American Institute of Mathematical Sciences

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

Global and blowup solutions for general Lotka-Volterra systems

 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.
Citation: Shaohua Chen, Runzhang Xu, Hongtao Yang. Global and blowup solutions for general Lotka-Volterra systems. Communications on Pure and 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. [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. [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. [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. [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. [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. [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. [8] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1967. [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. [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. [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. [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. [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. [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. [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.

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