doi: 10.3934/dcds.2021060

Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity

1. 

3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

2. 

3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

* Corresponding author

Received  June 2020 Revised  December 2020 Published  March 2021

We are concerned with blow-up mechanisms in a semilinear heat equation:
$ u_t = \Delta u + |x|^{2a} u^p , \quad x \in \textbf{R}^N , \, t>0, $
where
$ p>1 $
and
$ a>-1 $
are constants. As for the Fujita equation, which corresponds to
$ a = 0 $
, a well-known result due to M. A. Herrero and J. J. L. Velázquez, C. R. Acad. Sci. Paris Sér. I Math. (1994), states that if
$ N\geq 11 $
and
$ p> 1 + 4/(N-4-2\sqrt{N-1}) $
, then there exist radial blow-up solutions
$ u_{\ell, {\rm HV}}(x, t) $
,
$ \ell \in \bf{N} $
, such that
$ \lim\limits_{t\to T} \left( T-t \right)^{1/(p-1)} \| u_{\ell, {\rm HV}}(\cdot, t) \|_{L^{\infty}(\textbf{R}^N )} = \infty, $
where
$ T $
is the blow-up time. We revisit the idea of their construction and obtain refined estimates for such solutions by the techniques developed in recent works and elaborate estimates of the heat semigroup in backward similarity variables. Our method is naturally extended to the case
$ a\not = 0 $
. As a consequence, we obtain an example of solutions that blow up at
$ x = 0 $
, the zero point of potential
$ |x|^{2a} $
with
$ a>0 $
, and behave in non-self-similar manner for
$ N > 10 + 8a $
. This last result is in contrast to backward self-similar solutions previously obtained for
$ N < 10 + 8a $
, which blow up at
$ x = 0 $
.
Citation: 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, doi: 10.3934/dcds.2021060
References:
[1]

P. Biernat and Y. Seki, Type II blow-up mechanism in supercritical harmonic map heat flow, Int. Math. Res. Not., 2 (2019), 407-456.  doi: 10.1093/imrn/rnx122.  Google Scholar

[2]

P. Biernat and Y. Seki, Transition of blow-up mechanisms in $k$-equivariant harmonic map heat flow, Nonlinearity, 33 (2020), 2756-2796.  doi: 10.1088/1361-6544/ab74f4.  Google Scholar

[3]

J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity, 7 (1994), 539-575.  doi: 10.1088/0951-7715/7/2/011.  Google Scholar

[4]

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

[5]

C. CollotF. Merle and P. Raphaël, Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions, Comm. Math. Phys., 352 (2017), 215-285.  doi: 10.1007/s00220-016-2795-4.  Google Scholar

[6]

C. CollotF. Merle and P. Raphaël, Strongly anisotropic type II blow-up at isolated points, J. Amer. Math. Soc., 33 (2020), 527-607.  doi: 10.1090/jams/941.  Google Scholar

[7]

M. del PinoM. Musso and J. Wei, Type II blow-up in the 5-dimensional energy critical heat equation, Acta. Math. Sinica, 35 (2019), 1027-1042.  doi: 10.1007/s10114-019-8341-5.  Google Scholar

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M. del Pino, M. Musso and J. Wei, Geometry driven type II higher dimensional blow-up for the critical heat equation, J. Funct. Anal., 280 (2021), 108788, 49pp. doi: 10.1016/j.jfa.2020.108788.  Google Scholar

[9]

Z. Erbol, Blow-up rate estimates of sign-changing solutions for nonlinear parabolic system, Master thesis (in Japanese), Tohoku University, 2019. Google Scholar

[10]

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., 456 (2000), 2957-2982.  doi: 10.1098/rspa.2000.0648.  Google Scholar

[11]

S. Filippas and A. Tertikas, On similarity solutions of a heat equation with a nonhomogeneous nonlinearity, J. Differential Equations, 165 (2000), 468-492.  doi: 10.1006/jdeq.2000.3789.  Google Scholar

[12]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.   Google Scholar

[13]

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

[14]

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

[15]

J.-S. GuoC.-S. Lin and M. Shimojo, Blow-up behavior for a parabolic equation with spatially dependent coefficient, Dynam. Systems Appl., 19 (2010), 415-433.   Google Scholar

[16]

J.-S. GuoC.-S. Lin and M. Shimojo, Blow-up for a reaction-diffusion equation with variable coefficient, Appl. Math. Lett., 26 (2013), 150-153.  doi: 10.1016/j.aml.2012.07.017.  Google Scholar

[17]

J.-S. Guo and M. Shimojo, Blowing up at zero points of potential for an initial boundary value problem, Commun. Pure Appl. Anal., 10 (2011), 161-177.  doi: 10.3934/cpaa.2011.10.161.  Google Scholar

[18]

J.-S. Guo and P. Souplet, Excluding blowup at zero points of the potential by means of Liouville-type theorems, J. Differential Equations, 265 (2018), 4942-4964.  doi: 10.1016/j.jde.2018.06.025.  Google Scholar

[19]

J.-S. Guo and C.-C. Wu, Finite time dead-core rate for the heat equation with a strong absorption, Tohoku Math. J. (2), 60 (2008), 37-70.  doi: 10.2748/tmj/1206734406.  Google Scholar

[20]

J. Harada, Non self-similar blow-up solutions to the heat equation with nonlinear boundary conditions, Nonlinear Anal., 102 (2014), 36-83.  doi: 10.1016/j.na.2014.01.028.  Google Scholar

[21]

J. Harada, Construction of type II blow-up solutions for a semilinear parabolic system with higher dimension,, Calc. Var., 56 (2017), Paper No. 121, 36 pp. doi: 10.1007/s00526-017-1213-x.  Google Scholar

[22]

J. Harada, A higher speed type II blowup for the five dimensional energy critical heat equation, Ann. Inst. Henri. Poincare, Analyse Non Linéaire., 37 (2020), 309-341.  doi: 10.1016/j.anihpc.2019.09.006.  Google Scholar

[23]

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

[24]

M. A. Herrero and J. J. L. Velázquez, Explosion de solutions d'équations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 141-145.   Google Scholar

[25]

L. A. Lepin, Self-similar solutions of a semilinear heat equation, Mat. Model., 2 (1990), 63-74.   Google Scholar

[26]

Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x) u^p = 0$ in $\mathbb{R}^n$, J. Differential Equations, 95 (1992), 304-330.  doi: 10.1016/0022-0396(92)90034-K.  Google Scholar

[27]

H. Matano, Blow-up in nonlinear heat equations with supercritical power nonlinearity,, in Perspectives in Nonlinear Partial Differential Equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 446 (2007), 385–412. doi: 10.1090/conm/446/08641.  Google Scholar

[28]

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

[29]

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

[30]

H. Matano and F. Merle, Threshold and generic type I behaviors for a supercritical nonlinear heat equation, J. Funct. Anal., 261 (2011), 717-748.  doi: 10.1016/j.jfa.2011.02.025.  Google Scholar

[31]

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., 86 (1997), 143-195.   Google Scholar

[32]

N. Mizoguchi, Type-II blowup for a semilinear heat equation, Adv. Differential Equations, 9 (2004), 1279-1316.   Google Scholar

[33]

N. Mizoguchi, Rate of type II blowup for a semilinear heat equation, Math. Ann., 339 (2007), 839-877.  doi: 10.1007/s00208-007-0133-z.  Google Scholar

[34]

N. Mizoguchi, Blow-up rate of type II and the braid group theory, Trans. Amer. Math. Soc., 363 (2011), 1419-1443.  doi: 10.1090/S0002-9947-2010-04784-1.  Google Scholar

[35]

N. Mizoguchi, Nonexistence of type II blow-up solution for a semilinear heat equation, J. Differential Equations, 250 (2011), 26-32.  doi: 10.1016/j.jde.2010.10.012.  Google Scholar

[36]

N. Mizoguchi and P. Souplet., Optimal condition for blow-up of the critical $L^q$ norm for the semilinear heat equation, Adv. Math., 355 (2019), 106763, 24pp. doi: 10.1016/j.aim.2019.106763.  Google Scholar

[37]

Y. Naito and T. Senba., Existence of peaking solutions for semilinear heat equations with blow-up profile above the singular steady state, Nonlinear Anal., 181 (2019), 265-293.  doi: 10.1016/j.na.2018.12.001.  Google Scholar

[38]

Q. H. Phan, Blow-up rate estimates and Liouville type theorems for a semilinear heat equation with weighted source, J. Dynam. Differential Equations, 29 (2017), 1131-1144.  doi: 10.1007/s10884-015-9489-z.  Google Scholar

[39]

R. G. Pinsky, Existence and nonexistence of global solutions for $u_t = \Delta u+a(x)u^p$ in ${\bf{R}}^d$, J. Differential Equations, 133 (1997), 152-177.  doi: 10.1006/jdeq.1996.3196.  Google Scholar

[40]

P. Poláčik and P. Quittner, On the multiplicity of self-similar solutions of the semilinear heat equation, Nonlinear Anal., 191 (2020), 111639, 23pp. doi: 10.1016/j.na.2019.111639.  Google Scholar

[41]

P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Second edition, Birkhäuser Advanced Texts, Basel, 2019. doi: 10.1007/978-3-030-18222-9.  Google Scholar

[42]

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

[43]

Y. Seki, On exact dead-core rates for a semilinear heat equation with strong absorption, Comm. Contemp. Math., 13 (2011), 1-52.  doi: 10.1142/S0219199711004154.  Google Scholar

[44]

Y. Seki, Type II blow-up mechanisms in a semilinear heat equation with critical Joseph–Lundgren exponent, J. Funct. Anal., 275 (2018), 3380-3456.  doi: 10.1016/j.jfa.2018.05.008.  Google Scholar

[45]

Y. Seki, Type II blow-up mechanisms in a semilinear heat equation with Lepin exponent, J. Differential Equations, 268 (2020), 853-900.  doi: 10.1016/j.jde.2019.08.026.  Google Scholar

[46]

R. Suzuki, Existence and nonexistence of global solutions of quasilinear parabolic equations, J. Math. Soc. Japan, 54 (2002), 747-792.  doi: 10.2969/jmsj/1191591992.  Google Scholar

[47]

G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ., New York, 1939.  Google Scholar

[48]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549–590. doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar

[49]

J. J. L. Velázquez, Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21 (1994), 595-628.   Google Scholar

show all references

References:
[1]

P. Biernat and Y. Seki, Type II blow-up mechanism in supercritical harmonic map heat flow, Int. Math. Res. Not., 2 (2019), 407-456.  doi: 10.1093/imrn/rnx122.  Google Scholar

[2]

P. Biernat and Y. Seki, Transition of blow-up mechanisms in $k$-equivariant harmonic map heat flow, Nonlinearity, 33 (2020), 2756-2796.  doi: 10.1088/1361-6544/ab74f4.  Google Scholar

[3]

J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity, 7 (1994), 539-575.  doi: 10.1088/0951-7715/7/2/011.  Google Scholar

[4]

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

[5]

C. CollotF. Merle and P. Raphaël, Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions, Comm. Math. Phys., 352 (2017), 215-285.  doi: 10.1007/s00220-016-2795-4.  Google Scholar

[6]

C. CollotF. Merle and P. Raphaël, Strongly anisotropic type II blow-up at isolated points, J. Amer. Math. Soc., 33 (2020), 527-607.  doi: 10.1090/jams/941.  Google Scholar

[7]

M. del PinoM. Musso and J. Wei, Type II blow-up in the 5-dimensional energy critical heat equation, Acta. Math. Sinica, 35 (2019), 1027-1042.  doi: 10.1007/s10114-019-8341-5.  Google Scholar

[8]

M. del Pino, M. Musso and J. Wei, Geometry driven type II higher dimensional blow-up for the critical heat equation, J. Funct. Anal., 280 (2021), 108788, 49pp. doi: 10.1016/j.jfa.2020.108788.  Google Scholar

[9]

Z. Erbol, Blow-up rate estimates of sign-changing solutions for nonlinear parabolic system, Master thesis (in Japanese), Tohoku University, 2019. Google Scholar

[10]

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., 456 (2000), 2957-2982.  doi: 10.1098/rspa.2000.0648.  Google Scholar

[11]

S. Filippas and A. Tertikas, On similarity solutions of a heat equation with a nonhomogeneous nonlinearity, J. Differential Equations, 165 (2000), 468-492.  doi: 10.1006/jdeq.2000.3789.  Google Scholar

[12]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.   Google Scholar

[13]

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

[14]

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

[15]

J.-S. GuoC.-S. Lin and M. Shimojo, Blow-up behavior for a parabolic equation with spatially dependent coefficient, Dynam. Systems Appl., 19 (2010), 415-433.   Google Scholar

[16]

J.-S. GuoC.-S. Lin and M. Shimojo, Blow-up for a reaction-diffusion equation with variable coefficient, Appl. Math. Lett., 26 (2013), 150-153.  doi: 10.1016/j.aml.2012.07.017.  Google Scholar

[17]

J.-S. Guo and M. Shimojo, Blowing up at zero points of potential for an initial boundary value problem, Commun. Pure Appl. Anal., 10 (2011), 161-177.  doi: 10.3934/cpaa.2011.10.161.  Google Scholar

[18]

J.-S. Guo and P. Souplet, Excluding blowup at zero points of the potential by means of Liouville-type theorems, J. Differential Equations, 265 (2018), 4942-4964.  doi: 10.1016/j.jde.2018.06.025.  Google Scholar

[19]

J.-S. Guo and C.-C. Wu, Finite time dead-core rate for the heat equation with a strong absorption, Tohoku Math. J. (2), 60 (2008), 37-70.  doi: 10.2748/tmj/1206734406.  Google Scholar

[20]

J. Harada, Non self-similar blow-up solutions to the heat equation with nonlinear boundary conditions, Nonlinear Anal., 102 (2014), 36-83.  doi: 10.1016/j.na.2014.01.028.  Google Scholar

[21]

J. Harada, Construction of type II blow-up solutions for a semilinear parabolic system with higher dimension,, Calc. Var., 56 (2017), Paper No. 121, 36 pp. doi: 10.1007/s00526-017-1213-x.  Google Scholar

[22]

J. Harada, A higher speed type II blowup for the five dimensional energy critical heat equation, Ann. Inst. Henri. Poincare, Analyse Non Linéaire., 37 (2020), 309-341.  doi: 10.1016/j.anihpc.2019.09.006.  Google Scholar

[23]

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

[24]

M. A. Herrero and J. J. L. Velázquez, Explosion de solutions d'équations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 141-145.   Google Scholar

[25]

L. A. Lepin, Self-similar solutions of a semilinear heat equation, Mat. Model., 2 (1990), 63-74.   Google Scholar

[26]

Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x) u^p = 0$ in $\mathbb{R}^n$, J. Differential Equations, 95 (1992), 304-330.  doi: 10.1016/0022-0396(92)90034-K.  Google Scholar

[27]

H. Matano, Blow-up in nonlinear heat equations with supercritical power nonlinearity,, in Perspectives in Nonlinear Partial Differential Equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 446 (2007), 385–412. doi: 10.1090/conm/446/08641.  Google Scholar

[28]

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

[29]

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

[30]

H. Matano and F. Merle, Threshold and generic type I behaviors for a supercritical nonlinear heat equation, J. Funct. Anal., 261 (2011), 717-748.  doi: 10.1016/j.jfa.2011.02.025.  Google Scholar

[31]

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., 86 (1997), 143-195.   Google Scholar

[32]

N. Mizoguchi, Type-II blowup for a semilinear heat equation, Adv. Differential Equations, 9 (2004), 1279-1316.   Google Scholar

[33]

N. Mizoguchi, Rate of type II blowup for a semilinear heat equation, Math. Ann., 339 (2007), 839-877.  doi: 10.1007/s00208-007-0133-z.  Google Scholar

[34]

N. Mizoguchi, Blow-up rate of type II and the braid group theory, Trans. Amer. Math. Soc., 363 (2011), 1419-1443.  doi: 10.1090/S0002-9947-2010-04784-1.  Google Scholar

[35]

N. Mizoguchi, Nonexistence of type II blow-up solution for a semilinear heat equation, J. Differential Equations, 250 (2011), 26-32.  doi: 10.1016/j.jde.2010.10.012.  Google Scholar

[36]

N. Mizoguchi and P. Souplet., Optimal condition for blow-up of the critical $L^q$ norm for the semilinear heat equation, Adv. Math., 355 (2019), 106763, 24pp. doi: 10.1016/j.aim.2019.106763.  Google Scholar

[37]

Y. Naito and T. Senba., Existence of peaking solutions for semilinear heat equations with blow-up profile above the singular steady state, Nonlinear Anal., 181 (2019), 265-293.  doi: 10.1016/j.na.2018.12.001.  Google Scholar

[38]

Q. H. Phan, Blow-up rate estimates and Liouville type theorems for a semilinear heat equation with weighted source, J. Dynam. Differential Equations, 29 (2017), 1131-1144.  doi: 10.1007/s10884-015-9489-z.  Google Scholar

[39]

R. G. Pinsky, Existence and nonexistence of global solutions for $u_t = \Delta u+a(x)u^p$ in ${\bf{R}}^d$, J. Differential Equations, 133 (1997), 152-177.  doi: 10.1006/jdeq.1996.3196.  Google Scholar

[40]

P. Poláčik and P. Quittner, On the multiplicity of self-similar solutions of the semilinear heat equation, Nonlinear Anal., 191 (2020), 111639, 23pp. doi: 10.1016/j.na.2019.111639.  Google Scholar

[41]

P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Second edition, Birkhäuser Advanced Texts, Basel, 2019. doi: 10.1007/978-3-030-18222-9.  Google Scholar

[42]

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

[43]

Y. Seki, On exact dead-core rates for a semilinear heat equation with strong absorption, Comm. Contemp. Math., 13 (2011), 1-52.  doi: 10.1142/S0219199711004154.  Google Scholar

[44]

Y. Seki, Type II blow-up mechanisms in a semilinear heat equation with critical Joseph–Lundgren exponent, J. Funct. Anal., 275 (2018), 3380-3456.  doi: 10.1016/j.jfa.2018.05.008.  Google Scholar

[45]

Y. Seki, Type II blow-up mechanisms in a semilinear heat equation with Lepin exponent, J. Differential Equations, 268 (2020), 853-900.  doi: 10.1016/j.jde.2019.08.026.  Google Scholar

[46]

R. Suzuki, Existence and nonexistence of global solutions of quasilinear parabolic equations, J. Math. Soc. Japan, 54 (2002), 747-792.  doi: 10.2969/jmsj/1191591992.  Google Scholar

[47]

G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ., New York, 1939.  Google Scholar

[48]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549–590. doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar

[49]

J. J. L. Velázquez, Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21 (1994), 595-628.   Google Scholar

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