Asymptotic properties of blow-up solutions in reaction–diffusion equations with nonlocal boundary flux



This paper deals with a reaction–diffusion problem with coupled nonlinear inner sources and nonlocal boundary flux. Firstly, we propose the critical exponents on nonsimultaneous blow-up under some conditions on the initial data. Secondly, we combine the scaling technique and the Green’s identity method to determine four kinds of simultaneous blow-up rates. Thirdly, the lower and the upper bounds of blow-up time are derived by using Sobolev-type differential inequalities.


Reaction–diffusion system Nonsimultaneous blow-up Blow-up rate Blow-up time 

Mathematics Subject Classification

35K55 35B40 35K15 35B33 



The authors would like to thank the anonymous Referees and the Editor for valuable suggestions improving the first version of this paper. This paper is supported by NNSF of China; Shandong Provincial Natural Science Foundation, China (ZR2016AM12, ZR2017LA003); the Fundamental Research Funds for the Central Universities.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ScienceChina University of PetroleumQingdaoPeople’s Republic of China

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