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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 & Continuous Dynamical Systems, 2015, 35 (10) : 4859-4887. doi: 10.3934/dcds.2015.35.4859
References:
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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

H. Brezis and L. Veron, Removable singularities for some nonlinear elliptic equations,, Arch. Rational Mech. Anal., 75 (): 1.  doi: 10.1007/BF00284616.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions to the generalized porous medium equation, Rev. Mat. Iberoamericana, 2 (1986), 267-305.  Google Scholar

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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.  Google Scholar

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P. Daskalopoulos, M. del Pino and N. Sesum, Type II ancient compact solutions to the Yamabe flow,, , ().   Google Scholar

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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.  Google Scholar

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E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. Y. Hsu, Existence of singular solutions of a degenerate equation in $\mathbbR^2$, Math. Ann., 334 (2006), 153-197. doi: 10.1007/s00208-005-0714-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

K. M. Hui, On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential Integral Equations, 15 (2002), 769-804.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

J. L. Vazquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl., 71 (1992), 503-526.  Google Scholar

[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.  Google Scholar

[36]

J. L. Vazquez, The porous medium equation-Mathematical Theory, Oxford Mathematical Monographs, Oxford University Press, 2007.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

H. Brezis and L. Veron, Removable singularities for some nonlinear elliptic equations,, Arch. Rational Mech. Anal., 75 (): 1.  doi: 10.1007/BF00284616.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions to the generalized porous medium equation, Rev. Mat. Iberoamericana, 2 (1986), 267-305.  Google Scholar

[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.  Google Scholar

[12]

P. Daskalopoulos, M. del Pino and N. Sesum, Type II ancient compact solutions to the Yamabe flow,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. Y. Hsu, Existence of singular solutions of a degenerate equation in $\mathbbR^2$, Math. Ann., 334 (2006), 153-197. doi: 10.1007/s00208-005-0714-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

K. M. Hui, On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential Integral Equations, 15 (2002), 769-804.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

J. L. Vazquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl., 71 (1992), 503-526.  Google Scholar

[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.  Google Scholar

[36]

J. L. Vazquez, The porous medium equation-Mathematical Theory, Oxford Mathematical Monographs, Oxford University Press, 2007.  Google Scholar

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