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.
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
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
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
Galaktionov V A, Kurdyumov S P, Samarski A A. A parabolic system of quasilinear equations I. Differ Uravn, 1983, 19: 2123–2143MathSciNetGoogle Scholar
Galaktionov V A, Kurdyumov S P, Samarski A A. A parabolic system of quasilinear equations II. Differ Uravn, 1985, 21: 1544–1559MathSciNetGoogle Scholar
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
Gu Y G, Wang M X. A semilinear parabolic system arising in the nuclear reactors. Chinese Sci Bull, 1994, 39: 1588–1592zbMATHGoogle Scholar
Hoshino H, Yamada Y. Solvability and smoothing effect for semilinear parabolic equations. Funkcial Ekvac, 1991, 34: 475–494MathSciNetzbMATHGoogle Scholar
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
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
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
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