Asymptotic Analysis - Volume 131, issue 2

Authors: Bahrouni, Anouar | Missaoui, Hlel | Ounaies, Hichem

Article Type: Research Article

Abstract: In this paper, we study the existence of least-energy nodal (sign-changing) weak solutions for a class of fractional Orlicz equations given by ( − △ g ) α u + g ( u ) = K ( x ) f ( u ) , in R N , where N ⩾ 3 , ( − △ g ) α is the fractional Orlicz g -Laplace operator, while f …∈ C 1 ( R ) and K is a positive and continuous function. Under a suitable conditions on f and K , we prove a compact embeddings result for weighted fractional Orlicz–Sobolev spaces. Next, by a minimization argument on Nehari manifold and a quantitative deformation lemma, we show the existence of at least one nodal (sign-changing) weak solution. Show more

Keywords: Nodal solutions, Fractional Orlicz–Sobolev spaces, Nehari manifold method, least energy

DOI: 10.3233/ASY-221770

Citation: Asymptotic Analysis, vol. 131, no. 2, pp. 145-183, 2023

Authors: Truong Xuan, Pham | Thi Van Anh, Nguyen

Article Type: Research Article

Abstract: The primary objective of this paper is to investigate the modified Leray-alpha equation on the two-dimensional sphere S 2 , the square torus T 2 and the three-torus T 3 . In the strategy, we prove the existence and the uniqueness of the weak solutions and also the existence of the global attractor for the equation. Then we establish the upper and lower bounds of the Hausdorff and fractal dimensions of the global attractor on both S 2 …and T 2 . Our method is based on the estimates for the vorticity scalar equations and the stationary solutions around the invariant manifold that are constructed by using the Kolmogorov flows. Finally, we will use the results on T 2 to study the lower bound for attractor’s dimensions in the case of T 3 . Show more

Keywords: Modified Leray-alpha equation, 2-dimensional sphere, square torus, three-torus, global attractor, Hausdorff (fractal) dimension, Kolmogorov flows

DOI: 10.3233/ASY-221771

Citation: Asymptotic Analysis, vol. 131, no. 2, pp. 185-207, 2023

Authors: Anza Hafsa, Omar | Mandallena, Jean-Philippe

Article Type: Research Article

Abstract: We study stochastic homogenization by Γ-convergence of nonconvex integrals of the calculus of variations in the space of functions of bounded deformation.

Keywords: Stochastic homogenization, Γ-convergence, nonconvex integrand, space of functions of bounded deformation

DOI: 10.3233/ASY-221772

Citation: Asymptotic Analysis, vol. 131, no. 2, pp. 209-232, 2023

Authors: Polito, Andrea

Article Type: Research Article

Abstract: We study existence and regularity of weak solutions for a class of boundary value problems, whose form is − div ( log ( 1 + | ∇ u | ) | ∇ u | m ( x ) ∇ u ) + u | ∇ u | log ( 1 + | ∇ u | ) = f ( x ) , in  Ω u = 0 , on  ∂ Ω where both the principal …part and the lower order term have a logarithmic growth with respect to the gradient of the solutions. We prove that the solutions, due to the regularizing effect given by the lower order term, belong to the Orlicz–Sobolev space generated by the function s log ( 1 + | s | ) even for L 1 ( Ω ) data. Show more

Keywords: Nonlinear partial differential equations, Orlicz–Sobolev spaces, Logarithmic growth

DOI: 10.3233/ASY-221773

Citation: Asymptotic Analysis, vol. 131, no. 2, pp. 233-254, 2023

Authors: Laurençot, Philippe | Matioc, Bogdan-Vasile

Article Type: Research Article

Abstract: The singular limit of the thin film Muskat problem is performed when the density (and possibly the viscosity) of the lighter fluid vanishes and the porous medium equation is identified as the limit problem. In particular, the height of the denser fluid is shown to converge towards the solution to the porous medium equation and an explicit rate for this convergence is provided in space dimension d ⩽ 4 . Moreover, the limit of the height of the lighter fluid is determined in a certain regime and is given by the corresponding initial condition.

Keywords: Thin film Muskat problem, porous medium equation, singular limit, convergence

DOI: 10.3233/ASY-221774

Citation: Asymptotic Analysis, vol. 131, no. 2, pp. 255-271, 2023

Authors: Qin, Yuming | Cai, Qitao

Article Type: Research Article

Abstract: In this paper, we mainly study the upper semicontinuity of pullback D -attractors for a nonclassical diffusion equation with delay term b ( t , u t ) which contains some hereditary characteristics. Under a critical nonlinearity f , a time-dependent force g ( t , x ) with exponential growth and a delayed force term b ( t , u t ) , using the asymptotic a priori estimate method, we prove the upper semicontinuity of pullback D -attractor { A …ε ( t ) } t ∈ R to equation (1.1 ) with ε ∈ [ 0 , 1 ] . Show more

Keywords: Nonclassical diffusion equations, upper semicontinuity, pullback attractors, delay

DOI: 10.3233/ASY-221782

Citation: Asymptotic Analysis, vol. 131, no. 2, pp. 273-296, 2023