October  2021, 41(10): 4847-4885. 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  October 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (10) : 4847-4885. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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. 

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

M. A. Herrero and J. J. L. Velázquez, A blow up result for semilinear heat equations in the supercritical case, unpublished preprint.

[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. 

[25]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[32]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[47]

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

[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.

[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. 

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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. 

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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. 

[25]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[32]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[47]

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

[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.

[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. 

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