Journal of Mathematical Sciences, Vol. 254, No. 2, April, 2021 UNIFORM ATTRACTOR FOR AN N -DIMENSIONAL PARABOLIC SYSTEM WITH IMPULSIVE PERTURBATION O. V. Kapustyan 1 ; 2 , F. A. Asrorov 3 , and V. V. Sobchuk 4 UDC 517.9 We consider a weakly nonlinear N -dimensional parabolic system whose solutions are subjected to im- pulsive perturbations upon attainment of a certain fixed subset in the phase space. For broad classes of impulsive perturbations, it is proved that the system generates an impulsive semiflow with the minimal compact uniformly attracting set (uniform attractor) in the phase space. The invariance and stability of the nonimpulsive part of the uniform attractor are established under additional restrictions imposed on impulsive parameters. Introduction The qualitative theory of differential equations with impulsive perturbations was developed in [1–5]. For impulsive dynamical systems in finite-dimensional phase spaces, this theory was presented in [6–11]. For the infinite-dimensional phase space, the qualitative behavior of dissipative systems was studied within the framework of the theory of global attractors in [12–16]. In [17–19], the main notions and results of the theory of attractors were generalized to the case of infinite-dimensional impulsive dynamical systems. Moreover, the main object of investigations is the minimal compact uniformly attracting set (uniform attractor). The problems of existence, structure, and invariance of uniform attractors for different classes of infinite-dimensional impulsive systems were investigated in [18–22]. Conditions for an impulsive semiflow guaranteeing stability of the nonimpulsive part of a uniform attractor were proposed for the first time in [23]. In the present paper, we formulate these conditions in a more correct form and apply them for the investigation of stability of the uniform attractor of a weakly nonlinear N -dimensional parabolic system with impulsive perturbations. 1. Statement of the Problem In a bounded domain  ⇢ R n ;n � 1; for the unknown vector function u.t;x/ D .u 1 .t;x/;:::;u N .t;x// T ; .t;x/ 2  ⇥ .0; C1/; we consider a parabolic system 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ : @u 1 @t D a 1 Åu 1 � "f 1 .u 1 ;:::;u N /; ::::::::::::::::::::::::::::::::: @u N @t D a N Åu N � "f N .u 1 ;:::;u N /; u 1 j @ D ::: D u N j @ D 0; (1) 1 T. Shevchenko Kyiv National University, Volodymyrs’ka Str., 64, Kyiv, 01601, Ukraine; e-mail: alexkap@univ.kiev.ua. 2 Corresponding author. 3 T. Shevchenko Kyiv National University, Volodymyrs’ka Str., 64, Kyiv, 01601, Ukraine; e-mail: far@ukr.net. 4 L. Ukrainka East-European National University, Volya Ave., 13, Lutsk, 43025, Ukraine; e-mail: v.v.sobchuk@gmail.com. Translated from Neliniini Kolyvannya, Vol. 22, No. 4, pp. 474–481, October–December, 2019. Original article submitted October 3, 2019; revision submitted October 18, 2019. 1072-3374/21/2542–0219 c � 2021 Springer Science+Business Media, LLC 219 DOI 10.1007/s10958-021-05299-1