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, 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

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.  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

[1]

Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021060

[2]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032

[3]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011

[4]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[5]

Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194

[6]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243

[7]

G. Deugoué, B. Jidjou Moghomye, T. Tachim Medjo. Approximation of a stochastic two-phase flow model by a splitting-up method. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1135-1170. doi: 10.3934/cpaa.2021010

[8]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

[9]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222

[10]

Yumi Yahagi. Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021099

[11]

Xiaozhong Yang, Xinlong Liu. Numerical analysis of two new finite difference methods for time-fractional telegraph equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3921-3942. doi: 10.3934/dcdsb.2020269

[12]

Azmeer Nordin, Mohd Salmi Md Noorani. Counting finite orbits for the flip systems of shifts of finite type. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021046

[13]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002

[14]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073

[15]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016

[16]

Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248

[17]

Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, , () : -. doi: 10.3934/era.2021024

[18]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025

[19]

Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021115

[20]

Bruno Premoselli. Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021069

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (286)
  • HTML views (312)
  • Cited by (1)

[Back to Top]