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Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species
Blow-up scaling and global behaviour of solutions of the bi-Laplace equation via pencil operators
1. | Universidad Carlos III de Madrid, Av. Universidad 30, 28911-Leganés |
2. | Department of Mathematical Sciences, University of Bath, Bath BA2 7AY |
References:
[1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Acad. Press, New York-London, 1975. |
[2] |
V. Adamyan and V. Pivovarchik, On spectra of some classes of quadratic operator pencils, Contributions to operator theory in spaces with an indefinite metric (Vienna, 1995), Oper. Theory Adv. Appl., 106, Birkhäuser, Basel, (1998), 23-36.
doi: 10.1007/978-3-0348-8812-7_2. |
[3] |
P. Álvarez-Caudevilla and V. A. Galaktionov, The $p$-Laplace equation in domains with multiple crack section via pencil operators, Advances Nonlinear Studies, 15 (2015) no. 1, 91-116. |
[4] |
J. W. Dettman, Mathematical Methods in Physics and Engineering, Mc-Graw-Hill, New York, 1969. |
[5] |
Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohozaev, Global solutions of higher-order semilinear parabolic equations in the supercritical range, Adv. Differ. Equat., 9 (2004), 1009-1038. |
[6] |
D. Funaro, Polynomial Approximation of Differential Equations, Springer-Verlag, Berlin/Tokyo, 1992. |
[7] |
V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Chapman \& Hall/CRC, Boca Raton-Washington D.C., Florida, 2004.
doi: 10.1201/9780203998069. |
[8] |
V. A. Galaktionov, On extensions of Hardy's inequalities, Comm. Cont. Math., 7 (2005), 97-120.
doi: 10.1142/S0219199705001659. |
[9] |
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, 1976. |
[10] |
V. A. Kondrat'ev, Boundary value problems for parabolic equations in closed regions, Trans. Moscow Math. Soc., 15 (1966), 400-451. |
[11] |
V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., 16 (1967), 209-292. |
[12] |
M. Krein and H. Langer, On some mathematical principles in the linear theory of damped oscillations of continua. I, II, Int. Equat. Oper. Theory, 1 (1978), 364-399, 539-566.
doi: 10.1007/BF01682844. |
[13] |
A. Lemenant, On the homogeneity of global minimizers for the Mumford-Shah functional when $K$ is a smooth cone, Rend. Sem. Mat. Univ. Padova, 122 (2009), 129-159.
doi: 10.4171/RSMUP/122-9. |
[14] |
A. S. Markus, Introduction to Spectral Theory of Polynomial Operator Pencils, Translated from the Russian by H. H. McFaden. Translation edited by Ben Silver. With an appendix by M. V. Keldysh. Transl. of Math. Mon., 71, Amer. Math. Soc., Providence, RI, 1988. |
[15] |
C. Sturm, Mémoire sur une classe d'équations à différences partielles, J. Math. Pures Appl., 1 (1836), 373-444. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Acad. Press, New York-London, 1975. |
[2] |
V. Adamyan and V. Pivovarchik, On spectra of some classes of quadratic operator pencils, Contributions to operator theory in spaces with an indefinite metric (Vienna, 1995), Oper. Theory Adv. Appl., 106, Birkhäuser, Basel, (1998), 23-36.
doi: 10.1007/978-3-0348-8812-7_2. |
[3] |
P. Álvarez-Caudevilla and V. A. Galaktionov, The $p$-Laplace equation in domains with multiple crack section via pencil operators, Advances Nonlinear Studies, 15 (2015) no. 1, 91-116. |
[4] |
J. W. Dettman, Mathematical Methods in Physics and Engineering, Mc-Graw-Hill, New York, 1969. |
[5] |
Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohozaev, Global solutions of higher-order semilinear parabolic equations in the supercritical range, Adv. Differ. Equat., 9 (2004), 1009-1038. |
[6] |
D. Funaro, Polynomial Approximation of Differential Equations, Springer-Verlag, Berlin/Tokyo, 1992. |
[7] |
V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Chapman \& Hall/CRC, Boca Raton-Washington D.C., Florida, 2004.
doi: 10.1201/9780203998069. |
[8] |
V. A. Galaktionov, On extensions of Hardy's inequalities, Comm. Cont. Math., 7 (2005), 97-120.
doi: 10.1142/S0219199705001659. |
[9] |
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, 1976. |
[10] |
V. A. Kondrat'ev, Boundary value problems for parabolic equations in closed regions, Trans. Moscow Math. Soc., 15 (1966), 400-451. |
[11] |
V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., 16 (1967), 209-292. |
[12] |
M. Krein and H. Langer, On some mathematical principles in the linear theory of damped oscillations of continua. I, II, Int. Equat. Oper. Theory, 1 (1978), 364-399, 539-566.
doi: 10.1007/BF01682844. |
[13] |
A. Lemenant, On the homogeneity of global minimizers for the Mumford-Shah functional when $K$ is a smooth cone, Rend. Sem. Mat. Univ. Padova, 122 (2009), 129-159.
doi: 10.4171/RSMUP/122-9. |
[14] |
A. S. Markus, Introduction to Spectral Theory of Polynomial Operator Pencils, Translated from the Russian by H. H. McFaden. Translation edited by Ben Silver. With an appendix by M. V. Keldysh. Transl. of Math. Mon., 71, Amer. Math. Soc., Providence, RI, 1988. |
[15] |
C. Sturm, Mémoire sur une classe d'équations à différences partielles, J. Math. Pures Appl., 1 (1836), 373-444. |
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