Super fast vanishing solutions of the fast diffusion equation

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September 2020, 40(9): 5383-5414. doi: 10.3934/dcds.2020232

Super fast vanishing solutions of the fast diffusion equation

Department of Mathematics, National Chung Cheng University, 168 University Road, Min-Hsiung, Chia-Yi 621, Taiwan

Received November 2019 Revised April 2020 Published June 2020

We will extend a recent result of B. Choi, P. Daskalopoulos and J. King [5]. For any $ n\ge 3 $, $ 0<m<\frac{n-2}{n+2} $ and $ \gamma>0 $, we will construct subsolutions and supersolutions of the fast diffusion equation $ u_t = \frac{n-1}{m}\Delta u^m $ in $ \mathbb{R}^n\times (t_0, T) $, $ t_0<T $, which decay at the rate $ (T-t)^{\frac{1+\gamma}{1-m}} $ as $ t\nearrow T $. As a consequence we obtain the existence of unique solution of the Cauchy problem $ u_t = \frac{n-1}{m}\Delta u^m $ in $ \mathbb{R}^n\times (t_0, T) $, $ u(x, t_0) = u_0(x) $ in $ \mathbb{R}^n $, which decay at the rate $ (T-t)^{\frac{1+\gamma}{1-m}} $ as $ t\nearrow T $ when $ u_0 $ satisfies appropriate decay condition.

Keywords: Existence, Cauchy problem, subsolution, supersolution, super fast vanishing solution, fast diffusion equation.

Mathematics Subject Classification: Primary: 35K55; Secondary: 35A01, 35B44.

Citation: Shu-Yu Hsu. Super fast vanishing solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5383-5414. doi: 10.3934/dcds.2020232

References:

[1]	D. G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems, Lecture Notes in Math., 1224, Springer, Berlin, 1986, 1–46. doi: 10.1007/BFb0072687.
[2]	S. Brendle, Convergence of the Yamabe flow for arbitrary initial energy, J. Differential Geom., 69 (2005), 217-278. doi: 10.4310/jdg/1121449107.
[3]	S. Brendle, Convergence of the Yamabe flow in dimension 6 and higher, Invent. Math., 170 (2007), 541-576. doi: 10.1007/s00222-007-0074-x.
[4]	B. Choi and P. Daskalopoulos, Yamabe flow: Steady solutions and type Ⅱ singularities, Nonlinear Anal., 173 (2018), 1-18. doi: 10.1016/j.na.2018.03.008.
[5]	B. Choi, P. Daskalopoulos and J. King, Type Ⅱ singularities on complete non-compact Yamabe flow, preprint, arXiv: 1809.05281v1.
[6]	B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions of generalized porous medium equations, Rev. Mat. Iberoamericana, 2 (1986), 267-305. doi: 10.4171/RMI/34.
[7]	P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, Comm. Anal. Geom., 27 (2013). doi: 10.4310/CAG.2019.v27.n8.a4.
[8]	P. Daskalopoulos, M. del Pino, J. King and N. Sesum, Type Ⅰ ancient compact solutions of the Yamabe flow, Nonlinear Anal., 137 (2016), 338-356. doi: 10.1016/j.na.2015.12.005.
[9]	P. Daskalopoulos, M. del Pino, J. King and N. Sesum, New type Ⅰ ancient compact solutions of the Yamabe flow, Math. Res. Lett., 24 (2017), 1667-1691. doi: 10.4310/MRL.2017.v24.n6.a5.
[10]	P. Daskalopoulos and C. E. Kenig, Degenerate Diffusions. Initial Value Problems and Local Regularity Theory, EMS Tracts in Mathematics, 1, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/033.
[11]	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.
[12]	V. A. Galaktionov and L. A. Peletier, Asymptotic behaviour near finite-time extinction for the fast diffusion equation, Arch. Rational Mech. Anal., 139 (1997), 83-98. doi: 10.1007/s002050050048.
[13]	M. A. Herrero and M. Pierre, The Cauchy problem for $u_t = \Delta u^m$ for $0<m<1$, Trans. Amer. Math. Soc., 291 (1985), 145-158. doi: 10.1090/S0002-9947-1985-0797051-0.
[14]	S.-Y. Hsu, Singular limit and exact decay rate of a nonlinear elliptic equation, Nonlinear Anal., 75 (2012), 3443-3455. doi: 10.1016/j.na.2012.01.009.
[15]	S.-Y. Hsu, Existence and asymptotic behaviour of solutions of the very fast diffusion, Manuscripta Math., 140 (2013), 441-460. doi: 10.1007/s00229-012-0576-8.
[16]	S. Y. Hsu, Some properties of the Yamabe soliton and the related nonlinear elliptic equation, Calc. Var. Partial Differential Equations, 49 (2014), 307-321. doi: 10.1007/s00526-012-0583-3.
[17]	S.-Y. Hsu, Exact decay rate of a nonlinear elliptic equation related to the Yamabe flow, Proc. Amer. Math. Soc., 142 (2014), 4239-4249. doi: 10.1090/S0002-9939-2014-12152-6.
[18]	S.-Y. Hsu, Global behaviour of solutions of the fast diffusion equation, Manuscripta Math., 158 (2019), 103-117. doi: 10.1007/s00229-018-1008-1.
[19]	K. M. Hui and S. Kim, Vanishing time behavior of the solutions of the fast diffusion equation, preprint, arXiv: 1811.04410.
[20]	L. A. Peletier, The porous medium equation, in Nonlinear Diffusion Problems, Lecture Notes in Mathematics, 1224, Springer, Berlin, Heidelberg, 1986. doi: 10.1007/BFb0072687.
[21]	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.
[22]	J. L. Vazquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures. Appl. (9), 71 (1992), 503-526.
[23]	J. L. Vazquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001.

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References:

[1]	D. G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems, Lecture Notes in Math., 1224, Springer, Berlin, 1986, 1–46. doi: 10.1007/BFb0072687.
[2]	S. Brendle, Convergence of the Yamabe flow for arbitrary initial energy, J. Differential Geom., 69 (2005), 217-278. doi: 10.4310/jdg/1121449107.
[3]	S. Brendle, Convergence of the Yamabe flow in dimension 6 and higher, Invent. Math., 170 (2007), 541-576. doi: 10.1007/s00222-007-0074-x.
[4]	B. Choi and P. Daskalopoulos, Yamabe flow: Steady solutions and type Ⅱ singularities, Nonlinear Anal., 173 (2018), 1-18. doi: 10.1016/j.na.2018.03.008.
[5]	B. Choi, P. Daskalopoulos and J. King, Type Ⅱ singularities on complete non-compact Yamabe flow, preprint, arXiv: 1809.05281v1.
[6]	B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions of generalized porous medium equations, Rev. Mat. Iberoamericana, 2 (1986), 267-305. doi: 10.4171/RMI/34.
[7]	P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, Comm. Anal. Geom., 27 (2013). doi: 10.4310/CAG.2019.v27.n8.a4.
[8]	P. Daskalopoulos, M. del Pino, J. King and N. Sesum, Type Ⅰ ancient compact solutions of the Yamabe flow, Nonlinear Anal., 137 (2016), 338-356. doi: 10.1016/j.na.2015.12.005.
[9]	P. Daskalopoulos, M. del Pino, J. King and N. Sesum, New type Ⅰ ancient compact solutions of the Yamabe flow, Math. Res. Lett., 24 (2017), 1667-1691. doi: 10.4310/MRL.2017.v24.n6.a5.
[10]	P. Daskalopoulos and C. E. Kenig, Degenerate Diffusions. Initial Value Problems and Local Regularity Theory, EMS Tracts in Mathematics, 1, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/033.
[11]	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.
[12]	V. A. Galaktionov and L. A. Peletier, Asymptotic behaviour near finite-time extinction for the fast diffusion equation, Arch. Rational Mech. Anal., 139 (1997), 83-98. doi: 10.1007/s002050050048.
[13]	M. A. Herrero and M. Pierre, The Cauchy problem for $u_t = \Delta u^m$ for $0<m<1$, Trans. Amer. Math. Soc., 291 (1985), 145-158. doi: 10.1090/S0002-9947-1985-0797051-0.
[14]	S.-Y. Hsu, Singular limit and exact decay rate of a nonlinear elliptic equation, Nonlinear Anal., 75 (2012), 3443-3455. doi: 10.1016/j.na.2012.01.009.
[15]	S.-Y. Hsu, Existence and asymptotic behaviour of solutions of the very fast diffusion, Manuscripta Math., 140 (2013), 441-460. doi: 10.1007/s00229-012-0576-8.
[16]	S. Y. Hsu, Some properties of the Yamabe soliton and the related nonlinear elliptic equation, Calc. Var. Partial Differential Equations, 49 (2014), 307-321. doi: 10.1007/s00526-012-0583-3.
[17]	S.-Y. Hsu, Exact decay rate of a nonlinear elliptic equation related to the Yamabe flow, Proc. Amer. Math. Soc., 142 (2014), 4239-4249. doi: 10.1090/S0002-9939-2014-12152-6.
[18]	S.-Y. Hsu, Global behaviour of solutions of the fast diffusion equation, Manuscripta Math., 158 (2019), 103-117. doi: 10.1007/s00229-018-1008-1.
[19]	K. M. Hui and S. Kim, Vanishing time behavior of the solutions of the fast diffusion equation, preprint, arXiv: 1811.04410.
[20]	L. A. Peletier, The porous medium equation, in Nonlinear Diffusion Problems, Lecture Notes in Mathematics, 1224, Springer, Berlin, Heidelberg, 1986. doi: 10.1007/BFb0072687.
[21]	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.
[22]	J. L. Vazquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures. Appl. (9), 71 (1992), 503-526.
[23]	J. L. Vazquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001.

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