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Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network
Asymptotic large time behavior of singular solutions of the fast diffusion equation
1. | Institute of Mathematics, Academia Sinica, Taipei, Taiwan |
2. | Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China |
$u_t=Δ u^m$ |
$({\mathbb R}^n\setminus\{0\})×(0, ∞)$ |
$0<m<\frac{n-2}{n}$ |
$n≥3$ |
$u$ |
$t^{-α} f_i(t^{-β}x)$ |
$i=1, 2$ |
$u_0$ |
$A_1|x|^{-γ}≤ u_0≤ A_2|x|^{-γ}$ |
$A_2>A_1>0$ |
$\frac{2}{1-m}<γ<\frac{n-2}{m}$ |
$β:=\frac{1}{2-γ(1-m)}$, $α:=\frac{2\beta-1}{1-m}, $ |
$f_i$ |
$Δ f^m+α f+β x· \nabla f=0 \,\,\,\,\,\,\mbox{ in ${\mathbb R}^n\setminus\{0\}$}$ |
$\tilde u(y, τ):= t^{\, α} u(t^{\, β} y, t),\,\,\,\,\,\, { τ:=\log t}, $ |
References:
[1] |
D. G. Aronson, The porous medium equation, Nonlinear diffusion problems, (Montecatini Terme, 1985), 1-46, Lecture Notes in Math., 1224, Springer, Berlin, 1986.
doi: 10.1007/BFb0072687. |
[2] |
A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez,
Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal., 191 (2009), 347-385.
doi: 10.1007/s00205-008-0155-z. |
[3] |
M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez,
Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proc. Natl. Acad. Sci. USA, 107 (2010), 16459-16464.
doi: 10.1073/pnas.1003972107. |
[4] |
E. Chasseigne and J. L. Vázquez,
Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities, Arch. Ration. Mech. Anal., 164 (2002), 133-187.
doi: 10.1007/s00205-002-0210-0. |
[5] |
P. Daskalopoulos and C. E. Kenig,
Degenerate Diffusion: Initial Value Problems and Local Regularity Theory, EMS Tracts in Mathematics, 1. European Mathematical Society (EMS), Zürich, 2007.
doi: 10.4171/033. |
[6] |
P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, arXiv: 1306.0859. |
[7] |
P. Daskalopoulos, M. del Pino and N. Sesum, Type Ⅱ ancient compact solutions to the Yamabe flow, J. Reine Angew. Math., (2015), http://dx.doi.org/10.1515/crelle-2015-0048 in press.
doi: 10.1515/crelle-2015-0048. |
[8] |
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. |
[9] |
P. Daskalopoulos and N. Sesum,
The classification of locally conformally flat Yamabe solitons, Adv. Math., 240 (2013), 346-369.
doi: 10.1016/j.aim.2013.03.011. |
[10] |
M. Fila, J. L. Vázquez, M. Winkler and E. Yanagida,
Rate of convergence to Barenblatt profiles for the fast diffusion equation, Arch. Ration. Mech. Anal., 204 (2012), 599-625.
doi: 10.1007/s00205-011-0486-z. |
[11] |
M. Fila and M. Winkler,
Optimal rates of convergence to the singular Barenblatt profile for the fast diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 309-324.
doi: 10.1017/S0308210515000554. |
[12] |
M. Fila and M. Winkler,
Rate of convergence to separable solutions of the fast diffusion equation, Israel J. Math., 213 (2016), 1-32.
doi: 10.1007/s11856-016-1319-4. |
[13] |
M. Fila and M. Winkler,
Slow growth of solutions of superfast diffusion equations with unbounded initial data, J. London Math. Soc.(2), 95 (2017), 659-683.
doi: 10.1112/jlms.12029. |
[14] |
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. |
[15] |
S.Y. Hsu,
Asymptotic profile of solutions of a singular diffusion equation as $t \to∞$, Nonlinear Anal., 48 (2002), 781-790.
doi: 10.1016/S0362-546X(00)00214-5. |
[16] |
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. |
[17] |
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. |
[18] |
K. M. Hui,
On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential Integral Equations, 15 (2002), 769-804.
|
[19] |
K. M. Hui,
Singular limit of solutions of the very fast diffusion equation, Nonlinear Anal., 68 (2008), 1120-1147.
doi: 10.1016/j.na.2006.12.009. |
[20] |
K. M. Hui,
Asymptotic behaviour of solutions of the fast diffusion equation near its extinction time, J. Math. Anal. Appl., 454 (2017), 695-715.
doi: 10.1016/j.jmaa.2017.05.006. |
[21] |
T. Kato,
Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer-Verlag, Berlin, New York, 1976. |
[22] |
O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva,
Linear and Quasilinear Equations of Parabolic Type, (Russian) Transl. Math. Mono. vol. 23, Amer. Math. Soc., Providence, R. I., U. S. A., 1968. |
[23] |
S. J. Osher and J. V. Ralston,
L1 stability of traveling waves with applications to convective porous media flow, Comm. Pure Appl. Math., 35 (1982), 737-749.
doi: 10.1002/cpa.3160350602. |
[24] |
M. del Pino and M. Sáez,
On the extinction profile for solutions of $u_t=\Delta u^{\frac{N-2}{N+2}}$, Indiana Univ. Math. J., 50 (2001), 611-628.
doi: 10.1512/iumj.2001.50.1876. |
[25] |
J. L. Vázquez,
Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl.(9), 71 (1992), 503-526.
|
[26] |
J. L. Vázquez,
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. |
[27] |
J. L. Vázquez and M. Winkler,
The evolution of singularities in fast diffusion equations: Infinite time blow-down, SIAM J. Math. Anal., 43 (2011), 1499-1535.
doi: 10.1137/100809465. |
[28] |
R. Ye,
Global existence and convergence of Yamabe flow, J. Differential Geom., 39 (1994), 35-50.
doi: 10.4310/jdg/1214454674. |
show all references
References:
[1] |
D. G. Aronson, The porous medium equation, Nonlinear diffusion problems, (Montecatini Terme, 1985), 1-46, Lecture Notes in Math., 1224, Springer, Berlin, 1986.
doi: 10.1007/BFb0072687. |
[2] |
A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez,
Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal., 191 (2009), 347-385.
doi: 10.1007/s00205-008-0155-z. |
[3] |
M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez,
Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proc. Natl. Acad. Sci. USA, 107 (2010), 16459-16464.
doi: 10.1073/pnas.1003972107. |
[4] |
E. Chasseigne and J. L. Vázquez,
Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities, Arch. Ration. Mech. Anal., 164 (2002), 133-187.
doi: 10.1007/s00205-002-0210-0. |
[5] |
P. Daskalopoulos and C. E. Kenig,
Degenerate Diffusion: Initial Value Problems and Local Regularity Theory, EMS Tracts in Mathematics, 1. European Mathematical Society (EMS), Zürich, 2007.
doi: 10.4171/033. |
[6] |
P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, arXiv: 1306.0859. |
[7] |
P. Daskalopoulos, M. del Pino and N. Sesum, Type Ⅱ ancient compact solutions to the Yamabe flow, J. Reine Angew. Math., (2015), http://dx.doi.org/10.1515/crelle-2015-0048 in press.
doi: 10.1515/crelle-2015-0048. |
[8] |
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. |
[9] |
P. Daskalopoulos and N. Sesum,
The classification of locally conformally flat Yamabe solitons, Adv. Math., 240 (2013), 346-369.
doi: 10.1016/j.aim.2013.03.011. |
[10] |
M. Fila, J. L. Vázquez, M. Winkler and E. Yanagida,
Rate of convergence to Barenblatt profiles for the fast diffusion equation, Arch. Ration. Mech. Anal., 204 (2012), 599-625.
doi: 10.1007/s00205-011-0486-z. |
[11] |
M. Fila and M. Winkler,
Optimal rates of convergence to the singular Barenblatt profile for the fast diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 309-324.
doi: 10.1017/S0308210515000554. |
[12] |
M. Fila and M. Winkler,
Rate of convergence to separable solutions of the fast diffusion equation, Israel J. Math., 213 (2016), 1-32.
doi: 10.1007/s11856-016-1319-4. |
[13] |
M. Fila and M. Winkler,
Slow growth of solutions of superfast diffusion equations with unbounded initial data, J. London Math. Soc.(2), 95 (2017), 659-683.
doi: 10.1112/jlms.12029. |
[14] |
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. |
[15] |
S.Y. Hsu,
Asymptotic profile of solutions of a singular diffusion equation as $t \to∞$, Nonlinear Anal., 48 (2002), 781-790.
doi: 10.1016/S0362-546X(00)00214-5. |
[16] |
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. |
[17] |
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. |
[18] |
K. M. Hui,
On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential Integral Equations, 15 (2002), 769-804.
|
[19] |
K. M. Hui,
Singular limit of solutions of the very fast diffusion equation, Nonlinear Anal., 68 (2008), 1120-1147.
doi: 10.1016/j.na.2006.12.009. |
[20] |
K. M. Hui,
Asymptotic behaviour of solutions of the fast diffusion equation near its extinction time, J. Math. Anal. Appl., 454 (2017), 695-715.
doi: 10.1016/j.jmaa.2017.05.006. |
[21] |
T. Kato,
Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer-Verlag, Berlin, New York, 1976. |
[22] |
O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva,
Linear and Quasilinear Equations of Parabolic Type, (Russian) Transl. Math. Mono. vol. 23, Amer. Math. Soc., Providence, R. I., U. S. A., 1968. |
[23] |
S. J. Osher and J. V. Ralston,
L1 stability of traveling waves with applications to convective porous media flow, Comm. Pure Appl. Math., 35 (1982), 737-749.
doi: 10.1002/cpa.3160350602. |
[24] |
M. del Pino and M. Sáez,
On the extinction profile for solutions of $u_t=\Delta u^{\frac{N-2}{N+2}}$, Indiana Univ. Math. J., 50 (2001), 611-628.
doi: 10.1512/iumj.2001.50.1876. |
[25] |
J. L. Vázquez,
Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl.(9), 71 (1992), 503-526.
|
[26] |
J. L. Vázquez,
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. |
[27] |
J. L. Vázquez and M. Winkler,
The evolution of singularities in fast diffusion equations: Infinite time blow-down, SIAM J. Math. Anal., 43 (2011), 1499-1535.
doi: 10.1137/100809465. |
[28] |
R. Ye,
Global existence and convergence of Yamabe flow, J. Differential Geom., 39 (1994), 35-50.
doi: 10.4310/jdg/1214454674. |
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