract: We present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Posing controls on the porous part of the medium (distributed control), we set up a convex minimization problem. The main objective of this article is to characterize an arbitrary control to be an optimal control. We establish a relation between the optimal control and the corresponding adjoint state. At first, we analyse the microscopic description of the controlled system, then we homogenised the system by rigorous two-scale convergence method and periodic unfolding method.

Keywords: Diffusion–reaction–precipitation equations, optimal control problem, homogenisation, periodic porous medium, existence of solution, asymptotic expansion, two-scale convergence

DOI: 10.3233/ASY-241905

Citation: Asymptotic Analysis, vol. 139, no. 3-4, pp. 183-215, 2024

Authors: Chipot, Michel

Article Type: Research Article

Abstract: The goal of this paper is to explore the asymptotic behaviour of anisotropic problems governed by operators of the pseudo p -Laplacian type when the size of the domain goes to infinity in different directions.

Keywords: Anisotropic operators, nonlinear elliptic operators, pseudo p-Laplacian, asymptotic behaviour, cylinder like domains

DOI: 10.3233/ASY-241906

Citation: Asymptotic Analysis, vol. 139, no. 3-4, pp. 217-243, 2024

Authors: Rawat, Rama | Roy, Haripada | Roy, Prosenjit

Article Type: Research Article

Abstract: The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p -Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal. 85 (3–4) (2013) 199–227) where the linear case p = 2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.

Keywords: p-Laplacian, uniform elliptcity, Poincaré inequality, Krasnoselskii’s genus

DOI: 10.3233/ASY-241907

Citation: Asymptotic Analysis, vol. 139, no. 3-4, pp. 245-277, 2024