# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - S, 2012, 5 (3) : 449-472. doi: 10.3934/dcdss.2012.5.449
##### References:
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Equ., 7 (2007), 471. Google Scholar [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. Google Scholar [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. doi: 10.3934/dcds.2003.9.1105. Google Scholar [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. Google Scholar [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. Google Scholar [11] T. Cazenave, F. Dickstein and F. B. Weissler, A solution of the heat equation with a continuum of decay rates,, in, 63 (2005), 135. Google Scholar [12] T. Cazenave, F. Dickstein and F. B. Weissler, Multiscale asymptotic behavior of a solution of the heat equation in $\R^N$,, in, 66 (2006), 185. Google Scholar [13] T. Cazenave, F. Dickstein and F. B. Weissler, A solution of the constant coefficient heat equation on $\R$ with exceptional asymptotic behavior: An explicit constuction,, J. Math. Pures Appl. (9), 85 (2006), 119. doi: 10.1016/j.matpur.2005.08.006. Google Scholar [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. doi: 10.1007/s10884-007-9076-z. Google Scholar [15] T. Cazenave and F. B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations,, Math. Z., 228 (1998), 83. doi: 10.1007/PL00004606. Google Scholar [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. doi: 10.1512/iumj.2006.55.2664. Google Scholar [17] R. L. Devaney, Overview: Dynamics of Simple Maps,, in, 39 (1988), 1. Google Scholar [18] C. Dohmen and M. Hirose, Structure of positive radial solutions to the Haraux-Weissler equation,, Nonlinear Anal., 33 (1998), 51. doi: 10.1016/S0362-546X(97)00542-7. Google Scholar [19] H. Emamirad, G. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation,, Proc. Amer. Math. Soc, (2011), 0002. Google Scholar [20] M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation,, Nonlinear Anal., 11 (1987), 1103. doi: 10.1016/0362-546X(87)90001-0. Google Scholar [21] M. Escobedo and O. Kavian, Asymptotic behavior of positive solutions of a nonlinear heat equation,, Houston J. Math., 14 (1988), 39. Google Scholar [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. Google Scholar [23] D. Fang, J. Xie and T. Cazenave, Multiscale asymptotic behavior of the Schrödinger equation,, Funk. Ekva., 54 (2011), 69. doi: 10.1619/fesi.54.69. Google Scholar [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. Google Scholar [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 (2010). Google Scholar [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. Google Scholar [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. doi: 10.1016/0022-0396(84)90042-1. Google Scholar [28] A. Haraux and F. B. Weissler, Nonuniqueness for a semilinear initial value problem,, Indiana Univ. Math. J., 31 (1982), 167. doi: 10.1512/iumj.1982.31.31016. Google Scholar [29] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations,, Proc. Japan Acad., 49 (1973), 503. doi: 10.3792/pja/1195519254. Google Scholar [30] L. Herraiz, Asymptotic behaviour of solutions of some semilinear parabolic problems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 49. Google Scholar [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. Google Scholar [32] O. Kavian, Remarks on the time behaviour of a nonlinear diffusion equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 423. Google Scholar [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. Google Scholar [34] K. Kobayashi, T. Sirao and H. Tanaka, On the growing up problem for semilinear heat equations,, J. Math. Soc. Japan, 29 (1977), 407. doi: 10.2969/jmsj/02930407. Google Scholar [35] M. Kwak, A semilinear heat equation with singular initial data,, Proc. Royal Soc. Edinburgh Sect. A, 128 (1998), 745. Google Scholar [36] M. Marcus and L. Véron, Initial trace of positive solutions of some nonlinear parabolic equations,, Comm. Partial Differential Equations, 24 (1999), 1445. Google Scholar [37] F. Ribaud, "Analyse de Littlewood Paley pour la Résolution d'Équations Paraboliques Semi-Linéaires,", Ph.D Thesis, (1996). Google Scholar [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. Google Scholar [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. doi: 10.1007/PL00004498. Google Scholar [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. doi: 10.1142/S0252959902000274. Google Scholar [41] C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains,, Arch. Ration. Mech. Anal., 138 (1997), 279. doi: 10.1007/s002050050042. Google Scholar [42] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,, Israel J. Math., 38 (1981), 29. doi: 10.1007/BF02761845. Google Scholar [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. doi: 10.1007/BF00250744. Google Scholar [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. doi: 10.1007/BF00250743. Google Scholar [45] J. Xie, L. Zhang and T. Cazenave, A note on decay rates for Schrödinger's equation,, Proc. Amer. Math. Soc., 138 (2010), 199. doi: 10.1090/S0002-9939-09-10049-7. Google Scholar [46] E. Yanagida, Uniqueness of rapidly decaying solutions to the Haraux-Weissler equation,, J. Differential Equations, 127 (1996), 561. doi: 10.1006/jdeq.1996.0083. Google Scholar

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. doi: 10.1016/0001-8708(78)90130-5. Google Scholar [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. doi: 10.1007/BF00251357. Google Scholar [3] J. Bricmont and A. Kupiainen, Stable non-Gaussian diffusive profiles,, Nonlinear Anal., 26 (1996), 583. doi: 10.1016/0362-546X(94)00300-7. Google Scholar [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. doi: 10.1002/cpa.3160470606. Google Scholar [5] M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes,, in, (1994), 1993. Google Scholar [6] J. A. Carrillo and J. L. Vázquez, Asymptotic complexity in filtration equations,, J. Evol. Equ., 7 (2007), 471. Google Scholar [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. Google Scholar [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. doi: 10.3934/dcds.2003.9.1105. Google Scholar [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. Google Scholar [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. Google Scholar [11] T. Cazenave, F. Dickstein and F. B. Weissler, A solution of the heat equation with a continuum of decay rates,, in, 63 (2005), 135. Google Scholar [12] T. Cazenave, F. Dickstein and F. B. Weissler, Multiscale asymptotic behavior of a solution of the heat equation in $\R^N$,, in, 66 (2006), 185. Google Scholar [13] T. Cazenave, F. Dickstein and F. B. Weissler, A solution of the constant coefficient heat equation on $\R$ with exceptional asymptotic behavior: An explicit constuction,, J. Math. Pures Appl. (9), 85 (2006), 119. doi: 10.1016/j.matpur.2005.08.006. Google Scholar [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. doi: 10.1007/s10884-007-9076-z. Google Scholar [15] T. Cazenave and F. B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations,, Math. Z., 228 (1998), 83. doi: 10.1007/PL00004606. Google Scholar [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. doi: 10.1512/iumj.2006.55.2664. Google Scholar [17] R. L. Devaney, Overview: Dynamics of Simple Maps,, in, 39 (1988), 1. Google Scholar [18] C. Dohmen and M. Hirose, Structure of positive radial solutions to the Haraux-Weissler equation,, Nonlinear Anal., 33 (1998), 51. doi: 10.1016/S0362-546X(97)00542-7. Google Scholar [19] H. Emamirad, G. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation,, Proc. Amer. Math. Soc, (2011), 0002. Google Scholar [20] M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation,, Nonlinear Anal., 11 (1987), 1103. doi: 10.1016/0362-546X(87)90001-0. Google Scholar [21] M. Escobedo and O. Kavian, Asymptotic behavior of positive solutions of a nonlinear heat equation,, Houston J. Math., 14 (1988), 39. Google Scholar [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. Google Scholar [23] D. Fang, J. Xie and T. Cazenave, Multiscale asymptotic behavior of the Schrödinger equation,, Funk. Ekva., 54 (2011), 69. doi: 10.1619/fesi.54.69. Google Scholar [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. Google Scholar [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 (2010). Google Scholar [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. Google Scholar [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. doi: 10.1016/0022-0396(84)90042-1. Google Scholar [28] A. Haraux and F. B. Weissler, Nonuniqueness for a semilinear initial value problem,, Indiana Univ. Math. J., 31 (1982), 167. doi: 10.1512/iumj.1982.31.31016. Google Scholar [29] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations,, Proc. Japan Acad., 49 (1973), 503. doi: 10.3792/pja/1195519254. Google Scholar [30] L. Herraiz, Asymptotic behaviour of solutions of some semilinear parabolic problems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 49. Google Scholar [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. Google Scholar [32] O. Kavian, Remarks on the time behaviour of a nonlinear diffusion equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 423. Google Scholar [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. Google Scholar [34] K. Kobayashi, T. Sirao and H. Tanaka, On the growing up problem for semilinear heat equations,, J. Math. Soc. Japan, 29 (1977), 407. doi: 10.2969/jmsj/02930407. Google Scholar [35] M. Kwak, A semilinear heat equation with singular initial data,, Proc. Royal Soc. Edinburgh Sect. A, 128 (1998), 745. Google Scholar [36] M. Marcus and L. Véron, Initial trace of positive solutions of some nonlinear parabolic equations,, Comm. Partial Differential Equations, 24 (1999), 1445. Google Scholar [37] F. Ribaud, "Analyse de Littlewood Paley pour la Résolution d'Équations Paraboliques Semi-Linéaires,", Ph.D Thesis, (1996). Google Scholar [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. Google Scholar [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. doi: 10.1007/PL00004498. Google Scholar [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. doi: 10.1142/S0252959902000274. Google Scholar [41] C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains,, Arch. Ration. Mech. Anal., 138 (1997), 279. doi: 10.1007/s002050050042. Google Scholar [42] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,, Israel J. Math., 38 (1981), 29. doi: 10.1007/BF02761845. Google Scholar [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. doi: 10.1007/BF00250744. Google Scholar [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. doi: 10.1007/BF00250743. Google Scholar [45] J. Xie, L. Zhang and T. Cazenave, A note on decay rates for Schrödinger's equation,, Proc. Amer. Math. Soc., 138 (2010), 199. doi: 10.1090/S0002-9939-09-10049-7. Google Scholar [46] E. Yanagida, Uniqueness of rapidly decaying solutions to the Haraux-Weissler equation,, J. Differential Equations, 127 (1996), 561. doi: 10.1006/jdeq.1996.0083. Google Scholar
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