THE SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP CRITERIA FOR A DIFFUSION SYSTEM

doi:10.1016/S0252-9602(12)60200-8

Abstract

Abstract:

This paper investigates the finite time blow-up of nonnegative solutions for a nonlinear diffusion system with a more complicated source term, which is a product of localized source, local source, and weight function, and complemented by homogeneous Dirichlet boundary conditions. The criteria are proposed to identify simultaneous and non-simultaneous blow-up solutions. Moreover, the related classification for the four parameters in the model is optimal and complete. The results extend those in Zhang and Yang [12].

Key words: blow-up, simultaneous and non-simultaneous blow-up, critical exponent, dif-fusion system

CLC Number:

35B33

LING Zheng-Qiu, WANG Ze-Jia, ZHANG Guo-Qiang. THE SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP CRITERIA FOR A DIFFUSION SYSTEM[J].Acta mathematica scientia,Series B, 2013, 33(1): 139-149.

Trendmd

References

[1] Escobedo E, Herrero M A. Boundedness and blow-up for a semilinear reaction-diffusion system. J Diff Eqs, 1991, 89(1): 176–202

[2] Glansdorff P, Prigogine I. Thermodynamic Theory of Structure, Stability and Fluctuation. London: Wiley-Interscience, 1971

[3] Meinhardt H. Models of Biological Pattern Formation. London: Academic Press, 1982

[4] Escobedo M, Herrero M A. A semilinear parabolic system in a bounded domain. Ann Mate Pura Appl, 1993, CLXV(IV): 315–336

[5] Escobedo M, Levine H A. Critical blow-up and global existence numbers for a weakly coupled system of reaction-diffusion equations. Arch Rational Mech Anal, 1995, 129: 47–100

[6] Chen H W. Global existence and blow-up for a nonlinear reaction-diffusion system. J Math Anal Appl, 1997, 212: 481–492

[7] Wang M X. Global existence and finite time blow-up for a reaction-diffusion system. Z Angew Math Phys, 2000, 51: 160–167

[8] Zheng S N. Global existence global non-existence of solutions to a reaction-diffusion system. Nonlinear Anal, 2000, 39: 327–340

[9] Chen Q, Xiang Z Y, Mu C L. Blow-up profiles to a nonlocal reaction-diffusion system. Acta Math Sci, 2007, 27A(3): 420–427

[10] Ling Z Q, Lu W M, Zhou Z W. Uniform blow-up profile for a semilinear parabolic system with nonlocal boundaries. Acta Math Sci, 2012, 32A(2): 433–440

[11] Zhang H, Kong L H, Zheng S N. Propagations of singularities in a parabolic system with coupling nonlocal sources. Sci China Ser A: Math, 2009, 52: 181–194

[12] Zhang R, Yang Z D. Uniform blow-up rates and asymptotic estimates of solutions for diffusion systems with weighted localized sources. J Appl Math Comput, 2010, 32: 429–441

[13] Quir´os F, Rossi J D. Non-simultaneous blow-up in a nonlinear parabolic system. Z Angew Math Phys, 2001, 52: 342–346

[14] Zheng S N, Zhao L Z, Chen F. Blow-up rates in a parabolic system of ignition model. Nonlinear Anal, 2002, 51: 663–672

[15] Li F J, Liu B C, Zheng S N. Non-simultaneous blow-up criteria for localized parabolic equations. Comm Math Res, 2009, 25: 379–384

[16] Brändle C, Quir´os F, Rossi J D. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Commun Pure Appl Anal, 2005, 4: 523–536

[17] Brändle C, Quir´os F, Rossi J D. The role of non-linear diffusion in non-simultaneous blow-up. J Math Anal Appl, 2005, 308: 92–104

[18] Rossi J D, Souplet P. Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system. Differential Integral Equations, 2005, 18: 405–418