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Preface
Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring
1. | Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland |
References:
[1] |
N. Alikakos, An Application of the Invariance Principle to Reaction-Diffusion Equations, J. Differential Equations, 33 (1979), 201-225.
doi: 10.1016/0022-0396(79)90088-3. |
[2] |
P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. |
[3] |
P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal. TMA, 23 (1994), 1189-1209.
doi: 10.1016/0362-546X(94)90101-5. |
[4] |
P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles I, Colloq. Math., 66 (1994), 319-334. |
[5] |
J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. Blowup in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal. TMA, 75 (2012), 5215-5228.
doi: 10.1016/j.na.2012.04.038. |
[6] |
T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., (2013).
doi: 10.1007/s10440-013-9832-5. |
[7] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683. |
[8] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biology, 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[9] |
J. Moser, A sharp form of an inequality of N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092. |
[10] |
T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. |
[11] |
T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[12] |
T. Nagai, T. Senba, K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, 40 (1997), 411-433. |
[13] |
V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theoretical Biology, 42 (1973), 63-105.
doi: 10.1016/0022-5193(73)90149-5. |
[14] |
N. Trudinger, On imbeddings into Orlicz spaces and some applications, Indiana Univ. Math. J., 17 (1967), 473-483. |
[15] |
V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 138 (1961), 805-808. |
[16] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., (2013).
doi: 10.1016/j.matpur.2013.01.020. |
show all references
References:
[1] |
N. Alikakos, An Application of the Invariance Principle to Reaction-Diffusion Equations, J. Differential Equations, 33 (1979), 201-225.
doi: 10.1016/0022-0396(79)90088-3. |
[2] |
P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. |
[3] |
P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal. TMA, 23 (1994), 1189-1209.
doi: 10.1016/0362-546X(94)90101-5. |
[4] |
P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles I, Colloq. Math., 66 (1994), 319-334. |
[5] |
J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. Blowup in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal. TMA, 75 (2012), 5215-5228.
doi: 10.1016/j.na.2012.04.038. |
[6] |
T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., (2013).
doi: 10.1007/s10440-013-9832-5. |
[7] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683. |
[8] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biology, 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[9] |
J. Moser, A sharp form of an inequality of N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092. |
[10] |
T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. |
[11] |
T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[12] |
T. Nagai, T. Senba, K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, 40 (1997), 411-433. |
[13] |
V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theoretical Biology, 42 (1973), 63-105.
doi: 10.1016/0022-5193(73)90149-5. |
[14] |
N. Trudinger, On imbeddings into Orlicz spaces and some applications, Indiana Univ. Math. J., 17 (1967), 473-483. |
[15] |
V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 138 (1961), 805-808. |
[16] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., (2013).
doi: 10.1016/j.matpur.2013.01.020. |
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