Asymptotic Analysis - Volume 116, issue 2

Authors: Papageorgiou, Nikolaos S. | Zhang, Chao

Article Type: Research Article

Abstract: We consider a double phase problem with a reaction consisting of a parametric critical term plus a Caratheodory perturbation which is only locally restricted near zero. We show that for all big values of the parameter λ > 0 the problem has at least three nontrivial smooth solutions all with sign information (two of constant sign and the third nodal). Also we establish the asymptotic behavior of the solutions as λ → + ∞ . Finally if we improve the regularity of the perturbation term, we show that there is a second nodal solution for …a total of four nontrivial smooth solutions all with sign information. Show more

Keywords: Double phase equation, cut-off function, constant sign solutions, nodal solutions, nonlinear regularity, critical groups, critical term

DOI: 10.3233/ASY-191536

Citation: Asymptotic Analysis, vol. 116, no. 2, pp. 73-92, 2020

Authors: Blanc, X. | Josien, M. | Le Bris, C.

Article Type: Research Article

Abstract: We consider homogenization problems for linear elliptic equations in divergence form. The coefficients are assumed to be a local perturbation of some periodic background. We prove W 1 , p and Lipschitz convergence of the two-scale expansion, with explicit rates. For this purpose, we use a corrector adapted to this particular setting, and defined in (Comm. Partial Differential Equations 40 (2015 ) 2173–2236; Comm. Partial Differential Equations 43 (2018 ) 965–997), and apply the same strategy of proof as Avellaneda and Lin in (Comm. Pure Appl. Math. 40 …(1987 ) 803–847). We also propose an abstract setting generalizing our particular assumptions for which the same estimates hold. Show more

Keywords: Homogenization, elliptic PDE, periodic media, defects, convergence rate

DOI: 10.3233/ASY-191537

Citation: Asymptotic Analysis, vol. 116, no. 2, pp. 93-137, 2020

Authors: Zakharov, Sergei V.

Article Type: Research Article

Abstract: The long-time asymptotic behavior of the solution of the Cauchy problem for the evolutionary third-order Airy equation describing wave propagation in dispersive physical media is derived by using the auxiliary parameter method. For the solution in the form of the convolution of the initial data and the Airy function, an asymptotic Erdélyi expansion in inverse powers of the cube root of the time variable with the coefficients depending on a self-similar variable and the logarithm of time is obtained. To refine the asymptotics, a family of special function classes for its coefficients is introduced. It is pointed out how the …used method is connected with the geometrical optics approach and how the obtained result can be applied to nonlinear third-order PDEs. Show more

Keywords: Third-order Airy equation, linearized KdV equation, Cauchy problem, Airy function, convolution, auxiliary parameter method, long-time asymptotics, Poincaré and Erdélyi expansions, self-similarity

DOI: 10.3233/ASY-191541

Citation: Asymptotic Analysis, vol. 116, no. 2, pp. 139-148, 2020