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Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $
1. | Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom |
2. | Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada |
$ \begin{equation} \left\{ \begin{aligned} u_t & = \Delta u + u^3 {\quad\hbox{in } }\ \mathbb R^4 \times (0, T), \\ u(\cdot, 0) & = u_0 {\quad\hbox{in } } \mathbb R^4. \end{aligned}\right. ~~~~~~~~~~~~~~~~~~~~~~~(1)\end{equation} $ |
$ q_1, q_2, \ldots, q_k $ |
$ T>0 $ |
$ u_0 $ |
$ u(x, t) $ |
$ k $ |
$ (T-t)^{-\frac 12} $ |
$ \|u(\cdot, t)\|_\infty \sim (T-t)^{-1}\log^2(T-t) $ |
References:
[1] |
C. Collot,
Nonradial type Ⅱ blow up for the energy-supercritical semilinear heat equation, Anal. PDE, 10 (2017), 127-252.
doi: 10.2140/apde.2017.10.127. |
[2] |
C. Collot, F. Merle and P. Raphael, On strongly anisotropic type Ⅱ blow up, preprint, arXiv: 1709.04941. |
[3] |
C. Collot, P. Raphaël and J. Szeftel, On the Stability of Type I Blow Up for the Energy Super Critical Heat Equation, Mem. Amer. Math. Soc. 260. (2019), arXiv: 1605.07337.
doi: 10.1090/memo/1255. |
[4] |
C. Cortázar, M. del Pino and M. Musso, Green's function and infinite-time bubbling in the critical nonlinear heat equation, J. Eur. Math. Soc. (JEMS), to appear. |
[5] |
P. Daskalopoulos, M. del Pino and N. Sesum,
Type Ⅱ ancient compact solutions to the Yamabe flow, J. Reine Angew. Math., 738 (2018), 1-71.
doi: 10.1515/crelle-2015-0048. |
[6] |
J. Dávila, M. del Pino and J. Wei, Singularity formation for the two-dimensional harmonic map flow into S2, Invent. Math. arXiv: 1702.05801. |
[7] |
J. Dávila, M. del Pino, C. Pesce and J. Wei, Blow-up for the 3-dimensional axially symmetric harmonic map flow into $\mathbb{S}^2$, Discrete Contin. Dyn. Syst., to appear. |
[8] |
M. del Pino, M. Musso and J. Wei, Infinite time blow-up for the 3-dimensional energy critical heat equation, Anal. PDE, to appear. |
[9] |
M. del Pino, M. Musso and J. Wei, Geometry driven Type Ⅱ higher dimensional blow-up for the critical heat equation, preprint, arXiv: 1710.11461. |
[10] |
M. del Pino, M. Musso and J. C. Wei,
Type Ⅱ blow-up in the 5-dimensional energy critical heat equation, Acta Mathematica Sinica (Engl. Ser.), 35 (2019), 1027-1042.
doi: 10.1007/s10114-019-8341-5. |
[11] |
T. Duyckaerts, C. Kenig and F. Merle,
Universality of blow-up profile for small radial type Ⅱ blow-up solutions of the energy-critical wave equation, J. Eur. Math. Soc. (JEMS), 13 (2011), 533-599.
doi: 10.4171/JEMS/261. |
[12] |
C. J. Fan,
Log-log blow up solutions blow up at exactly m points, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1429-1482.
doi: 10.1016/j.anihpc.2016.11.002. |
[13] |
S. Filippas, M. A. Herrero and J. J. L. Velázquez,
Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 2957-2982.
doi: 10.1098/rspa.2000.0648. |
[14] |
H. Fujita,
On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+a}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.
|
[15] |
Y. Giga and R. V. Kohn,
Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319.
doi: 10.1002/cpa.3160380304. |
[16] |
Y. Giga and R. V. Kohn,
Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40.
doi: 10.1512/iumj.1987.36.36001. |
[17] |
Y. Giga, S. Matsui and S. Sasayama,
Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J., 53 (2004), 483-514.
doi: 10.1512/iumj.2004.53.2401. |
[18] |
M. A. Herrero and J. J. L. Velázquez,
Explosion de solutions d'equations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Paris Ser. I Math., 319 (1994), 141-145.
|
[19] |
M. A. Herrero and J. J. L. Velázquez, A blow up result for semilinear heat equations in the supercritical case, Unpublished. |
[20] |
J. Jendrej,
Construction of type Ⅱ blow-up solutions for the energy-critical wave equation in dimension 5, J. Funct. Anal., 272 (2017), 866-917.
doi: 10.1016/j.jfa.2016.10.019. |
[21] |
D. D. Joseph and T. S. Lundgren,
Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.
doi: 10.1007/BF00250508. |
[22] |
C. E. Kenig and F. Merle,
Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212.
doi: 10.1007/s11511-008-0031-6. |
[23] |
J. Krieger, W. Schlag and D. Tataru,
Slow blow-up solutions for the $H^{1}( \mathbb R^3)$ critical focusing semilinear wave equation, Duke Math. J., 147 (2009), 1-53.
doi: 10.1215/00127094-2009-005. |
[24] |
H. Matano and F. Merle,
On nonexistence of type Ⅱ blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math., 57 (2004), 1494-1541.
doi: 10.1002/cpa.20044. |
[25] |
H. Matano and F. Merle,
Classification of type Ⅰ and type Ⅱ behaviors for a supercritical nonlinear heat equation, J. Funct. Anal., 256 (2009), 992-1064.
doi: 10.1016/j.jfa.2008.05.021. |
[26] |
H. Matano and F. Merle,
Threshold and generic type Ⅰ behaviors for a supercritical nonlinear heat equation, J. Funct. Anal., 261 (2011), 716-748.
doi: 10.1016/j.jfa.2011.02.025. |
[27] |
F. Merle,
Solution of a nonlinear heat equation with arbitrarily given blow-up points, Commun. Pure Appl. Math., 45 (1992), 263-300.
doi: 10.1002/cpa.3160450303. |
[28] |
F. Merle and H. Zaag,
Stability of the blow-up profile for equations of the type $u_t = \Delta u+|u|^{p-1}u$, Duke Math. J., 86 (1997), 143-195.
doi: 10.1215/S0012-7094-97-08605-1. |
[29] |
N. Mizoguchi,
Nonexistence of type Ⅱ blowup solution for a semilinear heat equation, J. Differ. Equations, 250 (2011), 26-32.
doi: 10.1016/j.jde.2010.10.012. |
[30] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007. |
[31] |
P. Raphaël and R. Schweyer,
Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow, Comm. Pure Appl. Math., 66 (2013), 414-480.
doi: 10.1002/cpa.21435. |
[32] |
R. Schweyer,
Type Ⅱ blow-up for the four dimensional energy critical semi linear heat equation, J. Funct. Anal., 263 (2012), 3922-3983.
doi: 10.1016/j.jfa.2012.09.015. |
show all references
Dedicated to Professor Wei-Ming Ni on the occasion of his 70th birthday.
References:
[1] |
C. Collot,
Nonradial type Ⅱ blow up for the energy-supercritical semilinear heat equation, Anal. PDE, 10 (2017), 127-252.
doi: 10.2140/apde.2017.10.127. |
[2] |
C. Collot, F. Merle and P. Raphael, On strongly anisotropic type Ⅱ blow up, preprint, arXiv: 1709.04941. |
[3] |
C. Collot, P. Raphaël and J. Szeftel, On the Stability of Type I Blow Up for the Energy Super Critical Heat Equation, Mem. Amer. Math. Soc. 260. (2019), arXiv: 1605.07337.
doi: 10.1090/memo/1255. |
[4] |
C. Cortázar, M. del Pino and M. Musso, Green's function and infinite-time bubbling in the critical nonlinear heat equation, J. Eur. Math. Soc. (JEMS), to appear. |
[5] |
P. Daskalopoulos, M. del Pino and N. Sesum,
Type Ⅱ ancient compact solutions to the Yamabe flow, J. Reine Angew. Math., 738 (2018), 1-71.
doi: 10.1515/crelle-2015-0048. |
[6] |
J. Dávila, M. del Pino and J. Wei, Singularity formation for the two-dimensional harmonic map flow into S2, Invent. Math. arXiv: 1702.05801. |
[7] |
J. Dávila, M. del Pino, C. Pesce and J. Wei, Blow-up for the 3-dimensional axially symmetric harmonic map flow into $\mathbb{S}^2$, Discrete Contin. Dyn. Syst., to appear. |
[8] |
M. del Pino, M. Musso and J. Wei, Infinite time blow-up for the 3-dimensional energy critical heat equation, Anal. PDE, to appear. |
[9] |
M. del Pino, M. Musso and J. Wei, Geometry driven Type Ⅱ higher dimensional blow-up for the critical heat equation, preprint, arXiv: 1710.11461. |
[10] |
M. del Pino, M. Musso and J. C. Wei,
Type Ⅱ blow-up in the 5-dimensional energy critical heat equation, Acta Mathematica Sinica (Engl. Ser.), 35 (2019), 1027-1042.
doi: 10.1007/s10114-019-8341-5. |
[11] |
T. Duyckaerts, C. Kenig and F. Merle,
Universality of blow-up profile for small radial type Ⅱ blow-up solutions of the energy-critical wave equation, J. Eur. Math. Soc. (JEMS), 13 (2011), 533-599.
doi: 10.4171/JEMS/261. |
[12] |
C. J. Fan,
Log-log blow up solutions blow up at exactly m points, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1429-1482.
doi: 10.1016/j.anihpc.2016.11.002. |
[13] |
S. Filippas, M. A. Herrero and J. J. L. Velázquez,
Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 2957-2982.
doi: 10.1098/rspa.2000.0648. |
[14] |
H. Fujita,
On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+a}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.
|
[15] |
Y. Giga and R. V. Kohn,
Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319.
doi: 10.1002/cpa.3160380304. |
[16] |
Y. Giga and R. V. Kohn,
Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40.
doi: 10.1512/iumj.1987.36.36001. |
[17] |
Y. Giga, S. Matsui and S. Sasayama,
Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J., 53 (2004), 483-514.
doi: 10.1512/iumj.2004.53.2401. |
[18] |
M. A. Herrero and J. J. L. Velázquez,
Explosion de solutions d'equations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Paris Ser. I Math., 319 (1994), 141-145.
|
[19] |
M. A. Herrero and J. J. L. Velázquez, A blow up result for semilinear heat equations in the supercritical case, Unpublished. |
[20] |
J. Jendrej,
Construction of type Ⅱ blow-up solutions for the energy-critical wave equation in dimension 5, J. Funct. Anal., 272 (2017), 866-917.
doi: 10.1016/j.jfa.2016.10.019. |
[21] |
D. D. Joseph and T. S. Lundgren,
Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.
doi: 10.1007/BF00250508. |
[22] |
C. E. Kenig and F. Merle,
Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212.
doi: 10.1007/s11511-008-0031-6. |
[23] |
J. Krieger, W. Schlag and D. Tataru,
Slow blow-up solutions for the $H^{1}( \mathbb R^3)$ critical focusing semilinear wave equation, Duke Math. J., 147 (2009), 1-53.
doi: 10.1215/00127094-2009-005. |
[24] |
H. Matano and F. Merle,
On nonexistence of type Ⅱ blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math., 57 (2004), 1494-1541.
doi: 10.1002/cpa.20044. |
[25] |
H. Matano and F. Merle,
Classification of type Ⅰ and type Ⅱ behaviors for a supercritical nonlinear heat equation, J. Funct. Anal., 256 (2009), 992-1064.
doi: 10.1016/j.jfa.2008.05.021. |
[26] |
H. Matano and F. Merle,
Threshold and generic type Ⅰ behaviors for a supercritical nonlinear heat equation, J. Funct. Anal., 261 (2011), 716-748.
doi: 10.1016/j.jfa.2011.02.025. |
[27] |
F. Merle,
Solution of a nonlinear heat equation with arbitrarily given blow-up points, Commun. Pure Appl. Math., 45 (1992), 263-300.
doi: 10.1002/cpa.3160450303. |
[28] |
F. Merle and H. Zaag,
Stability of the blow-up profile for equations of the type $u_t = \Delta u+|u|^{p-1}u$, Duke Math. J., 86 (1997), 143-195.
doi: 10.1215/S0012-7094-97-08605-1. |
[29] |
N. Mizoguchi,
Nonexistence of type Ⅱ blowup solution for a semilinear heat equation, J. Differ. Equations, 250 (2011), 26-32.
doi: 10.1016/j.jde.2010.10.012. |
[30] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007. |
[31] |
P. Raphaël and R. Schweyer,
Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow, Comm. Pure Appl. Math., 66 (2013), 414-480.
doi: 10.1002/cpa.21435. |
[32] |
R. Schweyer,
Type Ⅱ blow-up for the four dimensional energy critical semi linear heat equation, J. Funct. Anal., 263 (2012), 3922-3983.
doi: 10.1016/j.jfa.2012.09.015. |
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