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Boundedness in a two-dimensional doubly degenerate nutrient taxis system.

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Title: Boundedness in a two-dimensional doubly degenerate nutrient taxis system.
Authors: Zhang, Zhiguang1,2 (AUTHOR) guangz_z@163.com, Li, Yuxiang2 (AUTHOR) lieyx@seu.edu.cn
Source: Mathematical Models & Methods in Applied Sciences. Mar2026, Vol. 36 Issue 3, p527-559. 33p.
Subjects: Bacterial population, Neumann problem, Parameterization, Fixed point theory, Nutrient uptake
Abstract: In this work, we study the doubly degenerate nutrient taxis system (*) u t = ∇ ⋅ (u v ∇ u) − χ ∇ ⋅ (u 2 v ∇ v) + ℓ u v , v t = Δ v − u v in a smoothly bounded domain Ω ⊂ ℝ 2 , where χ > 0 and ℓ > 0. This model was proposed by [J. F. Leyva, C. Málaga and R. G. Plaza, The effects of nutrient chemotaxis on bacterial aggregation patterns with nonlinear degenerate cross diffusion, Phys. A392 (2013) 5644–5662] to describe the experimentally observed complex pattern formation phenomena in bacterial populations. In this paper, we demonstrate that for all reasonably regular initial data, the corresponding homogeneous Neumann initial-boundary value problem (*) possesses a global bounded weak solution, which is continuous in its first component and essentially smooth in its second component. Compared to the existing works, we do not impose any smallness conditions on the initial data. Moreover, we identify a criterion regarding the initial smallness of the second component such that the solution stabilizes to a nonconstant steady state. [ABSTRACT FROM AUTHOR]
Copyright of Mathematical Models & Methods in Applied Sciences is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Boundedness in a two-dimensional doubly degenerate nutrient taxis system.
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  Data: <searchLink fieldCode="AR" term="%22Zhang%2C+Zhiguang%22">Zhang, Zhiguang</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> guangz_z@163.com</i><br /><searchLink fieldCode="AR" term="%22Li%2C+Yuxiang%22">Li, Yuxiang</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> lieyx@seu.edu.cn</i>
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  Data: <searchLink fieldCode="JN" term="%22Mathematical+Models+%26+Methods+in+Applied+Sciences%22">Mathematical Models & Methods in Applied Sciences</searchLink>. Mar2026, Vol. 36 Issue 3, p527-559. 33p.
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  Data: <searchLink fieldCode="DE" term="%22Bacterial+population%22">Bacterial population</searchLink><br /><searchLink fieldCode="DE" term="%22Neumann+problem%22">Neumann problem</searchLink><br /><searchLink fieldCode="DE" term="%22Parameterization%22">Parameterization</searchLink><br /><searchLink fieldCode="DE" term="%22Fixed+point+theory%22">Fixed point theory</searchLink><br /><searchLink fieldCode="DE" term="%22Nutrient+uptake%22">Nutrient uptake</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: In this work, we study the doubly degenerate nutrient taxis system (*) u t = ∇ ⋅ (u v ∇ u) − χ ∇ ⋅ (u 2 v ∇ v) + ℓ u v , v t = Δ v − u v in a smoothly bounded domain Ω ⊂ ℝ 2 , where χ > 0 and ℓ > 0. This model was proposed by [J. F. Leyva, C. Málaga and R. G. Plaza, The effects of nutrient chemotaxis on bacterial aggregation patterns with nonlinear degenerate cross diffusion, Phys. A392 (2013) 5644–5662] to describe the experimentally observed complex pattern formation phenomena in bacterial populations. In this paper, we demonstrate that for all reasonably regular initial data, the corresponding homogeneous Neumann initial-boundary value problem (*) possesses a global bounded weak solution, which is continuous in its first component and essentially smooth in its second component. Compared to the existing works, we do not impose any smallness conditions on the initial data. Moreover, we identify a criterion regarding the initial smallness of the second component such that the solution stabilizes to a nonconstant steady state. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Mathematical Models & Methods in Applied Sciences is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1142/S0218202526500077
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 33
        StartPage: 527
    Subjects:
      – SubjectFull: Bacterial population
        Type: general
      – SubjectFull: Neumann problem
        Type: general
      – SubjectFull: Parameterization
        Type: general
      – SubjectFull: Fixed point theory
        Type: general
      – SubjectFull: Nutrient uptake
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      – TitleFull: Boundedness in a two-dimensional doubly degenerate nutrient taxis system.
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            NameFull: Zhang, Zhiguang
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            – D: 01
              M: 03
              Text: Mar2026
              Type: published
              Y: 2026
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            – TitleFull: Mathematical Models & Methods in Applied Sciences
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