Attracting sets in sup-norms for mild solutions of impulsive-perturbed parabolic semilinear problems

Oleksiy Kapustyan; Svetlana Temesheva; Agila Tleulessova; Oleksiy Kapustyan; Svetlana Temesheva; Agila Tleulessova

doi:10.3934/math.2025857

AIMS Mathematics

2025, Volume 10, Issue 8: 19173-19188. doi: 10.3934/math.2025857

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Research article

Attracting sets in sup-norms for mild solutions of impulsive-perturbed parabolic semilinear problems

1.
Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
2.
Department of Mathematics, al-Farabi Kazakh National University, Almaty, Kazakhstan
3.
Department of Differential Equations, Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
4.
Department of Fundamental Mathematics, L.N. Gumilyov Eurasian National University, Astana, Kazakhstan

Received: 15 April 2025 Revised: 11 August 2025 Accepted: 18 August 2025 Published: 22 August 2025
MSC : 35B41, 35K58, 37L30, 34A37, 35B40

In this paper, we investigate the qualitative behavior of an evolutionary problem that consists of a semilinear parabolic equation whose trajectories undergo instantaneous impulsive perturbations at the moments when some integral functional reaches a certain threshold value. The key object is the uniform attractor of the corresponding impulsive infinite-dimensional dynamical system. The novelty of this study is the analysis of mild solutions in the phase space of continuous functions. Under general assumptions on the impulsive parameters, we prove that this problem generates an impulsive dynamical system, and its trajectories have a compact uniform attractor with respect to the supremum norm (sup-norm).
- impulsive dynamical systems,
- semilinear parabolic equations,
- mild solutions,
- uniform attractors,
- sup-norm,
- compact limit dynamics,
- impulsive perturbations,
- phase space,
- global attractors
Citation: Oleksiy Kapustyan, Svetlana Temesheva, Agila Tleulessova. Attracting sets in sup-norms for mild solutions of impulsive-perturbed parabolic semilinear problems[J]. AIMS Mathematics, 2025, 10(8): 19173-19188. doi: 10.3934/math.2025857

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Abstract

In this paper, we investigate the qualitative behavior of an evolutionary problem that consists of a semilinear parabolic equation whose trajectories undergo instantaneous impulsive perturbations at the moments when some integral functional reaches a certain threshold value. The key object is the uniform attractor of the corresponding impulsive infinite-dimensional dynamical system. The novelty of this study is the analysis of mild solutions in the phase space of continuous functions. Under general assumptions on the impulsive parameters, we prove that this problem generates an impulsive dynamical system, and its trajectories have a compact uniform attractor with respect to the supremum norm (sup-norm).

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