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Liouville theorems on the upper half space
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 |
We will extend a recent result of B. Choi, P. Daskalopoulos and J. King [
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. |
show all references
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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