June  2012, 5(3): 449-472. doi: 10.3934/dcdss.2012.5.449

Multi-scale multi-profile global solutions of parabolic equations in $\mathbb{R}^N $

1. 

Université Pierre et Marie Curie & CNRS, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris cedex 05, France

2. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21944–970 Rio de Janeiro, R.J.

3. 

Université Paris 13, CNRS UMR 7539 LAGA, 99 Avenue J.-B. Clément, F-93430 Villetaneuse, France

Received  September 2010 Revised  June 2011 Published  October 2011

This paper explores certain concepts which extend the notions of (forward) self-similar and asymptotically self-similar solutions. A self-similar solution of an evolution equation has the property of being invariant with respect to a certain group of space-time dilations. An asymptotically self-similar solution approaches (in an appropriate sense) a self-similar solution to first order approximation for large time. Such solutions have a definite long-time asymptotic behavior, with respect to a specific time dependent spatial rescaling. After reviewing these fundamental concepts and the basic known results for heat equations on $\mathbb{R}^N $, we examine the possibility that a global solution might not be asymptotically self-similar. More precisely, we show that the asymptotic form of a solution can evolve differently along different time sequences going to infinity. Indeed, there exist solutions which are asymptotic to infinitely many different self-similar solutions, along different time sequences, all with respect to the same time dependent rescaling. We exhibit an explicit relationship between this phenomenon and the spatial asymptotic behavior of the initial value under a related group of dilations. In addition, we show that a given solution can exhibit nontrivial asymptotic behavior along different time sequences going to infinity, and with respect to different time dependent rescalings.
Citation: Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Multi-scale multi-profile global solutions of parabolic equations in $\mathbb{R}^N $. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 449-472. doi: 10.3934/dcdss.2012.5.449
References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[2]

H. Brezis, L. A. Peletier and D. Terman, A very singular solution of the heat equation with absorption, Arch. Ration. Mech. Anal., 95 (1986), 185--209. doi: 10.1007/BF00251357.

[3]

J. Bricmont and A. Kupiainen, Stable non-Gaussian diffusive profiles, Nonlinear Anal., 26 (1996), 583-593. doi: 10.1016/0362-546X(94)00300-7.

[4]

J. Bricmont, A. Kupiainen and G. Lin, Renormalization group and asymptotics of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math., 47 (1994), 893-922. doi: 10.1002/cpa.3160470606.

[5]

M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes, in "Séminaire sur les Équations aux Dérives Partielles," 1993-1994, Exp. No. VIII, 12 pp., École Polytech., Palaiseau, 1994.

[6]

J. A. Carrillo and J. L. Vázquez, Asymptotic complexity in filtration equations, J. Evol. Equ., 7 (2007), 471-495.

[7]

T. Cazenave, F. Dickstein, M. Escobedo and F. B. Weissler, Self-similar solutions of a nonlinear heat equation, J. Math. Sci. Univ. Tokyo, 8 (2001), 501-540.

[8]

T. Cazenave, F. Dickstein and F. B. Weissler, Universal solutions of the heat equation on $\R^N $, Discrete Contin. Dynam. Systems, 9 (2003), 1105-1132. doi: 10.3934/dcds.2003.9.1105.

[9]

T. Cazenave, F. Dickstein and F. B. Weissler, Universal solutions of the nonlinear heat equation on $\R^N $, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 2 (2003), 77-117.

[10]

T. Cazenave, F. Dickstein and F. B. Weissler, Chaotic behavior of solutions of the Navier-Stokes system in $\R^N $, Adv. Differential Equations, 10 (2005), 361-398.

[11]

T. Cazenave, F. Dickstein and F. B. Weissler, A solution of the heat equation with a continuum of decay rates, in "Elliptic and Parabolic Problems: A Special Tribute to the Work of Haïm Brezis," Progress in Nonlinear Differential Equations and their Applications, 63, Birkhäuser, Basel, (2005), 135-138.

[12]

T. Cazenave, F. Dickstein and F. B. Weissler, Multiscale asymptotic behavior of a solution of the heat equation in $\R^N $, in "Nonlinear Differential Equations: A Tribute to D. G. de Figueiredo," Progress in Nonlinear Differential Equations and their Applications, 66, Birkhäuser, Basel, (2006), 185-194.

[13]

T. Cazenave, F. Dickstein and F. B. Weissler, A solution of the constant coefficient heat equation on $\mathbb{R}^{N}$ with exceptional asymptotic behavior: An explicit constuction, J. Math. Pures Appl. (9), 85 (2006), 119-150. doi: 10.1016/j.matpur.2005.08.006.

[14]

T. Cazenave, F. Dickstein and F. B. Weissler, Nonparabolic asymptotic limits of solutions of the heat equation on $\R^N $, J. Dynam. Differential Equations, 19 (2007), 789-818. doi: 10.1007/s10884-007-9076-z.

[15]

T. Cazenave and F. B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z., 228 (1998), 83-120. doi: 10.1007/PL00004606.

[16]

T. Cazenave and F. B. Weissler, Spatial decay and time-asymptotic profiles for solutions of Schrödinger equations, Indiana Univ. Math. J., 55 (2006), 75-118. doi: 10.1512/iumj.2006.55.2664.

[17]

R. L. Devaney, Overview: Dynamics of Simple Maps, in "Chaos and Fractals" (Providence, RI, 1988), 1-24, Proc. Symp. Appl. Math., 39, Amer. Math. Soc., Providence, RI, 1989.

[18]

C. Dohmen and M. Hirose, Structure of positive radial solutions to the Haraux-Weissler equation, Nonlinear Anal., 33 (1998), 51-69. doi: 10.1016/S0362-546X(97)00542-7.

[19]

H. Emamirad, G. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation, Proc. Amer. Math. Soc, S 0002-9939 (2011) 11069-4.

[20]

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal., 11 (1987), 1103-1133. doi: 10.1016/0362-546X(87)90001-0.

[21]

M. Escobedo and O. Kavian, Asymptotic behavior of positive solutions of a nonlinear heat equation, Houston J. Math., 14 (1988), 39-50.

[22]

M. Escobedo, O. Kavian and H. Matano, Large time behavior of solutions of a dissipative semilinear heat equation, Comm. Partial Differential Equations, 20 (1995), 1427-1452.

[23]

D. Fang, J. Xie and T. Cazenave, Multiscale asymptotic behavior of the Schrödinger equation, Funk. Ekva., 54 (2011), 69-94. doi: 10.1619/fesi.54.69.

[24]

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

[25]

M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations. Asymptotic Behavior of Solutions and Self-Similar Solutions," Progr. Nonlinear Differential Equations Appl., 79, Birkhäuser Boston, Inc., Boston, MA, 2010.

[26]

Y. Giga and T. Miyakawa, Navier-Stokes flow in $R^3$ with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations, 14 (1989), 577-618.

[27]

A. Gmira and L. Véron, Large time behaviour of the solutions of a semilinear parabolic equation in $\R^N $, J. Differential Equations, 53 (1984), 258-276. doi: 10.1016/0022-0396(84)90042-1.

[28]

A. Haraux and F. B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189. doi: 10.1512/iumj.1982.31.31016.

[29]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-505. doi: 10.3792/pja/1195519254.

[30]

L. Herraiz, Asymptotic behaviour of solutions of some semilinear parabolic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 49-105.

[31]

S. Kamin and L. A. Peletier, Large time behavior of solutions of the heat equation with absorption, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 12 (1985), 393-408.

[32]

O. Kavian, Remarks on the time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 423-452.

[33]

T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 1-15.

[34]

K. Kobayashi, T. Sirao and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424. doi: 10.2969/jmsj/02930407.

[35]

M. Kwak, A semilinear heat equation with singular initial data, Proc. Royal Soc. Edinburgh Sect. A, 128 (1998), 745-758.

[36]

M. Marcus and L. Véron, Initial trace of positive solutions of some nonlinear parabolic equations, Comm. Partial Differential Equations, 24 (1999), 1445-1499.

[37]

F. Ribaud, "Analyse de Littlewood Paley pour la Résolution d'Équations Paraboliques Semi-Linéaires," Ph.D Thesis, University of Paris XI, January, 1996.

[38]

S. Snoussi, S. Tayachi and F. B. Weissler, Asymptotically self-similar global solutions of a semilinear parabolic equation with a nonlinear gradient term, Proc. Royal Soc. Edinburgh Sect. A, 129 (1999), 1291-1307.

[39]

S. Snoussi, S. Tayachi and F. B. Weissler, Asymptotically self-similar global solutions of a general semilinear heat equation, Math. Ann., 321 (2001), 131-155. doi: 10.1007/PL00004498.

[40]

J. L. Vázquez and E. Zuazua, Complexity of large time behaviour of evolution equations with bounded data, Chinese Ann. Math. Ser. B, 23 (2002), 293-310. doi: 10.1142/S0252959902000274.

[41]

C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Ration. Mech. Anal., 138 (1997), 279-306. doi: 10.1007/s002050050042.

[42]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.

[43]

F. B. Weissler, Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations, Arch. Ration. Mech. Anal., 91 (1985), 247-266. doi: 10.1007/BF00250744.

[44]

F. B. Weissler, Asymptotic analysis of an ordinary differential equation and nonuniqueness for a semilinear partial differential equation, Arch. Ration. Mech. Anal., 91 (1985), 231-245. doi: 10.1007/BF00250743.

[45]

J. Xie, L. Zhang and T. Cazenave, A note on decay rates for Schrödinger's equation, Proc. Amer. Math. Soc., 138 (2010), 199-207. doi: 10.1090/S0002-9939-09-10049-7.

[46]

E. Yanagida, Uniqueness of rapidly decaying solutions to the Haraux-Weissler equation, J. Differential Equations, 127 (1996), 561-570. doi: 10.1006/jdeq.1996.0083.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[2]

H. Brezis, L. A. Peletier and D. Terman, A very singular solution of the heat equation with absorption, Arch. Ration. Mech. Anal., 95 (1986), 185--209. doi: 10.1007/BF00251357.

[3]

J. Bricmont and A. Kupiainen, Stable non-Gaussian diffusive profiles, Nonlinear Anal., 26 (1996), 583-593. doi: 10.1016/0362-546X(94)00300-7.

[4]

J. Bricmont, A. Kupiainen and G. Lin, Renormalization group and asymptotics of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math., 47 (1994), 893-922. doi: 10.1002/cpa.3160470606.

[5]

M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes, in "Séminaire sur les Équations aux Dérives Partielles," 1993-1994, Exp. No. VIII, 12 pp., École Polytech., Palaiseau, 1994.

[6]

J. A. Carrillo and J. L. Vázquez, Asymptotic complexity in filtration equations, J. Evol. Equ., 7 (2007), 471-495.

[7]

T. Cazenave, F. Dickstein, M. Escobedo and F. B. Weissler, Self-similar solutions of a nonlinear heat equation, J. Math. Sci. Univ. Tokyo, 8 (2001), 501-540.

[8]

T. Cazenave, F. Dickstein and F. B. Weissler, Universal solutions of the heat equation on $\R^N $, Discrete Contin. Dynam. Systems, 9 (2003), 1105-1132. doi: 10.3934/dcds.2003.9.1105.

[9]

T. Cazenave, F. Dickstein and F. B. Weissler, Universal solutions of the nonlinear heat equation on $\R^N $, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 2 (2003), 77-117.

[10]

T. Cazenave, F. Dickstein and F. B. Weissler, Chaotic behavior of solutions of the Navier-Stokes system in $\R^N $, Adv. Differential Equations, 10 (2005), 361-398.

[11]

T. Cazenave, F. Dickstein and F. B. Weissler, A solution of the heat equation with a continuum of decay rates, in "Elliptic and Parabolic Problems: A Special Tribute to the Work of Haïm Brezis," Progress in Nonlinear Differential Equations and their Applications, 63, Birkhäuser, Basel, (2005), 135-138.

[12]

T. Cazenave, F. Dickstein and F. B. Weissler, Multiscale asymptotic behavior of a solution of the heat equation in $\R^N $, in "Nonlinear Differential Equations: A Tribute to D. G. de Figueiredo," Progress in Nonlinear Differential Equations and their Applications, 66, Birkhäuser, Basel, (2006), 185-194.

[13]

T. Cazenave, F. Dickstein and F. B. Weissler, A solution of the constant coefficient heat equation on $\mathbb{R}^{N}$ with exceptional asymptotic behavior: An explicit constuction, J. Math. Pures Appl. (9), 85 (2006), 119-150. doi: 10.1016/j.matpur.2005.08.006.

[14]

T. Cazenave, F. Dickstein and F. B. Weissler, Nonparabolic asymptotic limits of solutions of the heat equation on $\R^N $, J. Dynam. Differential Equations, 19 (2007), 789-818. doi: 10.1007/s10884-007-9076-z.

[15]

T. Cazenave and F. B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z., 228 (1998), 83-120. doi: 10.1007/PL00004606.

[16]

T. Cazenave and F. B. Weissler, Spatial decay and time-asymptotic profiles for solutions of Schrödinger equations, Indiana Univ. Math. J., 55 (2006), 75-118. doi: 10.1512/iumj.2006.55.2664.

[17]

R. L. Devaney, Overview: Dynamics of Simple Maps, in "Chaos and Fractals" (Providence, RI, 1988), 1-24, Proc. Symp. Appl. Math., 39, Amer. Math. Soc., Providence, RI, 1989.

[18]

C. Dohmen and M. Hirose, Structure of positive radial solutions to the Haraux-Weissler equation, Nonlinear Anal., 33 (1998), 51-69. doi: 10.1016/S0362-546X(97)00542-7.

[19]

H. Emamirad, G. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation, Proc. Amer. Math. Soc, S 0002-9939 (2011) 11069-4.

[20]

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal., 11 (1987), 1103-1133. doi: 10.1016/0362-546X(87)90001-0.

[21]

M. Escobedo and O. Kavian, Asymptotic behavior of positive solutions of a nonlinear heat equation, Houston J. Math., 14 (1988), 39-50.

[22]

M. Escobedo, O. Kavian and H. Matano, Large time behavior of solutions of a dissipative semilinear heat equation, Comm. Partial Differential Equations, 20 (1995), 1427-1452.

[23]

D. Fang, J. Xie and T. Cazenave, Multiscale asymptotic behavior of the Schrödinger equation, Funk. Ekva., 54 (2011), 69-94. doi: 10.1619/fesi.54.69.

[24]

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

[25]

M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations. Asymptotic Behavior of Solutions and Self-Similar Solutions," Progr. Nonlinear Differential Equations Appl., 79, Birkhäuser Boston, Inc., Boston, MA, 2010.

[26]

Y. Giga and T. Miyakawa, Navier-Stokes flow in $R^3$ with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations, 14 (1989), 577-618.

[27]

A. Gmira and L. Véron, Large time behaviour of the solutions of a semilinear parabolic equation in $\R^N $, J. Differential Equations, 53 (1984), 258-276. doi: 10.1016/0022-0396(84)90042-1.

[28]

A. Haraux and F. B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189. doi: 10.1512/iumj.1982.31.31016.

[29]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-505. doi: 10.3792/pja/1195519254.

[30]

L. Herraiz, Asymptotic behaviour of solutions of some semilinear parabolic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 49-105.

[31]

S. Kamin and L. A. Peletier, Large time behavior of solutions of the heat equation with absorption, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 12 (1985), 393-408.

[32]

O. Kavian, Remarks on the time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 423-452.

[33]

T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 1-15.

[34]

K. Kobayashi, T. Sirao and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424. doi: 10.2969/jmsj/02930407.

[35]

M. Kwak, A semilinear heat equation with singular initial data, Proc. Royal Soc. Edinburgh Sect. A, 128 (1998), 745-758.

[36]

M. Marcus and L. Véron, Initial trace of positive solutions of some nonlinear parabolic equations, Comm. Partial Differential Equations, 24 (1999), 1445-1499.

[37]

F. Ribaud, "Analyse de Littlewood Paley pour la Résolution d'Équations Paraboliques Semi-Linéaires," Ph.D Thesis, University of Paris XI, January, 1996.

[38]

S. Snoussi, S. Tayachi and F. B. Weissler, Asymptotically self-similar global solutions of a semilinear parabolic equation with a nonlinear gradient term, Proc. Royal Soc. Edinburgh Sect. A, 129 (1999), 1291-1307.

[39]

S. Snoussi, S. Tayachi and F. B. Weissler, Asymptotically self-similar global solutions of a general semilinear heat equation, Math. Ann., 321 (2001), 131-155. doi: 10.1007/PL00004498.

[40]

J. L. Vázquez and E. Zuazua, Complexity of large time behaviour of evolution equations with bounded data, Chinese Ann. Math. Ser. B, 23 (2002), 293-310. doi: 10.1142/S0252959902000274.

[41]

C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Ration. Mech. Anal., 138 (1997), 279-306. doi: 10.1007/s002050050042.

[42]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.

[43]

F. B. Weissler, Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations, Arch. Ration. Mech. Anal., 91 (1985), 247-266. doi: 10.1007/BF00250744.

[44]

F. B. Weissler, Asymptotic analysis of an ordinary differential equation and nonuniqueness for a semilinear partial differential equation, Arch. Ration. Mech. Anal., 91 (1985), 231-245. doi: 10.1007/BF00250743.

[45]

J. Xie, L. Zhang and T. Cazenave, A note on decay rates for Schrödinger's equation, Proc. Amer. Math. Soc., 138 (2010), 199-207. doi: 10.1090/S0002-9939-09-10049-7.

[46]

E. Yanagida, Uniqueness of rapidly decaying solutions to the Haraux-Weissler equation, J. Differential Equations, 127 (1996), 561-570. doi: 10.1006/jdeq.1996.0083.

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