The comparability of motions in dynamical systems and recurrent solutions of (S)PDEs

David Cheban^a, Zhenxin Liu<sup>b

a</sup> State University of Moldova, Faculty of Mathematics and Informatics, Department of Mathematics, A. Mateevich Street 60, MD–2009 Chişinău, Moldova
^b School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China

Abstract: Shcherbakov's comparability method is very useful to study recurrent solutions of differential equations. In this paper, we extend the method from metric spaces to uniform spaces, which applies well to dynamical systems in infinite-dimensional spaces. This generalized comparability method can be easily used to study recurrent solutions of (stochastic) partial differential equations under weaker conditions than in earlier results. We also show that the distribution of solutions of SDEs naturally generates a semiflow or skew-product semiflow on the space of probability measures, which is interesting in itself. As illustration, we give an application to semilinear stochastic partial differential equations.

Keywords and phrases: (Uniform) comparability, recurrent motions, uniform space, infinite-dimensional differential equations.

MSC: 34C25, 34C27, 37B20, 60H15

Received: 31.03.2024

Language: English

DOI: 10.56415/basm.y2024.i1-2.p53