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On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion
Existence of Neumann and singular solutions of the fast diffusion equation
1. | Institute of Mathematics, Academia Sinica, Taipei, 10617, Taiwan |
2. | Department of Mathematics, School of Natural Sciences, The Catholic University of Korea, 43 Jibong-ro, Wonmi-gu, Bucheon-si, Gyeonggi-do, 420-743, South Korea |
References:
[1] |
D. G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems, Lecture Notes in Mathematics, 1224, Springer-Verlag, New York, 1986, 1-46.
doi: 10.1007/BFb0072687. |
[2] |
M. Bonforte, G. Grillo and J. L. Vazquez, Fast diffusion flow on manifolds of nonpositive curvature, J. Evol. Eq., 8 (2008), 99-128.
doi: 10.1007/s00028-007-0345-4. |
[3] |
M. Bonforte, G. Grillo and J. L. Vazquez, Behaviour near extinction for the fast diffusion equation on bounded domains, J. Math. Pures Appl., 97 (2012), 1-38.
doi: 10.1016/j.matpur.2011.03.002. |
[4] |
M. Bonforte and J. L. Vazquez, Global positivity estimates and Harnack inequalities for the fast diffusion equation, J. Funct. Anal., 240 (2006), 399-428.
doi: 10.1016/j.jfa.2006.07.009. |
[5] |
M. Bonforte and J. L. Vazquez, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations, Advances in Math., 223 (2010), 529-578.
doi: 10.1016/j.aim.2009.08.021. |
[6] |
M. Bonforte and J. L. Vazquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Adv. in Math., 250 (2014), 242-284.
doi: 10.1016/j.aim.2013.09.018. |
[7] |
H. Brezis and L. Veron, Removable singularities for some nonlinear elliptic equations, Arch. Rational Mech. Anal., 75 (1980/81), 1-6.
doi: 10.1007/BF00284616. |
[8] |
E. Chasseigne and J. L. Vazquez, Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities, Arch. Rat. Mech. Anal., 164 (2002), 133-187.
doi: 10.1007/s00205-002-0210-0. |
[9] |
Y. Z. Chen and E. Dibenedetto, On the local behavior of solutions of singular parabolic equations, Arch. Rat. Mech. Anal., 103 (1988), 319-345.
doi: 10.1007/BF00251444. |
[10] |
B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions to the generalized porous medium equation, Rev. Mat. Iberoamericana, 2 (1986), 267-305. |
[11] |
P. Daskalopoulos and C. E. Kenig, Degenerate Diffusion-Initial Value Problems and Local Regularity Theory, Tracts in Mathematics 1, European Mathematical Society, 2007.
doi: 10.4171/033. |
[12] |
P. Daskalopoulos, M. del Pino and N. Sesum, Type II ancient compact solutions to the Yamabe flow, arXiv:1209.5479v2. |
[13] |
P. Daskalopoulos and N. Sesum, On the extinction profile of solutions to fast diffusion, J. Reine Angew Math., 622 (2008), 95-119.
doi: 10.1515/CRELLE.2008.066. |
[14] |
P. Daskalopoulos and N. Sesum, The classification of locally conformally flat Yamabe solitons, Advances in Math., 240 (2013), 346-369.
doi: 10.1016/j.aim.2013.03.011. |
[15] |
E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[16] |
E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1584-8. |
[17] |
E. DiBenedetto, U. Gianazza and V. Vespri, Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9 (2010), 385-422. |
[18] |
E. DiBenedetto and Y. C. Kwong, Harnack estimates and extinction profile for weak solutions of certain singular parabolic equations, Trans. Amer. Math. Soc., 330 (1992), 783-811.
doi: 10.1090/S0002-9947-1992-1076615-7. |
[19] |
E. DiBenedetto, Y. C. Kwong and V. Vespri, Local space-analyticity of solutions of certain singular parabolic equations, Indiana Univ. Math. J., 40 (1991), 741-765.
doi: 10.1512/iumj.1991.40.40033. |
[20] |
M. Fila, J. L. Vazquez, M. Winkler and E. Yanagida, Rate of convergence to Barenblatt profiles for the fast diffusion equation, Arch. Rational Mech. Anal., 204 (2012), 599-625.
doi: 10.1007/s00205-011-0486-z. |
[21] |
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[22] |
M. A. Herrero and M. Pierre, The Cauchy problem for $u_t=\Delta u^m$ when $0 < m < 1$, Trans. Amer. Math. Soc., 291 (1985), 145-158.
doi: 10.1090/S0002-9947-1985-0797051-0. |
[23] |
S. Y. Hsu, Asymptotic behaviour of solution of the equation $u_t=\Delta \log u$ near the extinction time, Advances in Differential Equations, 8 (2003), 161-187. |
[24] |
S. Y. Hsu, Existence of singular solutions of a degenerate equation in $\mathbb{R}^2$, Math. Ann., 334 (2006), 153-197.
doi: 10.1007/s00208-005-0714-7. |
[25] |
S. Y. Hsu, Existence and asymptotic behaviour of solutions of the very fast diffusion equation, Manuscripta Math., 140 (2013), 441-460.
doi: 10.1007/s00229-012-0576-8. |
[26] |
K. M. Hui, Existence of solutions of the equation $u_t=\Delta \log u$, Nonlinear Anal. TMA, 37 (1999), 875-914.
doi: 10.1016/S0362-546X(98)00081-9. |
[27] |
K. M. Hui, On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential Integral Equations, 15 (2002), 769-804. |
[28] |
K. M. Hui, Singular limit of solutions of the very fast diffusion equation, Nonlinear Anal. TMA, 68 (2008), 1120-1147.
doi: 10.1016/j.na.2006.12.009. |
[29] |
O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Mono., Vol. 23, Amer. Math. Soc., Providence, R.I., U.S.A., 1968. |
[30] |
L. A. Peletier, The Porous Medium Equation, Applications of Nonlinear Analysis in the Physical Sciences (eds. H. Amann, N. Bazley and K. Kirchgassner), Pitman Advanced Publishing Program, Boston, 1981. |
[31] |
L. A. Peletier and H. Zhang, Self-similar solutions of a fast diffusion equation that do not conserve mass, Differential Integral Equations, 8 (1995), 2045-2064. |
[32] |
M. Del Pino and M. Sáez, On the extinction profile for solutions of $u_t=\Delta u^{(N-2)/(N+2)}$, Indiana Univ. Math. J., 50 (2001), 611-628.
doi: 10.1512/iumj.2001.50.1876. |
[33] |
P. E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Analysis TMA, 7 (1983), 387-409.
doi: 10.1016/0362-546X(83)90092-5. |
[34] |
J. L. Vazquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl., 71 (1992), 503-526. |
[35] |
J. L. Vazquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006.
doi: 10.1093/acprof:oso/9780199202973.001.0001. |
[36] |
J. L. Vazquez, The porous medium equation-Mathematical Theory, Oxford Mathematical Monographs, Oxford University Press, 2007. |
show all references
References:
[1] |
D. G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems, Lecture Notes in Mathematics, 1224, Springer-Verlag, New York, 1986, 1-46.
doi: 10.1007/BFb0072687. |
[2] |
M. Bonforte, G. Grillo and J. L. Vazquez, Fast diffusion flow on manifolds of nonpositive curvature, J. Evol. Eq., 8 (2008), 99-128.
doi: 10.1007/s00028-007-0345-4. |
[3] |
M. Bonforte, G. Grillo and J. L. Vazquez, Behaviour near extinction for the fast diffusion equation on bounded domains, J. Math. Pures Appl., 97 (2012), 1-38.
doi: 10.1016/j.matpur.2011.03.002. |
[4] |
M. Bonforte and J. L. Vazquez, Global positivity estimates and Harnack inequalities for the fast diffusion equation, J. Funct. Anal., 240 (2006), 399-428.
doi: 10.1016/j.jfa.2006.07.009. |
[5] |
M. Bonforte and J. L. Vazquez, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations, Advances in Math., 223 (2010), 529-578.
doi: 10.1016/j.aim.2009.08.021. |
[6] |
M. Bonforte and J. L. Vazquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Adv. in Math., 250 (2014), 242-284.
doi: 10.1016/j.aim.2013.09.018. |
[7] |
H. Brezis and L. Veron, Removable singularities for some nonlinear elliptic equations, Arch. Rational Mech. Anal., 75 (1980/81), 1-6.
doi: 10.1007/BF00284616. |
[8] |
E. Chasseigne and J. L. Vazquez, Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities, Arch. Rat. Mech. Anal., 164 (2002), 133-187.
doi: 10.1007/s00205-002-0210-0. |
[9] |
Y. Z. Chen and E. Dibenedetto, On the local behavior of solutions of singular parabolic equations, Arch. Rat. Mech. Anal., 103 (1988), 319-345.
doi: 10.1007/BF00251444. |
[10] |
B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions to the generalized porous medium equation, Rev. Mat. Iberoamericana, 2 (1986), 267-305. |
[11] |
P. Daskalopoulos and C. E. Kenig, Degenerate Diffusion-Initial Value Problems and Local Regularity Theory, Tracts in Mathematics 1, European Mathematical Society, 2007.
doi: 10.4171/033. |
[12] |
P. Daskalopoulos, M. del Pino and N. Sesum, Type II ancient compact solutions to the Yamabe flow, arXiv:1209.5479v2. |
[13] |
P. Daskalopoulos and N. Sesum, On the extinction profile of solutions to fast diffusion, J. Reine Angew Math., 622 (2008), 95-119.
doi: 10.1515/CRELLE.2008.066. |
[14] |
P. Daskalopoulos and N. Sesum, The classification of locally conformally flat Yamabe solitons, Advances in Math., 240 (2013), 346-369.
doi: 10.1016/j.aim.2013.03.011. |
[15] |
E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[16] |
E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1584-8. |
[17] |
E. DiBenedetto, U. Gianazza and V. Vespri, Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9 (2010), 385-422. |
[18] |
E. DiBenedetto and Y. C. Kwong, Harnack estimates and extinction profile for weak solutions of certain singular parabolic equations, Trans. Amer. Math. Soc., 330 (1992), 783-811.
doi: 10.1090/S0002-9947-1992-1076615-7. |
[19] |
E. DiBenedetto, Y. C. Kwong and V. Vespri, Local space-analyticity of solutions of certain singular parabolic equations, Indiana Univ. Math. J., 40 (1991), 741-765.
doi: 10.1512/iumj.1991.40.40033. |
[20] |
M. Fila, J. L. Vazquez, M. Winkler and E. Yanagida, Rate of convergence to Barenblatt profiles for the fast diffusion equation, Arch. Rational Mech. Anal., 204 (2012), 599-625.
doi: 10.1007/s00205-011-0486-z. |
[21] |
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[22] |
M. A. Herrero and M. Pierre, The Cauchy problem for $u_t=\Delta u^m$ when $0 < m < 1$, Trans. Amer. Math. Soc., 291 (1985), 145-158.
doi: 10.1090/S0002-9947-1985-0797051-0. |
[23] |
S. Y. Hsu, Asymptotic behaviour of solution of the equation $u_t=\Delta \log u$ near the extinction time, Advances in Differential Equations, 8 (2003), 161-187. |
[24] |
S. Y. Hsu, Existence of singular solutions of a degenerate equation in $\mathbb{R}^2$, Math. Ann., 334 (2006), 153-197.
doi: 10.1007/s00208-005-0714-7. |
[25] |
S. Y. Hsu, Existence and asymptotic behaviour of solutions of the very fast diffusion equation, Manuscripta Math., 140 (2013), 441-460.
doi: 10.1007/s00229-012-0576-8. |
[26] |
K. M. Hui, Existence of solutions of the equation $u_t=\Delta \log u$, Nonlinear Anal. TMA, 37 (1999), 875-914.
doi: 10.1016/S0362-546X(98)00081-9. |
[27] |
K. M. Hui, On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential Integral Equations, 15 (2002), 769-804. |
[28] |
K. M. Hui, Singular limit of solutions of the very fast diffusion equation, Nonlinear Anal. TMA, 68 (2008), 1120-1147.
doi: 10.1016/j.na.2006.12.009. |
[29] |
O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Mono., Vol. 23, Amer. Math. Soc., Providence, R.I., U.S.A., 1968. |
[30] |
L. A. Peletier, The Porous Medium Equation, Applications of Nonlinear Analysis in the Physical Sciences (eds. H. Amann, N. Bazley and K. Kirchgassner), Pitman Advanced Publishing Program, Boston, 1981. |
[31] |
L. A. Peletier and H. Zhang, Self-similar solutions of a fast diffusion equation that do not conserve mass, Differential Integral Equations, 8 (1995), 2045-2064. |
[32] |
M. Del Pino and M. Sáez, On the extinction profile for solutions of $u_t=\Delta u^{(N-2)/(N+2)}$, Indiana Univ. Math. J., 50 (2001), 611-628.
doi: 10.1512/iumj.2001.50.1876. |
[33] |
P. E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Analysis TMA, 7 (1983), 387-409.
doi: 10.1016/0362-546X(83)90092-5. |
[34] |
J. L. Vazquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl., 71 (1992), 503-526. |
[35] |
J. L. Vazquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006.
doi: 10.1093/acprof:oso/9780199202973.001.0001. |
[36] |
J. L. Vazquez, The porous medium equation-Mathematical Theory, Oxford Mathematical Monographs, Oxford University Press, 2007. |
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