@article{An2026, 
author = {Xiaowei An and Xianfa Song},
title = {Double thresholds for blowup and global existence of the solution to a system of parabolic equations},
year = {2026},
journal = {Electronic Research Archive},
volume = {34},
number = {1},
pages = {606-626},
keywords = {system of parabolic equations, global existence, blowup, double thresholds},
url = {https://www.sciopen.com/article/10.3934/era.2026028},
doi = {10.3934/era.2026028},
abstract = {We considered the following parabolic system:         {                                                u            t                    =                      d            1                    Δ          u          −          a          (          x          )          ⋅          ∇          u          +          f          (          u          ,          v          )          ,                    x          ∈          Ω          ,                    t          >          0          ,                                                          v            t                    =                      d            2                    Δ          v          −          b          (          x          )          ⋅          ∇          v          +          g          (          u          ,          v          )          ,                    x          ∈          Ω          ,                    t          >          0          ,                                              u          (          x          ,          0          )          =                      u            0                    (          x          )          ,                    v          (          x          ,          0          )          =                      v            0                    (          x          )          ,                    x          ∈          Ω          ,                        subject to Dirichlet (or Neumann) boundary conditions. Here    Ω  ⊂            R        N    (  N  ≥  1  ) is a bounded smooth domain. In addition to some results on blowup and global existence of the solution, we found some more interesting results as follows: (1) There exists double thresholds for blowup and global existence of the solution. Under certain conditions, if    f  (  u  ,  v  )  =      f    1    (  v  )      g    1    (  u  ) and    g  (  u  ,  v  )  =      f    2    (  v  )      g    2    (  u  ), then the first watershed is         ∫                  c        1                    +      ∞                  d      u                      g        1            (      u      )        =  +  ∞        a    n    d          ∫                  c        2                    +      ∞                  d      v                      f        2            (      v      )        =  +  ∞  ,and the second watershed is         ∫                                        c            ~                          1                    +      ∞                  d      U                                f          ~                    (                                    F            ~                                    −          1                    (      K                        G          ~                    (      U      )      )      )        =  +  ∞        a    n    d          ∫                                        c            ~                          2                    +      ∞                  d      V                                g          ~                    (                                    G            ~                                    −          1                    (              1        ϵ                              F          ~                    (      V      )      )      )        =  +  ∞  .Here              f      ~        ,            g      ~        ,            F      ~       and              G      ~       will be defined in Section 2.2. (2) If there exist nonnegative smooth functions    h  (  u  ),    l  (  v  ) and    H  (  s  ) such that     f  (  u  ,  v  )      h    ′    (  u  )  l  (  v  )  +  g  (  u  ,  v  )  h  (  u  )      l    ′    (  v  )  =  H  [  h  (  u  )  l  (  v  )  ]  ≥  0  ,then the watershed for blowup in finite time and global existence of the solution is         ∫    0          +      ∞                  d      s              H      (      s      )        =  +  ∞  .}
}