@article{Wang2025, 
author = {Chang-Jian Wang and Yuan-Hao Zang},
title = {Boundedness of solutions in a two-species chemotaxis system},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {5},
pages = {2862-2880},
keywords = {predator-prey model, classical solutions, global existence, nonlinear consumption},
url = {https://www.sciopen.com/article/10.3934/era.2025126},
doi = {10.3934/era.2025126},
abstract = {In this paper, we consider an initial-Neumann boundary value problem for a two-species chemotaxis system         {                                                      ∂              u                                      ∂              t                                =          Δ          u          −          χ          ∇          ⋅          (          u          ∇          w          )          +          u          (                      a                          1                                −                      b                          1                                            u                          m              −              1                                +                      c                          1                                v          )          ,                                                        (          x          ,          t          )          ∈          Ω          ×          (          0          ,                      T                          max                                )          ,                                                                ∂              v                                      ∂              t                                =          Δ          v          −          ξ          ∇          ⋅          (          v          ∇          w          )          +          v          (                      a                          2                                −                      b                          2                                            v                          l              −              1                                −                      c                          2                                u          )          ,                                                        (          x          ,          t          )          ∈          Ω          ×          (          0          ,                      T                          max                                )          ,                                                                ∂              w                                      ∂              t                                =          Δ          w          −          (                      u                          α                                +                      v                          β                                )          w          ,                                                        (          x          ,          t          )          ∈          Ω          ×          (          0          ,                      T                          max                                )          ,                        where the domain    Ω  ⊂            R              n        (  n  ≥  2  ) is bounded and smooth,        T          max        ∈  (  0  ,  ∞  ]  , and parameters        a          i        ,      b          i        ,      c          i        ,  m  ,  l  ,  α  ,    β  ,  χ  ,  ξ  >  0 with    m  ,  l  >  1  ,  i  =  1  ,  2. In the current work, we provide a sufficient condition of global classical solvability to the above system. More precisely, for some suitable initial data, if    m  >  max  {            α      (      n      +      2      )        2    ,  1  } and    l  >  max  {            β      (      n      +      2      )        2    ,  1  }  , then the system has a global classical solution. Compared to previous work, the existence result established here is more generalized, depending only on the nonlinear power exponents and spatial dimensions.}
}