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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

 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 $a_1,\dots, a_{i_0}$.
Citation: Kin Ming Hui, Sunghoon Kim. Existence of Neumann and singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4859-4887. doi: 10.3934/dcds.2015.35.4859
##### References:
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Kenig, Degenerate Diffusion-Initial Value Problems and Local Regularity Theory,, Tracts in Mathematics 1, (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. 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. doi: 10.1016/j.aim.2013.03.011. Google Scholar [15] E. DiBenedetto, Degenerate Parabolic Equations,, Universitext, (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, (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. 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. 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. 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. 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. 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. 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. Google Scholar [24] S. Y. Hsu, Existence of singular solutions of a degenerate equation in $\mathbbR^2$,, Math. Ann., 334 (2006), 153. 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. 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. 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. Google Scholar [28] K. M. Hui, Singular limit of solutions of the very fast diffusion equation,, Nonlinear Anal. TMA, 68 (2008), 1120. 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., (1968). Google Scholar [30] L. A. Peletier, The Porous Medium Equation,, Applications of Nonlinear Analysis in the Physical Sciences (eds. H. Amann, (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. 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. 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. 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. Google Scholar [35] J. L. Vazquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations,, Oxford Lecture Series in Mathematics and its Applications, (2006). doi: 10.1093/acprof:oso/9780199202973.001.0001. Google Scholar [36] J. L. Vazquez, The porous medium equation-Mathematical Theory,, Oxford Mathematical Monographs, (2007). Google Scholar

show all references

##### References:
 [1] D. G. Aronson, The porous medium equation,, in Nonlinear Diffusion Problems, (1224), 1. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [11] P. Daskalopoulos and C. E. Kenig, Degenerate Diffusion-Initial Value Problems and Local Regularity Theory,, Tracts in Mathematics 1, (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. 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. doi: 10.1016/j.aim.2013.03.011. Google Scholar [15] E. DiBenedetto, Degenerate Parabolic Equations,, Universitext, (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, (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. 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. 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. 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. 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. 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. 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. Google Scholar [24] S. Y. Hsu, Existence of singular solutions of a degenerate equation in $\mathbbR^2$,, Math. Ann., 334 (2006), 153. 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. 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. 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. Google Scholar [28] K. M. Hui, Singular limit of solutions of the very fast diffusion equation,, Nonlinear Anal. TMA, 68 (2008), 1120. 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., (1968). Google Scholar [30] L. A. Peletier, The Porous Medium Equation,, Applications of Nonlinear Analysis in the Physical Sciences (eds. H. Amann, (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. 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. 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. 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. Google Scholar [35] J. L. Vazquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations,, Oxford Lecture Series in Mathematics and its Applications, (2006). doi: 10.1093/acprof:oso/9780199202973.001.0001. Google Scholar [36] J. L. Vazquez, The porous medium equation-Mathematical Theory,, Oxford Mathematical Monographs, (2007). Google Scholar
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