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Global well-posedness of advective Lotka–Volterra competition systems with nonlinear diffusion

  • Qi Wang (a1), Jingyue Yang (a2) and Feng Yu (a3)

Abstract

This paper investigates the global well-posedness of a class of reaction–advection–diffusion models with nonlinear diffusion and Lotka–Volterra dynamics. We prove the existence and uniform boundedness of the global-in-time solutions to the fully parabolic systems under certain growth conditions on the diffusion and sensitivity functions. Global existence and uniform boundedness of the corresponding parabolic–elliptic system are also obtained. Our results suggest that attraction (positive taxis) inhibits blowups in Lotka–Volterra competition systems.

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1 Alikakos, N. D.. L p bounds of solutions of reaction-diffusion equations. Comm. Partial Diff. Equ. 4 (1979), 827868.
2Amann, H.. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. Funct. Spaces Diff. Operators Nonlinear Anal. Teubner, Stuttgart, Leipzig 133 (1993), 9126.
3Burger, M., Di Francesco, M., Dolak-Struss, Y.. The Keller–Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion. SIAM J. Math. Anal. 38 (2006), 12881315.
4Calvez, V., Carrillo, J. A.. Volume effects in the Keller–Segel model: energy estimates preventing blow-up. J. Math. Pures Appl. 86 (2006), 155175.
5Cantrell, R., Cosner, C. and Lou, Y.. Approximating the ideal free distribution via reaction–diffusion–advection equations. J. Diff. Equ. 245 (2008), 36873703.
6Cantrell, R., Cosner, C., Lou, Y. and Xie, C.. Random dispersal versus fitness-dependent dispersal. J. Diff. Equ. 254 (2013), 29052941.
7Chen, L. and Jüngel, A.. Analysis of a parabolic cross-diffusion population model without self-diffusion. J. Diff. Equ. 224 (2006), 3959.
8Choi, Y. S., Lui, R. and Yamada, Y.. Existence of global solutions for the Shigesada–Kawasaki–Teramoto model with weak cross-diffusion. Discrete Contin. Dyn. Syst. 9 (2003), 11931200.
9Choi, Y. S., Lui, R. and Yamada, Y.. Existence of global solutions for the Shigesada–Kawasaki–Teramoto model with strongly coupled cross-diffusion. Discrete Contin. Dyn. Syst. 10 (2004), 719730.
10Conway, E. and Smoller, J.. A comparison technique for systems of reaction–diffusion equations. Comm. Partial Diff. Equ. 2 (1977), 679697.
11Conway, E., Hoff, D. and Smoller, J.. Large time behavior of solutions of systems of nonlinear reaction–diffusion equations. SIAM J. Appl. Math. 35 (1978), 116.
12Cosner, C.. Reaction–diffusion–advection models for the effects and evolution of dispersal. Discrete Contin. Dyn. Syst. 34 (2014), 17011745.
13De Mottoni, P. and Rothe, F.. Convergence to homogeneous equilibrium state for generalized Volterra–Lotka systems with diffusion. SIAM J. Appl. Math. 37 (1979), 648663.
14Desvillettes, L., Lepoutre, T., Moussa, A. and Trescases, A.. On the entropic structure of reaction–cross diffusion systems. Comm. Partial Diff. Equ. 40 (2015), 17051747.
15Haroske, D. and Triebel, H.. Distributions, Sobolev Spaces, Elliptic equations (Zurich: European Mathematical Society, 2008).
16Hoang, L., Nguyen, T. and Phan, T.. Gradient estimates and global existence of smooth solutions to a cross-diffusion system. SIAM J. Math. Anal. 47 (2015), 21222177.
17Ishida, S., Seki, K. and Yokota, T.. Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains. J. Diff. Equ. 256 (2014), 29933010.
18Jüngel, A.. The boundedness-by-entropy principle for cross-diffusion systems. Nonlinearity 28 (2015), 19632001.
19Kishimoto, K. and Weinberger, H.. The spatial homogeneity of stable equilibria of some reaction-diffusion systems in convex domains. J. Diff. Equ. 58 (1985), 1521.
20Kolokolnikov, T. and Wei, J.. Stability of spiky solutions in a competition model with cross-diffusion. SIAM J. Appl. Math. 71 (2011), 14281457.
21Kuto, K.. Limiting structure of shrinking solutions to the stationary Shigesada–Kawasaki–Teramoto model with large cross-diffusion. SIAM J. Math. Anal. 47 (2015), 39934024.
22Kuto, K. and Tsujikawa, T.. Limiting structure of steady states to the Lotka–Volterra competition model with large diffusion and advection. J. Diff. Equ. 258 (2015), 18011858.
23Lankeit, J.. Chemotaxis can prevent thresholds on population density. Discrete Contin. Dyn. Syst., Ser. B 20 (2015), 14991527.
24Le, D.. Cross diffusion systems on n spatial dimensional domains. Indiana Univ. Math. J. 51 (2002), 625643.
25Le, D. and Nguyen, V.. Global solutions to cross diffusion parabolic systems on 2D domains. Proc. Amer. Math. Soc. 143 (2015), 29993010.
26Le, D. and Nguyen, V.. Global and blow up solutions to cross diffusion systems on 3D domains. Proc. Amer. Math. Soc. (2016), 144 (2016), 48454859.
27Le, D., Nguyen, L. and Nguyen, T.. Coexistence in cross diffusion systems. Indiana Univ. J. Math. 56 (2007), 17491791.
28Lou, Y. and Ni, W.-M.. Diffusion, self-diffusion and cross-diffusion. J. Diff. Equ. 131 (1996), 79131.
29Lou, Y., Ni, W.-M.. Diffusion vs cross-diffusion: an elliptic approach. J. Diff. Equ. 154 (1999), 157190.
30Lou, Y., Ni, W.-M. and Wu, Y.. On the global existence of a cross-diffusion system. Discrete Contin. Dynam. Systems 4 (1998), 193203.
31Lou, Y., Ni, W.-M. and Yotsutani, S.. On a limiting system in the Lotka–Volterra competition with cross-diffusion. Discrete Contin. Dyn. Syst. 10 (2004), 435458.
32Lou, Y., Winkler, M. and Tao, Y.. Approaching the ideal free distribution in two-species competition models with fitness-dependent dispersal. SIAM J. Math. Anal. 46 (2014), 12281262.
33Lou, Y., Ni, W.-M. and Yotsutani, S.. Pattern formation in a cross-diffusion system. Discrete Contin. Dyn. Syst. 35 (2015), 15891607.
34Matano, H. and Mimura, M.. Pattern formation in competition-diffusion systems in nonconvex domains. Publ. Res. Inst. Math. Sci. 19 (1983), 10491079.
35Mimura, M. and Kawasaki, K.. Spatial segregation in competitive interaction–diffusion equations. J. Math. Biol. 9 (1980), 4964.
36Mimura, M., Nishiura, Y., Tesei, A. and Tsujikawa, T.. Coexistence problem for two competing species models with density-dependent diffusion. Hiroshima Math. J. 14 (1984), 425449.
37Mimura, M., Ei, S.-I. and Fang, Q.. Effect of domain-shape on coexistence problems in a competition-diffusion system. J. Math. Biol. 29 (1991), 219237.
38Ni, W.-M., Wu, Y. and Xu, Q.. The existence and stability of nontrivial steady states for S–K–T competition model with cross diffusion. Discrete Contin. Dyn. Syst. 34 (2014), 52715298.
39Wang, Q., Song, Y. and Shao, L.. Boundedness and persistence of populations in advective Lotka–Volterra competition system. Discrete Contin. Dyn. Syst., Ser. B 23 (2018), 22452263.
40Shigesada, N., Kawasaki, K. and Teramoto, E.. Spatial segregation of interacting species. J. Theoret. Biol. 79 (1979), 8399.
41Shim, S.-A. Uniform boundedness and convergence of solutions to cross-diffusion systems. J. Diff. Equ. 185 (2002), 281305.
42Sugiyama, Y. and Kunii, H.. Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term. J. Diff. Equ. 227 (2006), 333364.
43Tao, Y. and Winkler, M.. Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity. J. Diff. Equ. 252 (2012), 692715.
44Tuoc, P.. Global existence of solutions to Shigesada–Kawasaki–Teramoto cross–diffusion systems on domains of arbitrary dimensions. Proc. Amer. Math. Soc. 135 (2007), 39333941.
45Tuoc, P. and Phan, V.. On global existence of solutions to a cross-diffusion system. J. Math. Anal. Appl. 343 (2008), 826834.
46Wang, Y.. Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. J. Diff. Equ. 260 (2016), 19751989.
47Wang, Q., Gai, C. and Yan, J.. Qualitative analysis of a Lotka–Volterra competition system with advection. Discrete Contin. Dyn. Syst. 35 (2015), 12391284.
48Wang, Q.. On the steady state of a shadow system to the SKT competition model. Discrete Contin. Dyn. Syst. Ser. B 19 (2014), 29412961.
49Wang, L., Mu, C. and Zhou, S.. Boundedness in a parabolic–parabolic chemotaxis system with nonlinear diffusion. Z. Angew. Math. Phys. 65 (2014), 11371152.
50Wang, Q. and Zhang, L.. On the multi-dimensional advective Lotka–Volterra competition systems. Nonlinear Anal., Real World Appl. 37 (2017), 329349.
51Winkler, M.. Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model. J. Diff. Equ. 248 (2010), 28892905.
52Winkler, M.. Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source. Comm. Partial Diff. Equ. 35 (2010), 15161537.
53Winkler, M.. Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction. J. Math. Anal. Appl. 384 (2011), 261272.
54Winkler, M.. How far can chemotactic cross-diffusion enforce exceeding carrying capacities?. J. Nonlinear Sci. 24 (2014), 809855.
55Wu, Y. and Xu, Q.. The existence and structure of large spiky steady states for SKT competition systems with cross-diffusion. Discrete Contin. Dyn. Syst. 29 (2011), 367385.
56Yamada, Y.. Global solutions for the Shigesada–Kawasaki–Teramoto model with cross-diffusion. In Recent progress on reaction-diffusion systems and viscosity solutions, (Edited by Du, Yihong, Ishii, Hitoshi and Lin, Wei-Yueh), pp. 282299 (NJ: World Scientific River Edge, 2009)
57Zhang, Q. and Li, Y.. Boundedness in a quasilinear fully parabolic Keller–Segel system with logistic source. Z. Angew. Math. Phys. 66 (2015), 24732484.
58Zheng, P., Mu, C. and Hu, X.. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete Contin. Dyn. Syst. 35 (2015), 22992323.

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