# American Institute of Mathematical Sciences

• Previous Article
Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains
• DCDS Home
• This Issue
• Next Article
Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems
January  2014, 34(1): 99-119. doi: 10.3934/dcds.2014.34.99

## Analysis of a degenerate biofilm model with a nutrient taxis term

 1 Department of Mathematics and Statistics, University of Guelph, Guelph, On, N1G 2W1, Canada 2 Helmholtz Zentrum München, Institute of Computational Biology, Ingolstädter Landstrasse1, D-85764 Neuherberg,, Germany 3 Inst. Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland 4 Centre for Mathematical Sciences, Technical University of Munich, Boltzmannstr. 3, 85748 Garching, Germany

Received  October 2012 Revised  March 2013 Published  June 2013

We introduce and analyze a prototype model for chemotactic effects in biofilm formation. The model is a system of quasilinear parabolic equations into which two thresholds are built in. One occurs at zero cell density level, the second one is related to the maximal density which the cells cannot exceed. Accordingly, both diffusion and taxis terms have degenerate or singular parts. This model extends a previously introduced degenerate biofilm model by combining it with a chemotaxis equation. We give conditions for existence and uniqueness of weak solutions and illustrate the model behavior in numerical simulations.
Citation: Hermann J. Eberl, Messoud A. Efendiev, Dariusz Wrzosek, Anna Zhigun. Analysis of a degenerate biofilm model with a nutrient taxis term. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 99-119. doi: 10.3934/dcds.2014.34.99
##### References:
 [1] R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis'' (eds. H. Triebel and H. J. Schmeisser), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126.  Google Scholar [3] D. G. Aronson, The porous medium equation, in "Nonlinear Diffusion Problems" (Montecatini Terme, 1985), Lecture Notes in Mathematics, 1224, Springer, Berlin, 1986. doi: 10.1007/BFb0072687.  Google Scholar [4] R. Denk, M. Hieber and J. Prüss, Optimal $L^p-L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.  Google Scholar [5] J. Dockery and I. Klapper, Finger formation in biofilm layers,, SIAM J. Appl. Math., 62 (): 853.  doi: 10.1137/S0036139900371709.  Google Scholar [6] H. J. Eberl and L. Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial ecology, in "Proceedings of the Sixth Mississippi State-UBA Conference on Differential Equations and Computational Simulations," El. J. Diff Equs Conf., 15, Southwest Texas State Univ., San Marcos, TX, (2007), 77-96.  Google Scholar [7] H. J. Eberl, D. F. Parker and M. C. M. Van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Med., 3 (2001), 161-176. doi: 10.1080/10273660108833072.  Google Scholar [8] M. A. Efendiev and S. Sonner, Verifying mathematical models with diffusion, transport and interaction, in "Current Advances in Nonlinear Analysis and Related Topics," GAKUTO Internat. Ser. Math. Sci. Appl., 32, Gakkōtosho, Tokyo, (2010), 41-67.  Google Scholar [9] M. A. Efendiev, R. Lasser and S. Sonner, Necessary and sufficient conditions for an infinite system of parabolic equations preserving the positive cone, Int. J. Biomath. & Biostats, 1 (2010), 47-52. Google Scholar [10] M. A. Efendiev, S. V. Zelik and H. J. Eberl, Existence and long time behaviour of a biofilm model, Comm. Pure and Appl. Analysis, 8 (2009), 509-531. doi: 10.3934/cpaa.2009.8.509.  Google Scholar [11] M. A. Efendiev and T. Senba, On the well-posedness of a class of PDEs including porous medium and chemotaxis effect, Adv. Differ. Equ., 16 (2011), 937-954.  Google Scholar [12] M. A. Efendiev and A. Zhigun, On a 'balance' condition for a class of PDEs including porous medium and chemotaxis effect: Non-autonomous case, Adv. Math. Sci. Appl., 21 (2011), 285-304.  Google Scholar [13] H. Fgaier, B. Feher, R. C. McKellar and H. J. Eberl, Predictive modeling of siderphore production by Pseudomonas fluorscens under iron limitation, J. Theor. Biol., 251 (2008), 348-362. doi: 10.1016/j.jtbi.2007.11.026.  Google Scholar [14] H. Fgaier and H. J. Eberl, Parameter identification and quantitative comparison of differential equations that describe physiological adaptation of a bacterial population under iron limitation, Disc. Cont. Dyn. Sys. Suppl., (2009), 230-239.  Google Scholar [15] M. R. Frederick, C. Kuttler, B. A. Hense, J. Müller and H. J. Eberl, A mathematical model of quorum sensing in patchy biofilm communities with slow background flow, Can. Appl. Math. Quart., 18 (2011), 267-298.  Google Scholar [16] R. Kowalczyk, A. Gamba and L. Preciosi, On the stability of homogeneous solutions to some aggregation models, Discrete Contin. Dynam. Systems-Series B, 4 (2004), 203-220.  Google Scholar [17] V. Gordon, Personal email communication, July 4, (2011). Google Scholar [18] Ph. Laurençot and D. Wrzosek, A chemotaxis model with threshold density and degenerate diffusion, in "Nonlinear Elliptic and Parabolic Problems," Progr. Nonlinear Differential Equations Appl., 64, Birkhäuser, Basel, (2005), 273-290. doi: 10.1007/3-7643-7385-7_16.  Google Scholar [19] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar [20] P. M. Lushnikov, N. Chen and M. Alber, Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact, Phys. Rev. E., 78 (2008), 061904, 12 pp. doi: 10.1103/PhysRevE.78.061904.  Google Scholar [21] M. A. Molina, J.-L. Ramos and M. Espinosa-Urgel, Plant-associated biofilms, Rev. Environ. Sci. Biotech., 2 (2003), 99-108. doi: 10.1023/B:RESB.0000040458.35960.25.  Google Scholar [22] N. Muhammad and H. J. Eberl, Model parameter uncertainties in a dual-species biofilm competition model affect ecological output parameters much stronger than morphological ones, Math. Biosci., 233 (2011), 1-18. doi: 10.1016/j.mbs.2011.05.006.  Google Scholar [23] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart., 10 (2002), 501-543.  Google Scholar [24] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [25] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar [26] R. Singh, D. Paul and R. K. Jain, Biofilms: Implications in bioremediation, TRENDS in Microbiol., 14 (2006), 389-397. doi: 10.1016/j.tim.2006.07.001.  Google Scholar [27] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258, Springer-Verlag, New York, 1994.  Google Scholar [28] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Diff. Equ., 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003.  Google Scholar [29] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar [30] Z. A Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297. doi: 10.1088/0951-7715/24/12/001.  Google Scholar [31] Z. A Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525. doi: 10.1137/110853972.  Google Scholar [32] O. Wanner, H. Eberl, E. Morgenroth, D. Noguera, C. Picioreanu, B. Rittmann and M. van Loosdrecht, "Mathematical Modeling of Biofilms," IWA Publishing, London, 2006. Google Scholar [33] D. Wrzosek, Chemotaxis models with a threshold cell density, in "Parabolic and Navier-Stokes Equations. Part 2," Banach Center Publ., 81, Polish Acad. Sci. Inst. Math., Warsaw, (2008), 553-566. doi: 10.4064/bc81-0-35.  Google Scholar [34] D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion, Nonl. Analysis TMA, 73 (2010), 338-349. doi: 10.1016/j.na.2010.02.047.  Google Scholar [35] D. Wrzosek, Volume filling effect in modelling chemotaxis, Math. Model. Nat. Phenom., 5 (2010), 123-147. doi: 10.1051/mmnp/20105106.  Google Scholar [36] A. Yagi, "Abstract Parabolic Evolution Equations and their Applications," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar [37] P. M. Yaryura, M. León, O. S. Correa, N. L. Kerber, N. L. Pucheu and A. F. García, Assessment of the role of chemotaxis and biofilm formation as requirement for colonization of roots and seed of soybean plants by Bacillus amyloliqufaciens BNM339, Curr. Microbiol., 56 (2008), 625-632. doi: 10.1007/s00284-008-9137-5.  Google Scholar

show all references

##### References:
 [1] R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis'' (eds. H. Triebel and H. J. Schmeisser), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126.  Google Scholar [3] D. G. Aronson, The porous medium equation, in "Nonlinear Diffusion Problems" (Montecatini Terme, 1985), Lecture Notes in Mathematics, 1224, Springer, Berlin, 1986. doi: 10.1007/BFb0072687.  Google Scholar [4] R. Denk, M. Hieber and J. Prüss, Optimal $L^p-L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.  Google Scholar [5] J. Dockery and I. Klapper, Finger formation in biofilm layers,, SIAM J. Appl. Math., 62 (): 853.  doi: 10.1137/S0036139900371709.  Google Scholar [6] H. J. Eberl and L. Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial ecology, in "Proceedings of the Sixth Mississippi State-UBA Conference on Differential Equations and Computational Simulations," El. J. Diff Equs Conf., 15, Southwest Texas State Univ., San Marcos, TX, (2007), 77-96.  Google Scholar [7] H. J. Eberl, D. F. Parker and M. C. M. Van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Med., 3 (2001), 161-176. doi: 10.1080/10273660108833072.  Google Scholar [8] M. A. Efendiev and S. Sonner, Verifying mathematical models with diffusion, transport and interaction, in "Current Advances in Nonlinear Analysis and Related Topics," GAKUTO Internat. Ser. Math. Sci. Appl., 32, Gakkōtosho, Tokyo, (2010), 41-67.  Google Scholar [9] M. A. Efendiev, R. Lasser and S. Sonner, Necessary and sufficient conditions for an infinite system of parabolic equations preserving the positive cone, Int. J. Biomath. & Biostats, 1 (2010), 47-52. Google Scholar [10] M. A. Efendiev, S. V. Zelik and H. J. Eberl, Existence and long time behaviour of a biofilm model, Comm. Pure and Appl. Analysis, 8 (2009), 509-531. doi: 10.3934/cpaa.2009.8.509.  Google Scholar [11] M. A. Efendiev and T. Senba, On the well-posedness of a class of PDEs including porous medium and chemotaxis effect, Adv. Differ. Equ., 16 (2011), 937-954.  Google Scholar [12] M. A. Efendiev and A. Zhigun, On a 'balance' condition for a class of PDEs including porous medium and chemotaxis effect: Non-autonomous case, Adv. Math. Sci. Appl., 21 (2011), 285-304.  Google Scholar [13] H. Fgaier, B. Feher, R. C. McKellar and H. J. Eberl, Predictive modeling of siderphore production by Pseudomonas fluorscens under iron limitation, J. Theor. Biol., 251 (2008), 348-362. doi: 10.1016/j.jtbi.2007.11.026.  Google Scholar [14] H. Fgaier and H. J. Eberl, Parameter identification and quantitative comparison of differential equations that describe physiological adaptation of a bacterial population under iron limitation, Disc. Cont. Dyn. Sys. Suppl., (2009), 230-239.  Google Scholar [15] M. R. Frederick, C. Kuttler, B. A. Hense, J. Müller and H. J. Eberl, A mathematical model of quorum sensing in patchy biofilm communities with slow background flow, Can. Appl. Math. Quart., 18 (2011), 267-298.  Google Scholar [16] R. Kowalczyk, A. Gamba and L. Preciosi, On the stability of homogeneous solutions to some aggregation models, Discrete Contin. Dynam. Systems-Series B, 4 (2004), 203-220.  Google Scholar [17] V. Gordon, Personal email communication, July 4, (2011). Google Scholar [18] Ph. Laurençot and D. Wrzosek, A chemotaxis model with threshold density and degenerate diffusion, in "Nonlinear Elliptic and Parabolic Problems," Progr. Nonlinear Differential Equations Appl., 64, Birkhäuser, Basel, (2005), 273-290. doi: 10.1007/3-7643-7385-7_16.  Google Scholar [19] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar [20] P. M. Lushnikov, N. Chen and M. Alber, Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact, Phys. Rev. E., 78 (2008), 061904, 12 pp. doi: 10.1103/PhysRevE.78.061904.  Google Scholar [21] M. A. Molina, J.-L. Ramos and M. Espinosa-Urgel, Plant-associated biofilms, Rev. Environ. Sci. Biotech., 2 (2003), 99-108. doi: 10.1023/B:RESB.0000040458.35960.25.  Google Scholar [22] N. Muhammad and H. J. Eberl, Model parameter uncertainties in a dual-species biofilm competition model affect ecological output parameters much stronger than morphological ones, Math. Biosci., 233 (2011), 1-18. doi: 10.1016/j.mbs.2011.05.006.  Google Scholar [23] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart., 10 (2002), 501-543.  Google Scholar [24] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [25] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar [26] R. Singh, D. Paul and R. K. Jain, Biofilms: Implications in bioremediation, TRENDS in Microbiol., 14 (2006), 389-397. doi: 10.1016/j.tim.2006.07.001.  Google Scholar [27] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258, Springer-Verlag, New York, 1994.  Google Scholar [28] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Diff. Equ., 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003.  Google Scholar [29] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar [30] Z. A Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297. doi: 10.1088/0951-7715/24/12/001.  Google Scholar [31] Z. A Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525. doi: 10.1137/110853972.  Google Scholar [32] O. Wanner, H. Eberl, E. Morgenroth, D. Noguera, C. Picioreanu, B. Rittmann and M. van Loosdrecht, "Mathematical Modeling of Biofilms," IWA Publishing, London, 2006. Google Scholar [33] D. Wrzosek, Chemotaxis models with a threshold cell density, in "Parabolic and Navier-Stokes Equations. Part 2," Banach Center Publ., 81, Polish Acad. Sci. Inst. Math., Warsaw, (2008), 553-566. doi: 10.4064/bc81-0-35.  Google Scholar [34] D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion, Nonl. Analysis TMA, 73 (2010), 338-349. doi: 10.1016/j.na.2010.02.047.  Google Scholar [35] D. Wrzosek, Volume filling effect in modelling chemotaxis, Math. Model. Nat. Phenom., 5 (2010), 123-147. doi: 10.1051/mmnp/20105106.  Google Scholar [36] A. Yagi, "Abstract Parabolic Evolution Equations and their Applications," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar [37] P. M. Yaryura, M. León, O. S. Correa, N. L. Kerber, N. L. Pucheu and A. F. García, Assessment of the role of chemotaxis and biofilm formation as requirement for colonization of roots and seed of soybean plants by Bacillus amyloliqufaciens BNM339, Curr. Microbiol., 56 (2008), 625-632. doi: 10.1007/s00284-008-9137-5.  Google Scholar
 [1] Kin Ming Hui, Sunghoon Kim. Existence of Neumann and singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4859-4887. doi: 10.3934/dcds.2015.35.4859 [2] Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 [3] Kin Ming Hui, Jinwan Park. Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5473-5508. doi: 10.3934/dcds.2021085 [4] Langhao Zhou, Liangwei Wang, Chunhua Jin. Global solvability to a singular chemotaxis-consumption model with fast and slow diffusion and logistic source. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021122 [5] Marcel Freitag. The fast signal diffusion limit in nonlinear chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1109-1128. doi: 10.3934/dcdsb.2019211 [6] Shu-Yu Hsu. Super fast vanishing solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5383-5414. doi: 10.3934/dcds.2020232 [7] Jie Zhao. A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021193 [8] Gui-Qiang Chen, Kenneth Hvistendahl Karlsen. Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients. Communications on Pure & Applied Analysis, 2005, 4 (2) : 241-266. doi: 10.3934/cpaa.2005.4.241 [9] Messoud Efendiev, Anna Zhigun. On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 651-673. doi: 10.3934/dcds.2018028 [10] Felipe Wallison Chaves-Silva, Sergio Guerrero, Jean Pierre Puel. Controllability of fast diffusion coupled parabolic systems. Mathematical Control & Related Fields, 2014, 4 (4) : 465-479. doi: 10.3934/mcrf.2014.4.465 [11] Hassan Khassehkhan, Messoud A. Efendiev, Hermann J. Eberl. A degenerate diffusion-reaction model of an amensalistic biofilm control system: Existence and simulation of solutions. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 371-388. doi: 10.3934/dcdsb.2009.12.371 [12] Mehdi Badra, Kaushik Bal, Jacques Giacomoni. Existence results to a quasilinear and singular parabolic equation. Conference Publications, 2011, 2011 (Special) : 117-125. doi: 10.3934/proc.2011.2011.117 [13] Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069 [14] Gabriela Marinoschi. Well posedness of a time-difference scheme for a degenerate fast diffusion problem. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 435-454. doi: 10.3934/dcdsb.2010.13.435 [15] Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073 [16] H. T. Liu. Impulsive effects on the existence of solutions for a fast diffusion equation. Conference Publications, 2001, 2001 (Special) : 248-253. doi: 10.3934/proc.2001.2001.248 [17] Marek Fila, Juan-Luis Vázquez, Michael Winkler. A continuum of extinction rates for the fast diffusion equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1129-1147. doi: 10.3934/cpaa.2011.10.1129 [18] Marek Fila, Hannes Stuke. Special asymptotics for a critical fast diffusion equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 725-735. doi: 10.3934/dcdss.2014.7.725 [19] Chunlai Mu, Jun Zhou, Yuhuan Li. Fast rate of dead core for fast diffusion equation with strong absorption. Communications on Pure & Applied Analysis, 2010, 9 (2) : 397-411. doi: 10.3934/cpaa.2010.9.397 [20] Kin Ming Hui. Collasping behaviour of a singular diffusion equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2165-2185. doi: 10.3934/dcds.2012.32.2165

2020 Impact Factor: 1.392