Mathematical challenges in the theory of chemotaxis
Abstract
We consider the simplest parabolic-elliptic model of chemotaxis in the whole space and in several space dimensions. Criteria either for the existence of radial global-in-time solutions or their blowup in terms of suitable Morrey spaces norms are discussed.
This is an extended version of the lecture presented at the University of Silesia on January 12, 2018, commemorating Professor Andrzej Lasota - great scholar, master of fine mathematics and applications to real world.
Keywords
chemotaxis; blowup of solutions; global existence of solutions
References
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5. Biler P., The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math. 114 (1995), 181–205.
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7. Biler P., Growth and accretion of mass in an astrophysical model, Appl. Math. (Warsaw) 23 (1995), 179–189.
8. Biler P., Local and global solvability of parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998), 715–743.
9. Biler P., Radially symmetric solutions of a chemotaxis model in the plane – the supercritical case, in: Rencławowicz J., Zajączkowski W.M. (eds.), Parabolic and Navier-Stokes Equations. Part 1, Banach Center Publications, 81, Polish Acad. Sci. Inst. Math., Warsaw, 2008, pp. 31–42.
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11. Biler P., Solvability for nonlinear heat equations with fractional diffusion. In preparation.
12. Biler P., Brandolese L., Global existence versus blow up for some models of interacting particles, Colloq. Math. 106 (2006), 293–303.
13. Biler P., Brandolese L., On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modelling chemotaxis, Studia Math. 193 (2009), 241–261.
14. Biler P., Cieślak T., Karch G., Zienkiewicz J., Local criteria for blowup in twodimensional chemotaxis models, Discrete Contin. Dyn. Syst. 37 (2017), 1841–1856.
15. Biler P., Corrias L., Dolbeault J., Large mass self-similar solutions of the parabolicparabolic Keller-Segel model of chemotaxis, J. Math. Biol. 63 (2011), 1–32.
16. Biler P., Dolbeault J., Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems, Ann. Henri Poincaré 1 (2000), 461–472.
17. Biler P., Guerra I., Karch G., Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane, Commun. Pure Appl. Anal. 14 (2015), 2117–2126.
18. Biler P., Hilhorst D., Nadzieja T., Existence and nonexistence of solutions for a model of gravitational interaction of particles. II, Colloq. Math. 67 (1994), 297–308.
19. Biler P., Karch G., Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ. 10 (2010), 247–262.
20. Biler P., Karch G., Solutions of fractional chemotaxis models. In preparation.
21. Biler P., Karch G., Laurençot Ph., Nadzieja T., The 8π-problem for radially symmetric solutions of a chemotaxis model in a disc, Topol. Methods Nonlinear Anal. 27 (2006), 133–147.
22. Biler P., Karch G., Laurençot Ph., Nadzieja T., The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Methods Appl. Sci. 29 (2006), 1563–1583.
23. Biler P., Karch G., Pilarczyk D., Global radial solutions in classical Keller-Segel chemotaxis model. In preparation.
24. Biler P., Karch G., Zienkiewicz J., Optimal criteria for blowup of radial and N-symmetric solutions of chemotaxis systems, Nonlinearity 28 (2015), 4369–4387.
25. Biler P., Karch G., Zienkiewicz J., Morrey spaces norms and criteria for blowup in chemotaxis models, Netw. Heterog. Media 11 (2016), 239–250.
26. Biler P., Karch G., Zienkiewicz J., Large global-in-time solutions to a nonlocal model of chemotaxis, Adv. Math. 330 (2018), 834–875.
27. Biler P., Nadzieja T., Existence and nonexistence of solutions for a model of gravitational interaction of particles. I, Colloq. Math. 66 (1994), 319–334.
28. Biler P., Zienkiewicz J., Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data, Bull. Pol. Acad. Sci. Math. 63 (2015), 41–51.
29. Biler P., Zienkiewicz J., Blowing up radial solutions in the minimal Keller-Segel chemotaxis model. In preparation.
30. Blanchet A., Carlen E.A., Carrillo J.A., Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, J. Funct. Anal. 262 (2012), 2142–2230.
31. Blanchet A., Carrillo J.A., Masmoudi N., Infinite time aggregation for the critical Patlak-Keller-Segel model in $R^2$, Comm. Pure Appl. Math. 61 (2008), 1449–1481.
32. Blanchet A., Dolbeault J., Perthame B., Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 2006, no. 44, 32 pp.
33. Brenner M.P., Constantin P., Kadanoff L.P., Schenkel A., Venkataramani S.C., Diffusion, attraction and collapse, Nonlinearity 12 (1999), 1071–1098.
34. Chandrasekhar S., Principles of Stellar Dynamics, University of Chicago Press, Chicago, 1942.
35. Chavanis P.H., Sommeria J., Robert R., Statistical mechanics of two-dimensional vortices and collisionless stellar systems, The Astrophys. Journal 471 (1996), 385–399.
36. Corrias L., Perthame B., Zaag H., Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math. 72 (2004), 1–28.
37. Debye P., Hückel E., Zur Theorie der Electrolyte. II, Phys. Zft. 24 (1923), 305–325.
38. Fujita H., On the blowing up of solutions of the Cauchy problem for $u_t=Δu+u^{1+α}$, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124.
39. Giga Y., Miyakawa T., Navier-Stokes flow in $R^3$ with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations 14 (1989), 577–618.
40. Giga Y., Mizoguchi N., Senba T., Asymptotic behavior of type I blowup solutions to a parabolic-elliptic system of drift-diffusion type, Arch. Ration. Mech. Anal. 201 (2011), 549–573.
41. Herrero M.A., Velázquez J.J.L., Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996), 583–623.
42. Herrero M.A., Velázquez J.J.L., Chemotactic collapse for the Keller-Segel model, J. Math. Biol. 35 (1996), 177–194.
43. Hillen T., Painter K.J., A users guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009), 183–217.
44. Horstmann D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. 105 (2003), 103–165.
45. Horstmann D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch. Math.-Verein. 106 (2004), 51–69.
46. Iwabuchi T., Global well-posedness for Keller-Segel system in Besov type spaces, J. Math. Anal. Appl. 379 (2011), 930–948.
47. Jäger W., Luckhaus S., On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819–824.
48. Karch G., Scaling in nonlinear parabolic equations, J. Math. Anal. Appl. 234 (1999), 534–558.
49. Kavallaris N.I., Souplet Ph., Grow-up rate and refined asymptotics for a twodimensional Patlak-Keller-Segel model in a disk, SIAM J. Math. Anal. 40 (2008/09), 1852–1881.
50. Keller E.F., Segel L.A., Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415.
51. Kozono H., Sugiyama Y., The Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}(R^n)$ and its application to self-similar solutions, Indiana Univ. Math. J. 57 (2008), 1467–1500.
52. Kurokiba M., Ogawa T., Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential Integral Equations 16 (2003), 427–452.
53. Lemarié-Rieusset P.-G., Small data in an optimal Banach space for the parabolicparabolic and parabolic-elliptic Keller-Segel equations in the whole space, Adv. Differential Equations 18 (2013), 1189–1208.
54. Mizoguchi N., Senba T., Type-II blowup of solutions to an elliptic-parabolic system, Adv. Math. Sci. Appl. 17 (2007), 505–545.
55. Mizoguchi N., Senba T., A sufficient condition for type I blowup in a parabolic-elliptic system, J. Differential Equations 250 (2011), 182–203.
56. Nagai T., Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl. 6 (2001), 37–55.
57. Naito Y., Senba T., Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains, Discrete Contin. Dyn. Syst. 32 (2012), 3691–3713.
58. Naito Y., Senba T., Bounded and unbounded oscillating solutions to a parabolic-elliptic system in two dimensional space, Commun. Pure Appl. Anal. 12 (2013), 1861–1880.
59. Pilarczyk D., Asymptotic stability of singular solution to nonlinear heat equation, Discrete Contin. Dyn. Syst. 25 (2009), 991–1001.
60. Pilarczyk D., Self-similar asymptotics of solutions to heat equation with inverse square potential, J. Evol. Equ. 13 (2013), 69–87.
61. Quittner P., Souplet Ph., Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2007.
62. Senba T., Blowup behavior of radial solutions to Jäger-Luckhaus system in high dimensional domains, Funkcial. Ekvac. 48 (2005), 247–271.
63. Souplet Ph., Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in $R^n$, J. Funct. Anal. 272 (2017), 2005–2037.
64. Suzuki T., Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser, Boston, 2005.
65. Taylor M.E., Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), 1407–1456.
2. Balinsky A.A., Evans W.D., Lewis R.T., The Analysis and Geometry of Hardy’s Inequality, Universitext, Springer, Cham, 2015.
3. Bedrossian J., Masmoudi N., Existence, uniqueness and Lipschitz dependence for Patlak-Keller-Segel and Navier-Stokes in $R^2$ with measure-valued initial data, Arch. Ration. Mech. Anal. 214 (2014), 717–801.
4. Bellomo N., Bellouquid A., Tao Y., Winkler M., Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015), 1663–1763.
5. Biler P., The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math. 114 (1995), 181–205.
6. Biler P., Existence and nonexistence of solutions for a model of gravitational interaction of particles. III, Colloq. Math. 68 (1995), 229–239.
7. Biler P., Growth and accretion of mass in an astrophysical model, Appl. Math. (Warsaw) 23 (1995), 179–189.
8. Biler P., Local and global solvability of parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998), 715–743.
9. Biler P., Radially symmetric solutions of a chemotaxis model in the plane – the supercritical case, in: Rencławowicz J., Zajączkowski W.M. (eds.), Parabolic and Navier-Stokes Equations. Part 1, Banach Center Publications, 81, Polish Acad. Sci. Inst. Math., Warsaw, 2008, pp. 31–42.
10. Biler P., Blowup versus global in time existence of solutions for nonlinear heat equations, Topol. Methods Nonlinear Anal. To appear. Available at arXiv:1705.03931v2.
11. Biler P., Solvability for nonlinear heat equations with fractional diffusion. In preparation.
12. Biler P., Brandolese L., Global existence versus blow up for some models of interacting particles, Colloq. Math. 106 (2006), 293–303.
13. Biler P., Brandolese L., On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modelling chemotaxis, Studia Math. 193 (2009), 241–261.
14. Biler P., Cieślak T., Karch G., Zienkiewicz J., Local criteria for blowup in twodimensional chemotaxis models, Discrete Contin. Dyn. Syst. 37 (2017), 1841–1856.
15. Biler P., Corrias L., Dolbeault J., Large mass self-similar solutions of the parabolicparabolic Keller-Segel model of chemotaxis, J. Math. Biol. 63 (2011), 1–32.
16. Biler P., Dolbeault J., Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems, Ann. Henri Poincaré 1 (2000), 461–472.
17. Biler P., Guerra I., Karch G., Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane, Commun. Pure Appl. Anal. 14 (2015), 2117–2126.
18. Biler P., Hilhorst D., Nadzieja T., Existence and nonexistence of solutions for a model of gravitational interaction of particles. II, Colloq. Math. 67 (1994), 297–308.
19. Biler P., Karch G., Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ. 10 (2010), 247–262.
20. Biler P., Karch G., Solutions of fractional chemotaxis models. In preparation.
21. Biler P., Karch G., Laurençot Ph., Nadzieja T., The 8π-problem for radially symmetric solutions of a chemotaxis model in a disc, Topol. Methods Nonlinear Anal. 27 (2006), 133–147.
22. Biler P., Karch G., Laurençot Ph., Nadzieja T., The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Methods Appl. Sci. 29 (2006), 1563–1583.
23. Biler P., Karch G., Pilarczyk D., Global radial solutions in classical Keller-Segel chemotaxis model. In preparation.
24. Biler P., Karch G., Zienkiewicz J., Optimal criteria for blowup of radial and N-symmetric solutions of chemotaxis systems, Nonlinearity 28 (2015), 4369–4387.
25. Biler P., Karch G., Zienkiewicz J., Morrey spaces norms and criteria for blowup in chemotaxis models, Netw. Heterog. Media 11 (2016), 239–250.
26. Biler P., Karch G., Zienkiewicz J., Large global-in-time solutions to a nonlocal model of chemotaxis, Adv. Math. 330 (2018), 834–875.
27. Biler P., Nadzieja T., Existence and nonexistence of solutions for a model of gravitational interaction of particles. I, Colloq. Math. 66 (1994), 319–334.
28. Biler P., Zienkiewicz J., Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data, Bull. Pol. Acad. Sci. Math. 63 (2015), 41–51.
29. Biler P., Zienkiewicz J., Blowing up radial solutions in the minimal Keller-Segel chemotaxis model. In preparation.
30. Blanchet A., Carlen E.A., Carrillo J.A., Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, J. Funct. Anal. 262 (2012), 2142–2230.
31. Blanchet A., Carrillo J.A., Masmoudi N., Infinite time aggregation for the critical Patlak-Keller-Segel model in $R^2$, Comm. Pure Appl. Math. 61 (2008), 1449–1481.
32. Blanchet A., Dolbeault J., Perthame B., Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 2006, no. 44, 32 pp.
33. Brenner M.P., Constantin P., Kadanoff L.P., Schenkel A., Venkataramani S.C., Diffusion, attraction and collapse, Nonlinearity 12 (1999), 1071–1098.
34. Chandrasekhar S., Principles of Stellar Dynamics, University of Chicago Press, Chicago, 1942.
35. Chavanis P.H., Sommeria J., Robert R., Statistical mechanics of two-dimensional vortices and collisionless stellar systems, The Astrophys. Journal 471 (1996), 385–399.
36. Corrias L., Perthame B., Zaag H., Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math. 72 (2004), 1–28.
37. Debye P., Hückel E., Zur Theorie der Electrolyte. II, Phys. Zft. 24 (1923), 305–325.
38. Fujita H., On the blowing up of solutions of the Cauchy problem for $u_t=Δu+u^{1+α}$, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124.
39. Giga Y., Miyakawa T., Navier-Stokes flow in $R^3$ with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations 14 (1989), 577–618.
40. Giga Y., Mizoguchi N., Senba T., Asymptotic behavior of type I blowup solutions to a parabolic-elliptic system of drift-diffusion type, Arch. Ration. Mech. Anal. 201 (2011), 549–573.
41. Herrero M.A., Velázquez J.J.L., Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996), 583–623.
42. Herrero M.A., Velázquez J.J.L., Chemotactic collapse for the Keller-Segel model, J. Math. Biol. 35 (1996), 177–194.
43. Hillen T., Painter K.J., A users guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009), 183–217.
44. Horstmann D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. 105 (2003), 103–165.
45. Horstmann D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch. Math.-Verein. 106 (2004), 51–69.
46. Iwabuchi T., Global well-posedness for Keller-Segel system in Besov type spaces, J. Math. Anal. Appl. 379 (2011), 930–948.
47. Jäger W., Luckhaus S., On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819–824.
48. Karch G., Scaling in nonlinear parabolic equations, J. Math. Anal. Appl. 234 (1999), 534–558.
49. Kavallaris N.I., Souplet Ph., Grow-up rate and refined asymptotics for a twodimensional Patlak-Keller-Segel model in a disk, SIAM J. Math. Anal. 40 (2008/09), 1852–1881.
50. Keller E.F., Segel L.A., Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415.
51. Kozono H., Sugiyama Y., The Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}(R^n)$ and its application to self-similar solutions, Indiana Univ. Math. J. 57 (2008), 1467–1500.
52. Kurokiba M., Ogawa T., Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential Integral Equations 16 (2003), 427–452.
53. Lemarié-Rieusset P.-G., Small data in an optimal Banach space for the parabolicparabolic and parabolic-elliptic Keller-Segel equations in the whole space, Adv. Differential Equations 18 (2013), 1189–1208.
54. Mizoguchi N., Senba T., Type-II blowup of solutions to an elliptic-parabolic system, Adv. Math. Sci. Appl. 17 (2007), 505–545.
55. Mizoguchi N., Senba T., A sufficient condition for type I blowup in a parabolic-elliptic system, J. Differential Equations 250 (2011), 182–203.
56. Nagai T., Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl. 6 (2001), 37–55.
57. Naito Y., Senba T., Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains, Discrete Contin. Dyn. Syst. 32 (2012), 3691–3713.
58. Naito Y., Senba T., Bounded and unbounded oscillating solutions to a parabolic-elliptic system in two dimensional space, Commun. Pure Appl. Anal. 12 (2013), 1861–1880.
59. Pilarczyk D., Asymptotic stability of singular solution to nonlinear heat equation, Discrete Contin. Dyn. Syst. 25 (2009), 991–1001.
60. Pilarczyk D., Self-similar asymptotics of solutions to heat equation with inverse square potential, J. Evol. Equ. 13 (2013), 69–87.
61. Quittner P., Souplet Ph., Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2007.
62. Senba T., Blowup behavior of radial solutions to Jäger-Luckhaus system in high dimensional domains, Funkcial. Ekvac. 48 (2005), 247–271.
63. Souplet Ph., Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in $R^n$, J. Funct. Anal. 272 (2017), 2005–2037.
64. Suzuki T., Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser, Boston, 2005.
65. Taylor M.E., Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), 1407–1456.
BilerP. (2018). Mathematical challenges in the theory of chemotaxis. Annales Mathematicae Silesianae, 32, 43-63. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13912
Piotr Biler
Piotr.Biler@math.uni.wroc.pl
Instytut Matematyki, Uniwersytet Wrocławski Poland
Instytut Matematyki, Uniwersytet Wrocławski Poland
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