[[1] Andreucci D., DiBenedetto E., On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), 363-441.]Search in Google Scholar
[[2] Balinsky A.A., Evans W.D., Lewis R.T., The Analysis and Geometry of Hardy’s Inequality, Universitext, Springer, Cham, 2015.10.1007/978-3-319-22870-9]Search in Google Scholar
[[3] Bedrossian J., Masmoudi N., Existence, uniqueness and Lipschitz dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with measure-valued initial data, Arch. Ration. Mech. Anal. 214 (2014), 717-801.10.1007/s00205-014-0796-z]Search in Google Scholar
[[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), 1663-1763.10.1142/S021820251550044X]Search in Google Scholar
[[5] Biler P., The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math. 114 (1995), 181-205.10.4064/sm-114-2-181-205]Search in Google Scholar
[[6] Biler P., Existence and nonexistence of solutions for a model of gravitational interaction of particles. III, Colloq. Math. 68 (1995), 229-239.10.4064/cm-68-2-229-239]Search in Google Scholar
[[7] Biler P., Growth and accretion of mass in an astrophysical model, Appl. Math. (Warsaw) 23 (1995), 179-189.10.4064/am-23-2-179-189]Search in Google Scholar
[[8] Biler P., Local and global solvability of parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998), 715-743.]Search in Google Scholar
[[9] Biler P., Radially symmetric solutions of a chemotaxis model in the plane - the supercritical case, in: Rencławowicz J., Zajaczkowski W.M. (eds.), Parabolic and Navier- Stokes Equations. Part 1, Banach Center Publications, 81, Polish Acad. Sci. Inst. Math., Warsaw, 2008, pp. 31-42.10.4064/bc81-0-2]Search in Google Scholar
[[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.]Search in Google Scholar
[[11] Biler P., Solvability for nonlinear heat equations with fractional diffusion. In preparation.]Search in Google Scholar
[[12] Biler P., Brandolese L., Global existence versus blow up for some models of interacting particles, Colloq. Math. 106 (2006), 293-303.10.4064/cm106-2-9]Search in Google Scholar
[[13] Biler P., Brandolese L., On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modelling chemotaxis, Studia Math. 193 (2009), 241-261.10.4064/sm193-3-2]Search in Google Scholar
[[14] Biler P., Cieslak T., Karch G., Zienkiewicz J., Local criteria for blowup in twodimensional chemotaxis models, Discrete Contin. Dyn. Syst. 37 (2017), 1841-1856.10.3934/dcds.2017077]Search in Google Scholar
[[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), 1-32.10.1007/s00285-010-0357-520730434]Open DOISearch in Google Scholar
[[16] Biler P., Dolbeault J., Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems, Ann. Henri Poincaré 1 (2000), 461-472.10.1007/s000230050003]Search in Google Scholar
[[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), 2117-2126.10.3934/cpaa.2015.14.2117]Search in Google Scholar
[[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), 297-308.10.4064/cm-67-2-297-308]Search in Google Scholar
[[19] Biler P., Karch G., Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ. 10 (2010), 247-262.10.1007/s00028-009-0048-0]Search in Google Scholar
[[20] Biler P., Karch G., Solutions of fractional chemotaxis models. In preparation.]Search in Google Scholar
[[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.]Search in Google Scholar
[[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), 1563-1583.10.1002/mma.743]Search in Google Scholar
[[23] Biler P., Karch G., Pilarczyk D., Global radial solutions in classical Keller-Segel chemotaxis model. In preparation.]Search in Google Scholar
[[24] Biler P., Karch G., Zienkiewicz J., Optimal criteria for blowup of radial and N-symmetric solutions of chemotaxis systems, Nonlinearity 28 (2015), 4369-4387.10.1088/0951-7715/28/12/4369]Search in Google Scholar
[[25] Biler P., Karch G., Zienkiewicz J., Morrey spaces norms and criteria for blowup in chemotaxis models, Netw. Heterog. Media 11 (2016), 239-250.10.3934/nhm.2016.11.239]Open DOISearch in Google Scholar
[[26] Biler P., Karch G., Zienkiewicz J., Large global-in-time solutions to a nonlocal model of chemotaxis, Adv. Math. 330 (2018), 834-875.10.1016/j.aim.2018.03.036]Search in Google Scholar
[[27] Biler P., Nadzieja T., Existence and nonexistence of solutions for a model of gravitational interaction of particles. I, Colloq. Math. 66 (1994), 319-334.10.4064/cm-66-2-319-334]Search in Google Scholar
[[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), 41-51.10.4064/ba63-1-6]Search in Google Scholar
[[29] Biler P., Zienkiewicz J., Blowing up radial solutions in the minimal Keller-Segel chemotaxis model. In preparation.]Search in Google Scholar
[[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), 2142-2230.10.1016/j.jfa.2011.12.012]Search in Google Scholar
[[31] Blanchet A., Carrillo J.A., Masmoudi N., Infinite time aggregation for the critical Patlak-Keller-Segel model in R2, Comm. Pure Appl. Math. 61 (2008), 1449-1481.10.1002/cpa.20225]Search in Google Scholar
[[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.]Search in Google Scholar
[[33] Brenner M.P., Constantin P., Kadanoff L.P., Schenkel A., Venkataramani S.C., Diffusion, attraction and collapse, Nonlinearity 12 (1999), 1071-1098.10.1088/0951-7715/12/4/320]Open DOISearch in Google Scholar
[[34] Chandrasekhar S., Principles of Stellar Dynamics, University of Chicago Press, Chicago, 1942.]Search in Google Scholar
[[35] Chavanis P.H., Sommeria J., Robert R., Statistical mechanics of two-dimensional vortices and collisionless stellar systems, The Astrophys. Journal 471 (1996), 385-399.10.1086/177977]Search in Google Scholar
[[36] Corrias L., Perthame B., Zaag H., Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math. 72 (2004), 1-28.10.1007/s00032-003-0026-x]Search in Google Scholar
[[37] Debye P., Hückel E., Zur Theorie der Electrolyte. II, Phys. Zft. 24 (1923), 305-325.]Search in Google Scholar
[[38] Fujita H., On the blowing up of solutions of the Cauchy problem for ut = Δu+u1+ α, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109-124.]Search in Google Scholar
[[39] Giga Y., Miyakawa T., Navier-Stokes flow in R3 with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations 14 (1989), 577-618.10.1080/03605308908820621]Open DOISearch in Google Scholar
[[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), 549-573.10.1007/s00205-010-0394-7]Search in Google Scholar
[[41] Herrero M.A., Velázquez J.J.L., Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996), 583-623.10.1007/BF01445268]Search in Google Scholar
[[42] Herrero M.A., Velázquez J.J.L., Chemotactic collapse for the Keller-Segel model, J. Math. Biol. 35 (1996), 177-194.10.1007/s0028500500499053436]Open DOISearch in Google Scholar
[[43] Hillen T., Painter K.J., A users guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009), 183-217.10.1007/s00285-008-0201-3]Open DOISearch in Google Scholar
[[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.]Search in Google Scholar
[[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.]Search in Google Scholar
[[46] Iwabuchi T., Global well-posedness for Keller-Segel system in Besov type spaces, J. Math. Anal. Appl. 379 (2011), 930-948.10.1016/j.jmaa.2011.02.010]Search in Google Scholar
[[47] Jäger W., Luckhaus S., On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819-824.10.1090/S0002-9947-1992-1046835-6]Search in Google Scholar
[[48] Karch G., Scaling in nonlinear parabolic equations, J. Math. Anal. Appl. 234 (1999), 534-558.10.1006/jmaa.1999.6370]Search in Google Scholar
[[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), 1852-1881.10.1137/080722229]Search in Google Scholar
[[50] Keller E.F., Segel L.A., Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415.10.1016/0022-5193(70)90092-5]Open DOISearch in Google Scholar
[[51] Kozono H., Sugiyama Y., The Keller-Segel system of parabolic-parabolic type with initial data in weak Ln/2(Rn) and its application to self-similar solutions, Indiana Univ. Math. J. 57 (2008), 1467-1500.10.1512/iumj.2008.57.3316]Search in Google Scholar
[[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), 427-452.10.57262/die/1356060652]Search in Google Scholar
[[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), 1189-1208.10.57262/ade/1378327383]Search in Google Scholar
[[54] Mizoguchi N., Senba T., Type-II blowup of solutions to an elliptic-parabolic system, Adv. Math. Sci. Appl. 17 (2007), 505-545.]Search in Google Scholar
[[55] Mizoguchi N., Senba T., A sufficient condition for type I blowup in a parabolic-elliptic system, J. Differential Equations 250 (2011), 182-203.10.1016/j.jde.2010.10.016]Search in Google Scholar
[[56] Nagai T., Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl. 6 (2001), 37-55.10.1155/S1025583401000042]Search in Google Scholar
[[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), 3691-3713.10.3934/dcds.2012.32.3691]Search in Google Scholar
[[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), 1861-1880.10.3934/cpaa.2013.12.1861]Open DOISearch in Google Scholar
[[59] Pilarczyk D., Asymptotic stability of singular solution to nonlinear heat equation, Discrete Contin. Dyn. Syst. 25 (2009), 991-1001.10.3934/dcds.2009.25.991]Search in Google Scholar
[[60] Pilarczyk D., Self-similar asymptotics of solutions to heat equation with inverse square potential, J. Evol. Equ. 13 (2013), 69-87.10.1007/s00028-012-0169-8]Open DOISearch in Google Scholar
[[61] Quittner P., Souplet Ph., Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2007.]Search in Google Scholar
[[62] Senba T., Blowup behavior of radial solutions to Jäger-Luckhaus system in high dimensional domains, Funkcial. Ekvac. 48 (2005), 247-271.10.1619/fesi.48.247]Search in Google Scholar
[[63] Souplet Ph., Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in Rn, J. Funct. Anal. 272 (2017), 2005-2037.10.1016/j.jfa.2016.09.002]Search in Google Scholar
[[64] Suzuki T., Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser, Boston, 2005.10.1007/0-8176-4436-9]Search in Google Scholar
[[65] Taylor M.E., Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), 1407-1456.10.1080/03605309208820892]Open DOISearch in Google Scholar