January  2021, 41(1): 413-438. doi: 10.3934/dcds.2020136

Entire and ancient solutions of a supercritical semilinear heat equation

1. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

2. 

Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava, Slovakia

* Corresponding author

Received  July 2019 Published  February 2020

Fund Project: The first author is supported in part by NSF grant DMS-1856491. The second author is supported in part by VEGA Grant 1/0347/18 and by the Slovak Research and Development Agency under the contracts No. APVV-14-0378 and APVV-18-0308

We consider the semilinear heat equation $ u_t = \Delta u+u^p $ on $ {\mathbb R}^N $. Assuming that $ N\ge 3 $ and $ p $ is greater than the Sobolev critical exponent $ (N+2)/(N-2) $, we examine entire solutions (classical solutions defined for all $ t\in {\mathbb R} $) and ancient solutions (classical solutions defined on $ (-\infty,T) $ for some $ T<\infty $). We prove a new Liouville-type theorem saying that if $ p $ is greater than the Lepin exponent $ p_L: = 1+6/(N-10) $ ($ p_L = \infty $ if $ N\le 10 $), then all positive bounded radial entire solutions are steady states. The theorem is not valid without the assumption of radial symmetry; in other ranges of supercritical $ p $ it is known not to be valid even in the class of radial solutions. Our other results include classification theorems for nonstationary entire solutions (when they exist) and ancient solutions, as well as some applications in the theory of blowup of solutions.

Citation: Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
References:
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M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964.  Google Scholar

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H. Amann, Linear and Quasilinear Parabolic Problems I, Birkhäuser, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

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T. BartschP. Poláčik and P. Quittner, Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations, J. European Math. Soc., 13 (2011), 219-247.  doi: 10.4171/JEMS/250.  Google Scholar

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W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

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X.-Y. Chen and P. Poláčik, Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball, J. Reine Angew. Math., 472 (1996), 17-51.   Google Scholar

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M. FilaH. Matano and P. Poláčik, Existence of $L^1$-connections between equilibria of a semilinear parabolic equation, J. Dynam. Differential Equations, 14 (2002), 463-491.  doi: 10.1023/A:1016507330323.  Google Scholar

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M. Fila and N. Mizoguchi, Multiple continuation beyond blow-up, Differential Integral Equations, 20 (2007), 671-680.   Google Scholar

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M. Fila and A. Pulkkinen, Backward selfsimilar solutions of supercritical parabolic equations, Appl. Math. Letters, 22 (2009), 897-901.  doi: 10.1016/j.aml.2008.07.018.  Google Scholar

[12]

M. Fila and E. Yanagida, Homoclinic and heteroclinic orbits for a semilinear parabolic equation, Tohoku Math. J., 63 (2011), 561-579.  doi: 10.2748/tmj/1325886281.  Google Scholar

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B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

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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

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C. GuiW.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in ${ {\mathbb R}}^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.  Google Scholar

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M. A. Herrero and J. J. L. Velázquez, A blow up result for semilinear heat equations in the supercritical case, preprint, 1994. Google Scholar

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L. A. Lepin, Countable spectrum of eigenfunctions of a nonlinear heat-con-duction equation with distributed parameters, Differentsial'nye Uravneniya, 24 (1988), 1226–1234; (English translation: Differential Equations, 24 (1988), 799–805).  Google Scholar

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

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H. Matano and F. Merle, On nonexistence of type Ⅱ blowup for a supercritical nonlinear heat equation, Commun. Pure Appl. Math., 57 (2004), 1494-1541.  doi: 10.1002/cpa.20044.  Google Scholar

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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

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F. Merle and H. Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math., 51 (1998), 139-196.  doi: 10.1002/(SICI)1097-0312(199802)51:2<139::AID-CPA2>3.0.CO;2-C.  Google Scholar

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N. Mizoguchi, Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation, J. Funct. Anal., 257 (2009), 2911-2937.  doi: 10.1016/j.jfa.2009.07.009.  Google Scholar

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N. Mizoguchi, On backward self-similar blow-up solutions to a supercritical semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 821-831.  doi: 10.1017/S0308210509000444.  Google Scholar

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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

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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

[28]

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

[29]

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

[30]

P. PoláčikP. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[31]

P. PoláčikP. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part Ⅱ: Parabolic equations, Indiana Univ. Math. J., 56 (2007), 879-908.  doi: 10.1512/iumj.2007.56.2911.  Google Scholar

[32]

P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann., 327 (2003), 745-771.  doi: 10.1007/s00208-003-0469-y.  Google Scholar

[33]

P. Poláčik and E. Yanagida, A Liouville property and quasiconvergence for a semilinear heat equation, J. Differential Equations, 208 (2005), 194-214.  doi: 10.1016/j.jde.2003.10.019.  Google Scholar

[34]

P. Quittner, Liouville theorems for scaling invariant superlinear parabol-ic problems with gradient structure, Math. Ann., 364 (2016), 269-292.  doi: 10.1007/s00208-015-1219-7.  Google Scholar

[35]

P. Quittner, Uniqueness of singular self-similar solutions of a semilinear parabolic equation, Differential Integral Equations, 31 (2018), 881-892.   Google Scholar

[36]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2007.  Google Scholar

[37]

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

[38]

W. C. Troy, The existence of bounded solutions of a semilinear heat equation, SIAM J. Math. Anal., 18 (1987), 332-336.  doi: 10.1137/0518026.  Google Scholar

[39]

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

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964.  Google Scholar

[2]

H. Amann, Linear and Quasilinear Parabolic Problems I, Birkhäuser, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[3]

T. BartschP. Poláčik and P. Quittner, Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations, J. European Math. Soc., 13 (2011), 219-247.  doi: 10.4171/JEMS/250.  Google Scholar

[4]

J. Bebernes and D. Eberly, A description of self-similar blow-up for dimensions $n\geq 3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 1-21.   Google Scholar

[5]

M.-F. Bidaut-Véron, Initial blow-up for the solutions of a semilinear parabolic equation with source term, in Equations Aux Dérivées Partielles et Applications, articles dédiés à Jacques-Louis Lions, Gauthier-Villars, Paris, (1998), 189–198.  Google Scholar

[6]

C. Budd and Y.-W. Qi, The existence of bounded solutions of a semilinear elliptic equation, J. Differential Equations, 82 (1989), 207-218.  doi: 10.1016/0022-0396(89)90131-9.  Google Scholar

[7]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[8]

X.-Y. Chen and P. Poláčik, Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball, J. Reine Angew. Math., 472 (1996), 17-51.   Google Scholar

[9]

M. FilaH. Matano and P. Poláčik, Existence of $L^1$-connections between equilibria of a semilinear parabolic equation, J. Dynam. Differential Equations, 14 (2002), 463-491.  doi: 10.1023/A:1016507330323.  Google Scholar

[10]

M. Fila and N. Mizoguchi, Multiple continuation beyond blow-up, Differential Integral Equations, 20 (2007), 671-680.   Google Scholar

[11]

M. Fila and A. Pulkkinen, Backward selfsimilar solutions of supercritical parabolic equations, Appl. Math. Letters, 22 (2009), 897-901.  doi: 10.1016/j.aml.2008.07.018.  Google Scholar

[12]

M. Fila and E. Yanagida, Homoclinic and heteroclinic orbits for a semilinear parabolic equation, Tohoku Math. J., 63 (2011), 561-579.  doi: 10.2748/tmj/1325886281.  Google Scholar

[13]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[14]

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

[15]

C. GuiW.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in ${ {\mathbb R}}^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.  Google Scholar

[16]

M. A. Herrero and J. J. L. Velázquez, A blow up result for semilinear heat equations in the supercritical case, preprint, 1994. Google Scholar

[17]

L. A. Lepin, Countable spectrum of eigenfunctions of a nonlinear heat-con-duction equation with distributed parameters, Differentsial'nye Uravneniya, 24 (1988), 1226–1234; (English translation: Differential Equations, 24 (1988), 799–805).  Google Scholar

[18]

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

[19]

H. Matano and F. Merle, On nonexistence of type Ⅱ blowup for a supercritical nonlinear heat equation, Commun. Pure Appl. Math., 57 (2004), 1494-1541.  doi: 10.1002/cpa.20044.  Google Scholar

[20]

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

[21]

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

[22]

J. Matos, Convergence of blow-up solutions of nonlinear heat equations in the supercritical case, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 1197-1227.  doi: 10.1017/S0308210500019351.  Google Scholar

[23]

F. Merle and H. Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math., 51 (1998), 139-196.  doi: 10.1002/(SICI)1097-0312(199802)51:2<139::AID-CPA2>3.0.CO;2-C.  Google Scholar

[24]

N. Mizoguchi, Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation, J. Funct. Anal., 257 (2009), 2911-2937.  doi: 10.1016/j.jfa.2009.07.009.  Google Scholar

[25]

N. Mizoguchi, On backward self-similar blow-up solutions to a supercritical semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 821-831.  doi: 10.1017/S0308210509000444.  Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

P. PoláčikP. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[31]

P. PoláčikP. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part Ⅱ: Parabolic equations, Indiana Univ. Math. J., 56 (2007), 879-908.  doi: 10.1512/iumj.2007.56.2911.  Google Scholar

[32]

P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann., 327 (2003), 745-771.  doi: 10.1007/s00208-003-0469-y.  Google Scholar

[33]

P. Poláčik and E. Yanagida, A Liouville property and quasiconvergence for a semilinear heat equation, J. Differential Equations, 208 (2005), 194-214.  doi: 10.1016/j.jde.2003.10.019.  Google Scholar

[34]

P. Quittner, Liouville theorems for scaling invariant superlinear parabol-ic problems with gradient structure, Math. Ann., 364 (2016), 269-292.  doi: 10.1007/s00208-015-1219-7.  Google Scholar

[35]

P. Quittner, Uniqueness of singular self-similar solutions of a semilinear parabolic equation, Differential Integral Equations, 31 (2018), 881-892.   Google Scholar

[36]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2007.  Google Scholar

[37]

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

[38]

W. C. Troy, The existence of bounded solutions of a semilinear heat equation, SIAM J. Math. Anal., 18 (1987), 332-336.  doi: 10.1137/0518026.  Google Scholar

[39]

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

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