@article{Wang2024, 
author = {Changjian Wang and Jiayue Zhu},
title = {Global dynamics to a quasilinear chemotaxis system under some critical parameter conditions},
year = {2024},
journal = {Electronic Research Archive},
volume = {32},
number = {3},
pages = {2180-2202},
keywords = {chemotaxis system, critical parameter conditions, boundedness, long time behavior},
url = {https://www.sciopen.com/article/10.3934/era.2024099},
doi = {10.3934/era.2024099},
abstract = {In this manuscript, the following chemotaxis system has been considered:         {                                        v                          t                                =          ∇          ⋅          (          ϕ          (          v          )          ∇          v          −          φ          (          v          )          ∇                      w                          1                                +          ψ          (          v          )          ∇                      w                          2                                )          +          a          v          −          b                      v                          κ                                ,                                                        x          ∈          Ω          ,                    t          >          0          ,                                      0          =          Δ                      w                          1                                +          α                      v                                          γ                                  1                                                              −          β                      w                          1                                ,                                                        x          ∈          Ω          ,                    t          >          0          ,                                      0          =          Δ                      w                          2                                +          γ                      v                                          γ                                  2                                                              −          δ                      w                          2                                ,                                                        x          ∈          Ω          ,                    t          >          0          ,                        where    Ω is a bounded smooth domain of              R              n        (  n  ≥  1  )  , the parameters    a  ,  b  ,  α  ,  β  ,  γ  ,  δ  ,      γ          1        ,      γ          2        >  0  ,  κ  >  1  , and nonnegative functions    ϕ  (  ϱ  )  =  (  ϱ  +  1      )          m        ,      φ  (  ϱ  )  =  χ  ϱ  (  ϱ  +  1      )          θ      −      1       and    ψ  (  ϱ  )  =  ξ  ϱ  (  ϱ  +  1      )          l      −      1       for    ϱ  ≥  0 with    m  ,  θ  ,  l  ∈      R   and    χ  ,  ξ  >  0. In the present work, we improve the boundedness criteria established in previous work and further show that under the corresponding critical cases, namely, assume that    θ  +      γ          1        =  max  {  l  +      γ          2        ,  κ  }  ≥  m  +      2    n    +  1 with    m  >  −      2    n    ,  n  ≥  3  , if one of the following conditions holds:(a) when    θ  +      γ          1        =  l  +      γ          2        =  κ  , if    θ  ≥  l  ≥  1 and              [      (      κ      −      1      −      m      )      n      −      2      ]      (      2      α      χ      −      γ      ξ      )              2      (      l      −      1      )      +      (      κ      −      1      −      m      )      n        =  b  , or    l  ≥  θ  ≥  1 and              2      α      χ      [      (      κ      −      1      −      m      )      n      −      2      ]              2      (      θ      −      1      )      +      (      κ      −      1      −      m      )      n        =  b  ;    (b) when    θ  +      γ          1        =  κ  >  l  +      γ          2        , if    θ  ≥  1 and              2      α      χ      [      (      κ      −      1      −      m      )      n      −      2      ]              2      (      θ      −      1      )      +      (      κ      −      1      −      m      )      n        =  b  ,then the system still possesses at least a global classical solution, which is bounded in    Ω  ×  (  0  ,  ∞  ). Additionally, we have also explored the long time behavior of the classical solution mentioned above.}
}