Asymptotic Analysis - Volume 121, issue 1

Authors: Czapla, Dawid | Horbacz, Katarzyna | Wojewódka-Ściążko, Hanna

Article Type: Research Article

Abstract: We propose certain conditions implying the functional law of the iterated logarithm (the Strassen invariance principle) for some general class of non-stationary Markov–Feller chains. This class may be briefly specified by the following two properties: firstly, the transition operator of the chain under consideration enjoys a non-linear Lyapunov-type condition, and secondly, there exists an appropriate Markovian coupling whose transition probability function can be decomposed into two parts, one of which is contractive and dominant in some sense. Our criterion may serve as a useful tool in verifying the functional law of the iterated logarithm for certain random dynamical systems, developed e.g. …in biology and population dynamics. In the final part of the paper we present an example application of our main theorem to a mathematical model describing stochastic dynamics of gene expression. Show more

Keywords: Markov chain, random dynamical system, invariant measure, law of the iterated logarithm, asymptotic coupling

DOI: 10.3233/ASY-191592

Citation: Asymptotic Analysis, vol. 121, no. 1, pp. 1-34, 2021

Authors: Helffer, B. | Hoffmann-Ostenhof, T. | Jauberteau, F. | Léna, C.

Article Type: Research Article

Abstract: We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R 2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity 3. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a smaller concentric disk in function of their size.

Keywords: Spectral theory, Laplacian, multiplicity

DOI: 10.3233/ASY-191594

Citation: Asymptotic Analysis, vol. 121, no. 1, pp. 35-57, 2021

Authors: Ikehata, Ryo

Article Type: Research Article

Abstract: We consider the Cauchy problem in R n for the so-called σ -evolution equations with damping terms. We derive asymptotic profiles of solutions with weighted L 1 , 1 ( R n ) initial data, and investigates the optimality of estimates of solutions in L 2 -sense. The obtained results will generalize and compensate those already known in (J. Math. Anal. Appl. 478 (2019 ) 476–498, J. Diff. Eqns 257 (2014 ) 2159–2177, Diff. Int. Eqns …30 (2017 ), 505–520). Show more

Keywords: σ-evolution equation, damping terms, optimal estimates, weighted L1-initial data

DOI: 10.3233/ASY-191595

Citation: Asymptotic Analysis, vol. 121, no. 1, pp. 59-74, 2021

Authors: Zhao, Mingxia | Yang, Xin-Guang | Yan, Xingjie | Cui, Xiaona

Article Type: Research Article

Abstract: This paper is concerned with the tempered pullback dynamics for a three dimensional Benjamin–Bona–Mahony equations with sublinear operator on bounded domain, which describes the long time behavior for long waves model in shallow water with friction. By virtue of a new retarded Gronwall inequality, and using the energy equation method from J.M. Ball (Disc. Cont. Dyn. Syst. 10 (2004 ) 31–52) to achieve asymptotic compactness for solution process, the minimal family of pullback attractors has been obtained, which reduces a single trajectory under a sufficient condition.

Keywords: Benjamin–Bona–Mahony equation, uniformly asymptotic stable, pullback attractors

DOI: 10.3233/ASY-201601

Citation: Asymptotic Analysis, vol. 121, no. 1, pp. 75-100, 2021