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April  2019, 12(2): 303-322. doi: 10.3934/krm.2019013

Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion

Marco Di Francesco and  Yahya Jaafra

Department of Information Engineering, Computer Science, and Mathematics, University of L'Aquila, Via Vetoio (Coppito), 67100, L'Aquila, Italy

Received  October 2017 Revised  August 2018 Published  November 2018

In this paper we consider a one-dimensional nonlocal interaction equation with quadratic porous-medium type diffusion in which the interaction kernels are attractive, nonnegative, and integrable on the real line. Earlier results in the literature have shown existence of nontrivial steady states if the $L^1$ norm of the kernel $G$ is larger than the diffusion constant $\varepsilon$. In this paper we aim at showing that this equation exhibits a 'multiple' behavior, in that solutions can either converge to the nontrivial steady states or decay to zero for large times. We prove the former situation holds in case the initial conditions are concentrated enough and 'close' to the steady state in the $∞$-Wasserstein distance. Moreover, we prove that solutions decay to zero for large times in the diffusion-dominated regime $\varepsilon≥ \|G\|_{L^1}$. Finally, we show two partial results suggesting that the large-time decay also holds in the complementary regime $\varepsilon < \|G\|_{L^1}$ for initial data with large enough second moment. We use numerical simulations both to validate our local asymptotic stability result and to support our conjecture on the large time decay.

Keywords: Quadratic diffusion, nonlocal interaction, asymptotic behavior, steady states, Wasserstein distance.
Mathematics Subject Classification: Primary: 35B30, 35B35, 35B40; Secondary: 35B36, 92D25.
Citation: Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic & Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005.  Google Scholar

[2]

J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Applied Mathematics Letters, 24 (2011), 1927-1932.  doi: 10.1016/j.aml.2011.05.022.  Google Scholar

[3]

J. Bedrossian, Large mass global solutions for a class of L1-critical nonlocal aggregation equations and parabolic-elliptic Patlak-Keller-Segel models, arXiv: 1403.4124. Google Scholar

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F. Berthelin, D. Chiron and M. Ribot, Stationary solutions with vacuum for a one-dimensional chemotaxis model with non-linear pressure, Commun. Math. Sci., 14 (2015), 147-186.  doi: 10.4310/CMS.2016.v14.n1.a6.  Google Scholar

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A. L. Bertozzi and J. Brandman, Finite-time blow-up of L infinity weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 45-65.  doi: 10.4310/CMS.2010.v8.n1.a4.  Google Scholar

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A. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in Rn, Comm. Math. Phys., 274 (2007), 717-735.  doi: 10.1007/s00220-007-0288-1.  Google Scholar

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A. L. Bertozzi, T. Laurent and J. Rosado, Lp theory for the multidimensional aggregation equation, Commun. Pure Appl. Math., 64 (2011), 45-83.  doi: 10.1002/cpa.20334.  Google Scholar

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A. Blanchet, J. A. Carrillo and P. Laurencot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Cal. Var. Partial Differential Equations, 35 (2009), 133-168.  doi: 10.1007/s00526-008-0200-7.  Google Scholar

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A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 2006 (2006), No. 44, 32 pp.  Google Scholar

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M. Bodnar and J. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differ. Equations, 222 (2006), 341-380.  doi: 10.1016/j.jde.2005.07.025.  Google Scholar

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S. Boi, V. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens, Nonlinear Anal, Real World Appl., 1 (2000), 163-176.  doi: 10.1016/S0362-546X(99)00399-5.  Google Scholar

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M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Networks and Heterogeneous Media, 3 (2008), 749-785.  doi: 10.3934/nhm.2008.3.749.  Google Scholar

[13]

M. Burger, M. Di Francesco and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Commun. Math. Sci., 11 (2012), 709-738.  doi: 10.4310/CMS.2013.v11.n3.a3.  Google Scholar

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M. Burger, R. C. Fetecau and Y. Huang, Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13 (2014), 397-424.  doi: 10.1137/130923786.  Google Scholar

[15]

V. Calvez, J. A. Carrillo and F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Analysis TMA, 159 (2017), 85-128.  doi: 10.1016/j.na.2017.03.008.  Google Scholar

[16]

J. A. Carrillo, D. Castorina and B. Volzone, Ground states for diffusion dominated free energies with logarithmic interaction, SIAM J. Math. Anal., 47 (2015), 1-25.  doi: 10.1137/140951588.  Google Scholar

[17]

J. A. Carrillo, A. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a.  Google Scholar

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J. A. Carrillo, M. Gualdani and G. Toscani, Finite speed of propagation in porous media by mass transportation methods, Comptes Rendus Mathematique, 338 (2004), 815-818.  doi: 10.1016/j.crma.2004.03.025.  Google Scholar

[19]

J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19 (2003), 971-1018.  doi: 10.4171/RMI/376.  Google Scholar

[20]

J. A. Carrillo and G. Toscani, Wasserstein metric and large-time asymptotics of nonlinear diffusion equations, New Trends in Mathematical Physics, (2004), 234-244.   Google Scholar

[21]

R. Choksi, R. C. Fetecau and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 32 (2015), 1283-1305.  doi: 10.1016/j.anihpc.2014.09.004.  Google Scholar

[22]

K. Fellner and G. Raoul, Stability of stationary states of non-local interaction equations, Mathematical and Computer Modelling, 53 (2011), 1436-1450.  doi: 10.1016/j.mcm.2010.03.021.  Google Scholar

[23]

L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for onedimensional filtration equations, SIAM Journal on Numerical Analysis, 43 (2006), 2590-2606.  doi: 10.1137/040608672.  Google Scholar

[24]

S. Gottlieb, C. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.  Google Scholar

[25]

D. Helbing, I. J. Farkas, P. Molnar and T. Vicsek, Simulation of pedestrian crowds in normal and evacuation situations, Pedestrian and Evacuation Dynamics, 21 (2002), 21-58.   Google Scholar

[26]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[27]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.  Google Scholar

[28]

G. Kaib, Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential, SIAM J. Math. Anal., 49 (2017), 272-296.  doi: 10.1137/16M1072450.  Google Scholar

[29]

I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2012), 568-602.  doi: 10.1137/110823584.  Google Scholar

[30]

H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Archive for rational mechanics and analysis, 172 (2004), 407-428.  doi: 10.1007/s00205-004-0307-8.  Google Scholar

[31]

E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684.  Google Scholar

[32]

P. L. Lions, The concentration-compactness principle in calculus of variations. The locally compact case, part 1, Ann. Inst. Henri Poincar e, Anal. Nonlin., 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[33]

R. McCann, A convexity principle for interacting gases, Advances in mathematics, 128 (1997), 153-179.  doi: 10.1006/aima.1997.1634.  Google Scholar

[34]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.  Google Scholar

[35]

D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: from individuals to populations, Journal of Mathematical Biology, 50 (2005), 49-66.  doi: 10.1007/s00285-004-0279-1.  Google Scholar

[36]

G. Russo, Deterministic diffusion of particles, Communications on Pure and Applied Mathematics, 43 (1990), 697-733.  doi: 10.1002/cpa.3160430602.  Google Scholar

[37]

L. Scardia, R. Peerlings, M. Geers and M. A. Peletier, Mechanics of dislocation pile-ups: A unification of scaling regimes, Journal of the Mechanics and Physics of Solids, 70 (2014), 42-61.  doi: 10.1016/j.jmps.2014.04.014.  Google Scholar

[38]

K. Sznajd-Weron and J. Sznajd. Opinion evolution in closed community, Opinion evolution in closed community, Int. J. Mod. Phys. C, 11 (2000), 1157-1166.   Google Scholar

[39]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bulletin of Mathematical Biology, 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[40]

G. Toscani, Kinetic and hydrodynamic models of nearly elastic granular flows, Monatsh. Math., 142 (2004), 179-192.  doi: 10.1007/s00605-004-0241-8.  Google Scholar

[41]

J. L. Vazquez, The Porous Medium Equation, Oxford Mathematical Monographs, 2007.  Google Scholar

[42]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, AMS, Providence, RI, 2003. doi: 10.1007/b12016.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005.  Google Scholar

[2]

J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Applied Mathematics Letters, 24 (2011), 1927-1932.  doi: 10.1016/j.aml.2011.05.022.  Google Scholar

[3]

J. Bedrossian, Large mass global solutions for a class of L1-critical nonlocal aggregation equations and parabolic-elliptic Patlak-Keller-Segel models, arXiv: 1403.4124. Google Scholar

[4]

F. Berthelin, D. Chiron and M. Ribot, Stationary solutions with vacuum for a one-dimensional chemotaxis model with non-linear pressure, Commun. Math. Sci., 14 (2015), 147-186.  doi: 10.4310/CMS.2016.v14.n1.a6.  Google Scholar

[5]

A. L. Bertozzi and J. Brandman, Finite-time blow-up of L infinity weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 45-65.  doi: 10.4310/CMS.2010.v8.n1.a4.  Google Scholar

[6]

A. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in Rn, Comm. Math. Phys., 274 (2007), 717-735.  doi: 10.1007/s00220-007-0288-1.  Google Scholar

[7]

A. L. Bertozzi, T. Laurent and J. Rosado, Lp theory for the multidimensional aggregation equation, Commun. Pure Appl. Math., 64 (2011), 45-83.  doi: 10.1002/cpa.20334.  Google Scholar

[8]

A. Blanchet, J. A. Carrillo and P. Laurencot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Cal. Var. Partial Differential Equations, 35 (2009), 133-168.  doi: 10.1007/s00526-008-0200-7.  Google Scholar

[9]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 2006 (2006), No. 44, 32 pp.  Google Scholar

[10]

M. Bodnar and J. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differ. Equations, 222 (2006), 341-380.  doi: 10.1016/j.jde.2005.07.025.  Google Scholar

[11]

S. Boi, V. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens, Nonlinear Anal, Real World Appl., 1 (2000), 163-176.  doi: 10.1016/S0362-546X(99)00399-5.  Google Scholar

[12]

M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Networks and Heterogeneous Media, 3 (2008), 749-785.  doi: 10.3934/nhm.2008.3.749.  Google Scholar

[13]

M. Burger, M. Di Francesco and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Commun. Math. Sci., 11 (2012), 709-738.  doi: 10.4310/CMS.2013.v11.n3.a3.  Google Scholar

[14]

M. Burger, R. C. Fetecau and Y. Huang, Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13 (2014), 397-424.  doi: 10.1137/130923786.  Google Scholar

[15]

V. Calvez, J. A. Carrillo and F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Analysis TMA, 159 (2017), 85-128.  doi: 10.1016/j.na.2017.03.008.  Google Scholar

[16]

J. A. Carrillo, D. Castorina and B. Volzone, Ground states for diffusion dominated free energies with logarithmic interaction, SIAM J. Math. Anal., 47 (2015), 1-25.  doi: 10.1137/140951588.  Google Scholar

[17]

J. A. Carrillo, A. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a.  Google Scholar

[18]

J. A. Carrillo, M. Gualdani and G. Toscani, Finite speed of propagation in porous media by mass transportation methods, Comptes Rendus Mathematique, 338 (2004), 815-818.  doi: 10.1016/j.crma.2004.03.025.  Google Scholar

[19]

J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19 (2003), 971-1018.  doi: 10.4171/RMI/376.  Google Scholar

[20]

J. A. Carrillo and G. Toscani, Wasserstein metric and large-time asymptotics of nonlinear diffusion equations, New Trends in Mathematical Physics, (2004), 234-244.   Google Scholar

[21]

R. Choksi, R. C. Fetecau and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 32 (2015), 1283-1305.  doi: 10.1016/j.anihpc.2014.09.004.  Google Scholar

[22]

K. Fellner and G. Raoul, Stability of stationary states of non-local interaction equations, Mathematical and Computer Modelling, 53 (2011), 1436-1450.  doi: 10.1016/j.mcm.2010.03.021.  Google Scholar

[23]

L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for onedimensional filtration equations, SIAM Journal on Numerical Analysis, 43 (2006), 2590-2606.  doi: 10.1137/040608672.  Google Scholar

[24]

S. Gottlieb, C. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.  Google Scholar

[25]

D. Helbing, I. J. Farkas, P. Molnar and T. Vicsek, Simulation of pedestrian crowds in normal and evacuation situations, Pedestrian and Evacuation Dynamics, 21 (2002), 21-58.   Google Scholar

[26]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[27]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.  Google Scholar

[28]

G. Kaib, Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential, SIAM J. Math. Anal., 49 (2017), 272-296.  doi: 10.1137/16M1072450.  Google Scholar

[29]

I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2012), 568-602.  doi: 10.1137/110823584.  Google Scholar

[30]

H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Archive for rational mechanics and analysis, 172 (2004), 407-428.  doi: 10.1007/s00205-004-0307-8.  Google Scholar

[31]

E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684.  Google Scholar

[32]

P. L. Lions, The concentration-compactness principle in calculus of variations. The locally compact case, part 1, Ann. Inst. Henri Poincar e, Anal. Nonlin., 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[33]

R. McCann, A convexity principle for interacting gases, Advances in mathematics, 128 (1997), 153-179.  doi: 10.1006/aima.1997.1634.  Google Scholar

[34]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.  Google Scholar

[35]

D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: from individuals to populations, Journal of Mathematical Biology, 50 (2005), 49-66.  doi: 10.1007/s00285-004-0279-1.  Google Scholar

[36]

G. Russo, Deterministic diffusion of particles, Communications on Pure and Applied Mathematics, 43 (1990), 697-733.  doi: 10.1002/cpa.3160430602.  Google Scholar

[37]

L. Scardia, R. Peerlings, M. Geers and M. A. Peletier, Mechanics of dislocation pile-ups: A unification of scaling regimes, Journal of the Mechanics and Physics of Solids, 70 (2014), 42-61.  doi: 10.1016/j.jmps.2014.04.014.  Google Scholar

[38]

K. Sznajd-Weron and J. Sznajd. Opinion evolution in closed community, Opinion evolution in closed community, Int. J. Mod. Phys. C, 11 (2000), 1157-1166.   Google Scholar

[39]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bulletin of Mathematical Biology, 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[40]

G. Toscani, Kinetic and hydrodynamic models of nearly elastic granular flows, Monatsh. Math., 142 (2004), 179-192.  doi: 10.1007/s00605-004-0241-8.  Google Scholar

[41]

J. L. Vazquez, The Porous Medium Equation, Oxford Mathematical Monographs, 2007.  Google Scholar

[42]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, AMS, Providence, RI, 2003. doi: 10.1007/b12016.  Google Scholar

Figure 1.  Two different asymptotic behaviors of the solution depending on the initial datum
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Figure 4.  Growing of the second moment of the solution to 3 in time in the regime $1/3<\varepsilon<\|G\|_{L^1}$. $\varepsilon = 0.5, $ $G(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}$, and initial datum 40 with $\delta = 0.05.$
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Figure 2.  Convergence of the solution to the steady state in the aggregation-dominated regime $0<\varepsilon<\|G\|_{L^1}$. $\varepsilon = 0.002, $ $G(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}.$ (a) $\rho_{0}(x) = \frac{93}{8}(1-\frac{961}{4}x^2), $ (b) $\rho_{0}(x) = \frac{21}{8}(1-\frac{49}{4}x^2).$
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Figure 3.  Decay of the solution to zero in the diffusion-dominated regime $\varepsilon\geq\|G\|_{L^{1}}$. $\varepsilon = 2, $ $G(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}, $ $\rho_{0}(x) = \frac{21}{8}(1-\frac{49}{4}x^2)$.
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Figure 5.  Convergence of the solution to the steady state in the aggregation-dominated regime $0<\varepsilon<\|G\|_{L^{1}}$. $\varepsilon = 0.5, $ $G(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}.$ (a) $\rho_{0}(x) = \frac{9}{8}(1-\frac{9}{4}x^2), $ (b) $\rho_{0}(x) = \frac{105}{400}(1-\frac{1225}{10000}x^2).$
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