June  2020, 40(6): 3327-3355. doi: 10.3934/dcds.2020052

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

* Corresponding author: Manuel del Pino

Dedicated to Professor Wei-Ming Ni on the occasion of his 70th birthday.

Received  April 2019 Revised  May 2019 Published  October 2019

Fund Project: The first author has been supported by a UK Royal Society Research Professorship and Grant PAI AFB-170001, Chile. The second author has been partly supported by Fondecyt grant 1160135, Chile. The research of the third author is partially supported by NSERC of Canada

We consider the Cauchy problem for the energy critical heat equation
$ \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} $
We find that for given points
$ q_1, q_2, \ldots, q_k $
and any sufficiently small
$ T>0 $
there is an initial condition
$ u_0 $
such that the solution
$ u(x, t) $
of (1) blows up at exactly those
$ k $
points with a type Ⅱ rate, namely larger than
$ (T-t)^{-\frac 12} $
. In fact
$ \|u(\cdot, t)\|_\infty \sim (T-t)^{-1}\log^2(T-t) $
. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin-Talenti bubbles.
Citation: Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052
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.  Google Scholar

[2]

C. Collot, F. Merle and P. Raphael, On strongly anisotropic type Ⅱ blow up, preprint, arXiv: 1709.04941. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

P. DaskalopoulosM. 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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

M. del PinoM. 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.  Google Scholar

[11]

T. DuyckaertsC. 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.  Google Scholar

[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.  Google Scholar

[13]

S. FilippasM. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Y. GigaS. 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.  Google Scholar

[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.   Google Scholar

[19]

M. A. Herrero and J. J. L. Velázquez, A blow up result for semilinear heat equations in the supercritical case, Unpublished. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. KriegerW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

C. Collot, F. Merle and P. Raphael, On strongly anisotropic type Ⅱ blow up, preprint, arXiv: 1709.04941. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

P. DaskalopoulosM. 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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

M. del PinoM. 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.  Google Scholar

[11]

T. DuyckaertsC. 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.  Google Scholar

[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.  Google Scholar

[13]

S. FilippasM. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Y. GigaS. 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.  Google Scholar

[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.   Google Scholar

[19]

M. A. Herrero and J. J. L. Velázquez, A blow up result for semilinear heat equations in the supercritical case, Unpublished. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. KriegerW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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