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 & 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,, \emph{J. Differential Equations}, 245 (2008), 1112.  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,, \emph{Comm. Pure Appl. Anal.}, 8 (2009), 587.  doi: 10.3934/cpaa.2009.8.587.  Google Scholar

[3]

S. Chen and K. MacDonald, Global and blowup solutions for general quasilinear parabolic systems,, \emph{Nonlinear Anal. RWA}, 14 (2013), 423.  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,, \emph{Nonlinear Anal.}, 60 (2005), 977.  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,, \emph{Nonlinear Anal. RWA}, 11 (2010), 3421.  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,, \emph{Discrete Contin. Dyn. Syst.}, 8 (2002), 399.  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,, \emph{Nonlinear Anal. RWA}, 7 (2006), 597.  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., (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,, \emph{Acta Math. Sci.}, 27B (2007), 92.  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,, \emph{Nonl. Anal.}, 95 (2014), 460.  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,, \emph{J. Math. Anal. Appl.}, 421 (2015), 1721.  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,, \emph{J. Differential Equations}, 255 (2013), 1515.  doi: 10.1016/j.jde.2013.05.015.  Google Scholar

[13]

M. Wang, Some degenerate and quasilinear parabolic systems not in divergence form,, \emph{J. Math. Anal. Appl.}, 274 (2002), 424.  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,, \emph{Z. Angew. Math. Phys.}, 55 (2004), 741.  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,, \emph{Commun. Pure Appl. Anal.}, 14 (2015), 1183.  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,, \emph{J. Differential Equations}, 245 (2008), 1112.  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,, \emph{Comm. Pure Appl. Anal.}, 8 (2009), 587.  doi: 10.3934/cpaa.2009.8.587.  Google Scholar

[3]

S. Chen and K. MacDonald, Global and blowup solutions for general quasilinear parabolic systems,, \emph{Nonlinear Anal. RWA}, 14 (2013), 423.  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,, \emph{Nonlinear Anal.}, 60 (2005), 977.  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,, \emph{Nonlinear Anal. RWA}, 11 (2010), 3421.  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,, \emph{Discrete Contin. Dyn. Syst.}, 8 (2002), 399.  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,, \emph{Nonlinear Anal. RWA}, 7 (2006), 597.  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., (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,, \emph{Acta Math. Sci.}, 27B (2007), 92.  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,, \emph{Nonl. Anal.}, 95 (2014), 460.  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,, \emph{J. Math. Anal. Appl.}, 421 (2015), 1721.  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,, \emph{J. Differential Equations}, 255 (2013), 1515.  doi: 10.1016/j.jde.2013.05.015.  Google Scholar

[13]

M. Wang, Some degenerate and quasilinear parabolic systems not in divergence form,, \emph{J. Math. Anal. Appl.}, 274 (2002), 424.  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,, \emph{Z. Angew. Math. Phys.}, 55 (2004), 741.  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,, \emph{Commun. Pure Appl. Anal.}, 14 (2015), 1183.  doi: 10.3934/cpaa.2015.14.1183.  Google Scholar

[1]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[2]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[3]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[4]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[5]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[6]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[7]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[8]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[9]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[10]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[11]

Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082

[12]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[13]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[14]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[15]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[16]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[17]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[18]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[19]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[20]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]