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September  2016, 15(5): 1757-1768. doi: 10.3934/cpaa.2016012

Global and blowup solutions for general Lotka-Volterra systems

Shaohua Chen 1, , Runzhang Xu 2, and  Hongtao Yang 2,

1. 

School of Science and Technology, Cape Breton University, Sydney, NS, Canada, B1P 6L2, Canada
2. 

College of Science, Harbin Engineering University, Harbin 150001, China, China

Received  September 2015 Revised  March 2016 Published  July 2016

This paper deals with global and blowup solutions of the degenerate parabolic system $u_t=\alpha(v)\nabla\cdot(u^{p}\nabla u)+f(u,v)$ and $v_t=\beta(u)\nabla\cdot(v^{q}\nabla v)+g(u,v)$ with homogeneous Dirichlet boundary conditions. We will give sufficient conditions such that the solutions either exist globally or blow up in a finite time. In special cases, a necessary and sufficient condition for the global existence is given.
Keywords: degenerate parabolic systems, Global and blowup solutions, Lotka-Volterra model..
Mathematics Subject Classification: Primary: 35K10, 35K55; Secondary: 35C5.
Citation: Shaohua Chen, Runzhang Xu, Hongtao Yang. Global and blowup solutions for general Lotka-Volterra systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1757-1768. doi: 10.3934/cpaa.2016012
References:
[1]

S. Chen, Global existence and nonexistence for some degenerate and quasilinear parabolic systems, J. Differential Equations, 245 (2008), 1112-1136. doi: 10.1016/j.jde.2007.11.008.  Google Scholar

[2]

S. Chen, Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms, Comm. Pure Appl. Anal., 8 (2009), 587-600. doi: 10.3934/cpaa.2009.8.587.  Google Scholar

[3]

S. Chen and K. MacDonald, Global and blowup solutions for general quasilinear parabolic systems, Nonlinear Anal. RWA, 14 (2013), 423-433. doi: 10.1016/j.nonrwa.2012.07.006.  Google Scholar

[4]

W. Deng, Global existence and finite time blow up for a degenerate reaction-diffusion system, Nonlinear Anal., 60 (2005), 977-991. doi: 10.1016/j.na.2004.10.016.  Google Scholar

[5]

Y. Han and W. Gao, A degenerate and strongly coupled quasilinear parabolic system with crosswise diffusion for a mutualistic model, Nonlinear Anal. RWA, 11 (2010), 3421-3430. doi: 10.1016/j.nonrwa.2009.12.002.  Google Scholar

[6]

V. A. Galaktionov and J. L. Vazquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst., 8 (2002), 399-433. doi: 10.3934/dcds.2002.8.399.  Google Scholar

[7]

K. Kim and Z. Lin, A degenerate parabolic system with self-diffusion for a mutualistic model in ecology, Nonlinear Anal. RWA, 7 (2006), 597-609. doi: 10.1016/j.nonrwa.2005.03.020.  Google Scholar

[8]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1967.  Google Scholar

[9]

C. Mu, X. Hu, Y. Li and Z. Cui, Blow-up and global existence for a coupled system of degenerte parabolic equations in a bounded domain, Acta Math. Sci., 27B (2007), 92-106. doi: 10.1016/S0252-9602(07)60008-3.  Google Scholar

[10]

C. V. Pao, A Lotka-Volterra cooperating reaction-diffusion system with degenerate density-dependent diffusion, Nonl. Anal., 95 (2014), 460-467. doi: 10.1016/j.na.2013.09.015.  Google Scholar

[11]

C. V. Pao, Dynamics of Lotka-Volterra competition reaction-diffusion systems with degenerate diffusion, J. Math. Anal. Appl., 421 (2015), 1721-1742. doi: 10.1016/j.jmaa.2014.07.070.  Google Scholar

[12]

C. V. Pao and W. H. Ruan, Quasilinear parabolic and elliptic systems with mixed quasimonotone functions, J. Differential Equations, 255 (2013), 1515-1553. doi: 10.1016/j.jde.2013.05.015.  Google Scholar

[13]

M. Wang, Some degenerate and quasilinear parabolic systems not in divergence form, J. Math. Anal. Appl., 274 (2002), 424-436. doi: 10.1016/S0022-247X(02)00347-5.  Google Scholar

[14]

M. Wang and C. Xie, A degenerate and strongly coupled quasilinear parabolic system not in divergence form, Z. Angew. Math. Phys., 55 (2004), 741-755. doi: 10.1007/s00033-004-1133-4.  Google Scholar

[15]

W. Yang, J. Wu and H. Nie, Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate, Commun. Pure Appl. Anal., 14 (2015), 1183-1204. doi: 10.3934/cpaa.2015.14.1183.  Google Scholar

show all references

References:
[1]

S. Chen, Global existence and nonexistence for some degenerate and quasilinear parabolic systems, J. Differential Equations, 245 (2008), 1112-1136. doi: 10.1016/j.jde.2007.11.008.  Google Scholar

[2]

S. Chen, Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms, Comm. Pure Appl. Anal., 8 (2009), 587-600. doi: 10.3934/cpaa.2009.8.587.  Google Scholar

[3]

S. Chen and K. MacDonald, Global and blowup solutions for general quasilinear parabolic systems, Nonlinear Anal. RWA, 14 (2013), 423-433. doi: 10.1016/j.nonrwa.2012.07.006.  Google Scholar

[4]

W. Deng, Global existence and finite time blow up for a degenerate reaction-diffusion system, Nonlinear Anal., 60 (2005), 977-991. doi: 10.1016/j.na.2004.10.016.  Google Scholar

[5]

Y. Han and W. Gao, A degenerate and strongly coupled quasilinear parabolic system with crosswise diffusion for a mutualistic model, Nonlinear Anal. RWA, 11 (2010), 3421-3430. doi: 10.1016/j.nonrwa.2009.12.002.  Google Scholar

[6]

V. A. Galaktionov and J. L. Vazquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst., 8 (2002), 399-433. doi: 10.3934/dcds.2002.8.399.  Google Scholar

[7]

K. Kim and Z. Lin, A degenerate parabolic system with self-diffusion for a mutualistic model in ecology, Nonlinear Anal. RWA, 7 (2006), 597-609. doi: 10.1016/j.nonrwa.2005.03.020.  Google Scholar

[8]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1967.  Google Scholar

[9]

C. Mu, X. Hu, Y. Li and Z. Cui, Blow-up and global existence for a coupled system of degenerte parabolic equations in a bounded domain, Acta Math. Sci., 27B (2007), 92-106. doi: 10.1016/S0252-9602(07)60008-3.  Google Scholar

[10]

C. V. Pao, A Lotka-Volterra cooperating reaction-diffusion system with degenerate density-dependent diffusion, Nonl. Anal., 95 (2014), 460-467. doi: 10.1016/j.na.2013.09.015.  Google Scholar

[11]

C. V. Pao, Dynamics of Lotka-Volterra competition reaction-diffusion systems with degenerate diffusion, J. Math. Anal. Appl., 421 (2015), 1721-1742. doi: 10.1016/j.jmaa.2014.07.070.  Google Scholar

[12]

C. V. Pao and W. H. Ruan, Quasilinear parabolic and elliptic systems with mixed quasimonotone functions, J. Differential Equations, 255 (2013), 1515-1553. doi: 10.1016/j.jde.2013.05.015.  Google Scholar

[13]

M. Wang, Some degenerate and quasilinear parabolic systems not in divergence form, J. Math. Anal. Appl., 274 (2002), 424-436. doi: 10.1016/S0022-247X(02)00347-5.  Google Scholar

[14]

M. Wang and C. Xie, A degenerate and strongly coupled quasilinear parabolic system not in divergence form, Z. Angew. Math. Phys., 55 (2004), 741-755. doi: 10.1007/s00033-004-1133-4.  Google Scholar

[15]

W. Yang, J. Wu and H. Nie, Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate, Commun. Pure Appl. Anal., 14 (2015), 1183-1204. doi: 10.3934/cpaa.2015.14.1183.  Google Scholar

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