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October  2015, 35(10): 4859-4887. doi: 10.3934/dcds.2015.35.4859

Existence of Neumann and singular solutions of the fast diffusion equation

Kin Ming Hui 1, and  Sunghoon Kim 2,

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

Received  June 2014 Revised  February 2015 Published  April 2015

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$, $n\ge 3$, $0 < m \le \frac{n-2}{n}$, $a_1,a_2,\dots, a_{i_0}\in\Omega$, $\delta_0 = \min_{1 \le i \le i_0} \mbox{dist} (a_i,∂\Omega)$ and let $\Omega_{\delta}=\Omega\setminus\cup_{i=1}^{i_0}B_{\delta}(a_i)$ and $\hat{\Omega}=\Omega\setminus\{a_1\,\dots,a_{i_0}\}$. For any $0<\delta<\delta_0$ we will prove the existence and uniqueness of positive solution of the Neumann problem for the equation $u_t=\Delta u^m$ in $\Omega_{\delta}\times (0,T)$ for some $T>0$. We will prove the existence of singular solutions of this equation in $\hat{\Omega}\times (0,T)$ for some $T>0$ that blow-up at the points $a_1,\dots, a_{i_0}$.
Keywords: existence, Fast diffusion equations, blow-up rate., singular solutions, Neumann solutions.
Mathematics Subject Classification: 35A01, 35K6.
Citation: Kin Ming Hui, Sunghoon Kim. Existence of Neumann and singular solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4859-4887. doi: 10.3934/dcds.2015.35.4859
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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