@article{Wang2023, 
author = {Chang-Jian Wang and Zi-Han Zheng},
title = {The effects of cross-diffusion and logistic source on the boundedness of solutions to a pursuit-evasion model},
year = {2023},
journal = {Electronic Research Archive},
volume = {31},
number = {6},
pages = {3362-3380},
keywords = {boundedness criteria, pursuit-evasion model, cross-diffusion, logistic source},
url = {https://www.sciopen.com/article/10.3934/era.2023170},
doi = {10.3934/era.2023170},
abstract = {We study the following quasilinear pursuit-evasion model:         {                                        u                          t                                =          Δ          u          −          χ          ∇          ⋅          (          u          (          u          +          1                      )                          α                                ∇          w          )          +          u          (                      λ                          1                                −                      μ                          1                                            u                                          r                                  1                                            −              1                                +          a          v          )          ,                                                        x          ∈          Ω          ,                    t          >          0          ,                                                  v                          t                                =          Δ          v          +          ξ          ∇          ⋅          (          v          (          v          +          1                      )                          β                                ∇          z          )          +          v          (                      λ                          2                                −                      μ                          2                                            v                                          r                                  2                                            −              1                                −          b          u          )          ,                                                        x          ∈          Ω          ,                    t          >          0          ,                                      0          =          Δ          w          −          w          +          v          ,                                                        x          ∈          Ω          ,                    t          >          0          ,                                      0          =          Δ          z          −          z          +          u          ,                                                        x          ∈          Ω          ,                    t          >          0          ,                        in a smooth and bounded domain    Ω  ⊂            R              n        (  n  ≥  1  )  , where    a  ,  b  ,  χ  ,  ξ  ,      λ          1        ,      λ          2        ,      μ          1        ,      μ          2        >  0  ,    α  ,  β  ∈      R    , and        r          1        ,      r          2        >  1. When        r          1        >  max  {  1  ,  1  +  α  }  ,      r          2        >  max  {  1  ,  1  +  β  }  , it has been proved that if    min  {  (      r          1        −  1  )  (      r          2        −  β  −  1  )  ,  (      r          1        −  α  −  1  )  (      r          2        −  β  −  1  )  }  >            (      n      −      2              )                  +                      n    , then for some suitable nonnegative initial data        u          0       and        v          0        , the system admits a unique globally classical solution which is bounded in    Ω  ×  (  0  ,  ∞  ).}
}