Pullback attractors and statistical solutions for the lattice Zakharov equations on time-dependent spaces

Anran Li; Caidi Zhao; Tomás Caraballo; Anran Li; Caidi Zhao; Tomás Caraballo

doi:10.3934/math.2026137

AIMS Mathematics

2026, Volume 11, Issue 2: 3367-3393. doi: 10.3934/math.2026137

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Research article

Pullback attractors and statistical solutions for the lattice Zakharov equations on time-dependent spaces

1.
Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province, 325035, China
2.
Facultad de Matemáticas, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain

Received: 20 November 2025 Revised: 19 January 2026 Accepted: 27 January 2026 Published: 04 February 2026
MSC : 34D35, 35B41, 76D06

In this paper, the authors investigate the probability distribution of solutions within the time-dependent phase spaces for the lattice Zakharov equations with varying coefficients via the pullback attractors and the notion of generalized Banach limits. They firstly show that the addressed initial value problem is globally well-posed and prove that the related evolution process has a time-dependent pullback attractor on the time-dependent phase spaces. Then they construct a family of invariant Borel probability measures with supports contained in the pullback attractor. Furthermore, they prove that the constructed family of invariant measures is a statistical solution for the addressed lattice Zakharov equations and that Liouville's theorem holds true.
- time-dependent pullback attractor,
- lattice Zakharov equations,
- varying coefficient,
- invariant Borel probability measure,
- statistical solution
Citation: Anran Li, Caidi Zhao, Tomás Caraballo. Pullback attractors and statistical solutions for the lattice Zakharov equations on time-dependent spaces[J]. AIMS Mathematics, 2026, 11(2): 3367-3393. doi: 10.3934/math.2026137

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Abstract

In this paper, the authors investigate the probability distribution of solutions within the time-dependent phase spaces for the lattice Zakharov equations with varying coefficients via the pullback attractors and the notion of generalized Banach limits. They firstly show that the addressed initial value problem is globally well-posed and prove that the related evolution process has a time-dependent pullback attractor on the time-dependent phase spaces. Then they construct a family of invariant Borel probability measures with supports contained in the pullback attractor. Furthermore, they prove that the constructed family of invariant measures is a statistical solution for the addressed lattice Zakharov equations and that Liouville's theorem holds true.

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