Global well-posedness of coupled parabolic systems

  • Runzhang XuEmail author
  • Wei Lian
  • Yi Niu


The initial boundary value problem of a class of reaction-diffusion systems (coupled parabolic systems) with nonlinear coupled source terms is considered in order to classify the initial data for the global existence, finite time blowup and long time decay of the solution. The whole study is conducted by considering three cases according to initial energy: the low initial energy case, critical initial energy case and high initial energy case. For the low initial energy case and critical initial energy case the suffcient initial conditions of global existence, long time decay and finite time blowup are given to show a sharp-like condition. In addition, for the high initial energy case the possibility of both global existence and finite time blowup is proved first, and then some suffcient initial conditions of finite time blowup and global existence are obtained, respectively.


reaction-diffusion systems coupled parabolic systems global existence asymptotic behavior finite time blowup 


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This work was supported by National Natural Science Foundation of China (Grant No. 11471087), the China Postdoctoral Science Foundation International Postdoctoral Exchange Fellowship Program, the Heilongjiang Postdoctoral Foundation (Grant No. LBH-Z13056) and the Fundamental Research Funds for the Central Universities. Part of this work was finished when the first author visited Professor Zhouping Xin in the Institute of Mathematical Sciences, the Chinese University of Hong Kong. The authors appreciate the referees' valuable suggestions, which help so much with the improving and modification of this paper. Thanks are also to Dr. Xingchang Wang and Dr. Yuxuan Chen for helping revise the paper.


  1. 1.
    Alaa N. Global existence for reaction-diffusion systems with mass control and critical growth with respect to the gradient. J Math Anal Appl, 2001, 253: 532–557MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bai X. Finite time blow-up for a reaction-diffusion system in bounded domain. Z Angew Math Phys, 2014, 65: 135–138MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ball J M. Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Q J Math, 1977, 28: 473–486MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bebernes J, Eberly D. Mathematical Problems from Combustion Theory. New York: Springer-Verlag, 1989CrossRefzbMATHGoogle Scholar
  5. 5.
    Bedjaoui N, Souplet P. Critical blowup exponents for a system of reaction-diffusion equations with absorption. Z Angew Math Phys, 2002, 53: 197–210MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cazenave T, Lions P L. Solutions globales d'équations de la chaleur semi linéaies. Comm Partial Differential Equations, 1984, 9: 955–978MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen H W. Global existence and blow-up for a nonlinear reaction-diffusion system. J Math Anal Appl, 1997, 212: 481–492MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dancer E N, Wang K, Zhang Z. Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species. J Differential Equations, 2011, 251: 2737–2769MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dickstein F, Escobedo M. A maximum principle for semilinear parabolic systems and applications. Nonlinear Anal, 2001, 45: 825–837MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duong A T, Phan D H. A Liouville-type theorem for cooperative parabolic systems. Discrete Contin Dyn Syst, 2018, 38: 823–833MathSciNetzbMATHGoogle Scholar
  11. 11.
    Escobedo M, Herrero M A. Boundedness and blow up for a semilinear reaction-diffusion system. J Differential Equations, 1991, 89: 176–202MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Escobedo M, Herrero M A. A semilinear parabolic system in a bounded domain. Ann Mat Pura Appl (4), 1993, 165: 315–336MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Escobedo M, Levine H A. Critical blowup and global existence numbers for a weakly coupled system of reactiondi ffusion equations. Arch Ration Mech Anal, 1995, 129: 47–100CrossRefzbMATHGoogle Scholar
  14. 14.
    Galaktionov V A, Kurdyumov S P, Samarski A A. A parabolic system of quasilinear equations I. Differ Uravn, 1983, 19: 2123–2143MathSciNetGoogle Scholar
  15. 15.
    Galaktionov V A, Kurdyumov S P, Samarski A A. A parabolic system of quasilinear equations II. Differ Uravn, 1985, 21: 1544–1559MathSciNetGoogle Scholar
  16. 16.
    Gazzola F, Weth T. Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level. Differential Integral Equations, 2005, 18: 961–990MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gu Y G, Wang M X. A semilinear parabolic system arising in the nuclear reactors. Chinese Sci Bull, 1994, 39: 1588–1592zbMATHGoogle Scholar
  18. 18.
    Hoshino H, Yamada Y. Solvability and smoothing effect for semilinear parabolic equations. Funkcial Ekvac, 1991, 34: 475–494MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kwembe T A, Zhang Z B. A semilinear parabolic system with generalized Wentzell boundary condition. Nonlinear Anal, 2012, 75: 3078–3091MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ladyzenskaja O A, Solonnikov V A, Ural'ceva N N. Linear and Quasilinear Equations of Parabolic Type. Providence: Amer Math Soc, 1968CrossRefGoogle Scholar
  21. 21.
    Levine H A. Instability and non-existence of global solutions to nonlinear wave equations of the form Putt = Au + F(u). Trans Amer Math Soc, 1974, 192: 1–21MathSciNetGoogle Scholar
  22. 22.
    Li H L, Wang M X. Critical exponents and lower bounds of blow-up rate for a reaction-diffusion system. Nonlinear Anal, 2005, 63: 1083–1093MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Liu W J. Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms. Nonlinear Anal, 2010, 73: 244–255MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liu Y C, Zhao J S. On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal, 2006, 64: 2665–2687MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pao C V. On nonlinear reaction-diffusion systems. J Math Anal Appl, 1982, 87: 165–198MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pao C V. Nonlinear Parabolic and Elliptic Equations. New York: Plenum Press, 1992Google Scholar
  27. 27.
    Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Israel J Math, 1975, 22: 273–303MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Quiros F, Rossi J D. Non-simultaneous blow-up in a semilinear parabolic system. Z Angew Math Phys, 2001, 52: 342–346MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Quittner P. Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems. Houston J Math, 2003, 29: 757–799MathSciNetzbMATHGoogle Scholar
  30. 30.
    Rossi J D, Souplet P. Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system. Differential Integral Equations, 2005, 18: 405–418MathSciNetzbMATHGoogle Scholar
  31. 31.
    Sato S. Life span of solutions with large initial data for a semilinear parabolic system. J Math Anal Appl, 2011, 380: 632–641MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Simon L. Asymptotics for a class of nonlinear evolution equations with applications to geometric problems. Ann of Math (2), 1983, 118: 525–571MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Souplet P, Tayachi S. Optimal condition for non-simultaneous blow-up in a reaction-diffusion system. J Math Soc Japan, 2004, 56: 571–584MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wang M X. Global existence and finite time blow up for a reaction-diffusion system. Z Angew Math Phys, 2000, 51: 160–167MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wu S T. Global existence, blow-up and asymptotic behavior of solutions for a class of coupled nonlinear Klein-Gordon equations with damping terms. Acta Appl Math, 2012, 11: 75–95MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Xu R Z. Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data. Quart Appl Math, 2010, 68: 459–468MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Xu R Z, Niu Y. Addendum toλobal existence and finite time blow-up for a class of semilinear pseudo-parabolic equations" [J Funct Anal, 2013, 264: 2732–2763]. J Funct Anal, 2016, 270: 4039–4041MathSciNetCrossRefGoogle Scholar
  38. 38.
    Xu R Z, Su J. Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J Funct Anal, 2013, 264: 2732–2763MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Xu X, Ye Z. Life span of solutions with large initial data for a class of coupled parabolic systems. Z Angew Math Phys, 2013, 64: 705–717MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yang J K, Cao Y, Zheng S N. Fujita phenomena in nonlinear pseudo-parabolic system. Sci China Math, 2014, 57: 555–568MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zhang Y. Uniform boundedness and convergence of global solutions to a strongly-coupled parabolic system with three competitive species. Appl Math Comput, 2013, 221: 720–726MathSciNetzbMATHGoogle Scholar
  42. 42.
    Zou H. Blow-up rates for semi-linear reaction-diffusion systems. J Differential Equations, 2014, 257: 843–867MathSciNetCrossRefzbMATHGoogle Scholar

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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of AutomationHarbin Engineering UniversityHarbinChina
  2. 2.College of ScienceHarbin Engineering UniversityHarbinChina
  3. 3.The Institute of Mathematical SciencesThe Chinese University of Hong KongHong KongChina
  4. 4.School of Information Science and EngineeringShandong Normal UniversityJinanChina

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