Universal solutions of a nonlinear heat equation on ${ℝ}^{N}$
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, p. 77-117

In this paper, we study the relationship between the long time behavior of a solution $u\left(t,x\right)$ of the nonlinear heat equation ${u}_{t}-\Delta u+{|u|}^{\alpha }u=0$ on ${ℝ}^{N}$ (where $\alpha >0$) and the asymptotic behavior as $|x|\to \infty$ of its initial value ${u}_{0}$. In particular, we show that if the sequence of dilations ${\lambda }_{n}^{2/\alpha }{u}_{0}\left({\lambda }_{n}·\right)$ converges weakly to $z\left(·\right)$ as ${\lambda }_{n}\to \infty$, then the rescaled solution ${t}^{1/\alpha }u\left(t,·\sqrt{t}\right)$ converges uniformly on ${ℝ}^{N}$ to $𝒰\left(1\right)z$ along the subsequence ${t}_{n}={\lambda }_{n}^{2}$, where $𝒰\left(t\right)$ is an appropriate flow. Moreover, we show there exists an initial value ${U}_{0}$ such that the set of all possible $z$ attainable in this fashion is a closed ball $B$ of a weighted ${L}^{\infty }$ space. The resulting “universal” solution is therefore asymptotically close along appropriate subsequences to all solutions with initial values in $B$. These results are restricted to positive solutions in the case $\alpha <2/N$.

Classification:  35K55,  35B40
@article{ASNSP_2003_5_2_1_77_0,
author = {Cazenave, Thierry and Dickstein, Fl\'avio and Weissler, Fred B.},
title = {Universal solutions of a nonlinear heat equation on $\mathbb {R}^N$},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 2},
number = {1},
year = {2003},
pages = {77-117},
zbl = {1170.35448},
mrnumber = {1990975},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2003_5_2_1_77_0}
}

Cazenave, Thierry; Dickstein, Flávio; Weissler, Fred B. Universal solutions of a nonlinear heat equation on $\mathbb {R}^N$. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, pp. 77-117. http://www.numdam.org/item/ASNSP_2003_5_2_1_77_0/

[1] C. Bardos - L. Tartar, Sur l'unicité rétrograde des équations paraboliques et quelques questions voisines, Arch. Rational Mech. Anal. 50 (1973), 10-25. | MR 338517 | Zbl 0258.35039

[2] H. Brezis - A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), 73-97. | MR 700049 | Zbl 0527.35043

[3] H. Brezis - L. A. Peletier - D. Terman, A very singular solution of the heat equation with absorption, Arch. Rational Mech. Anal. 95 (1986), 185-209. | MR 853963 | Zbl 0627.35046

[4] T. Cazenave - F. Dickstein - M. Escobedo - F. B. Weissler, Self-similar solutions of a nonlinear heat equation, J. Math. Sci. Univ. Tokyo 8 (2001), 501-540. | MR 1855457 | Zbl 0996.35031

[5] T. Cazenave - F. Dickstein - F. B. Weissler, Universal solutions of the heat equation, preprint 2001. | Zbl 1029.35105

[6] T. Cazenave - F. Dickstein - F. B. Weissler, Chaotic behavior of solutions of the Navier-Stokes system in ${ℝ}^{N}$, in preparation. | Zbl 1208.35101 | Zbl pre05056153

[7] T. Cazenave - F. B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z. 228 (1998), 83-120. | MR 1617975 | Zbl 0916.35109

[8] R. L. Devaney, “Overview: Dynamics of Simple Maps, in Chaos and Fractals”, Proc. Symp. Appl. Math. 39, Providence, 1989. | MR 1010233

[9] M. Escobedo - O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal. 11 (1987), 1103-1133. | MR 913672 | Zbl 0639.35038

[10] M. Escobedo - O. Kavian, Asymptotic behavior of positive solutions of a nonlinear heat equation, Houston J. Math. 13 (1987), 39-50. | MR 959221 | Zbl 0666.35046

[11] M. Escobedo - O. Kavian - H. Matano, Large time behavior of solutions of a dissipative semi-linear heat equation, Comm. Partial Differential Equations 20 (1995), 1427-1452. | MR 1335757 | Zbl 0838.35015

[12] J-M. Ghidaglia, Some backward uniqueness results, Nonlinear Anal. 10 (1986), 777-790. | MR 851146 | Zbl 0622.35029

[13] A. Gmira - L. Véron, Large time behaviour of the solutions of a semilinear parabolic equation in ${ℝ}^{N}$, J. Differential Equations 53 (1984), 258-276. | MR 748242 | Zbl 0529.35041

[14] A. Haraux, “Systèmes dynamiques dissipatifs et applications”, R.M.A. 17, P. G. Ciarlet et J.-L. Lions (eds.), Masson, Paris, 1991. | MR 1084372 | Zbl 0726.58001

[15] L. Herraiz, Asymptotic behaviour of solutions of some semilinear parabolic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), 49-105. | Numdam | MR 1668560 | Zbl 0918.35025

[16] S. Kamin - 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. | Numdam | MR 837255 | Zbl 0598.35050

[17] M. Kwak, A semilinear heat equation with singular initial data, Proc. Royal Soc. Edinburgh Sect. A 128 (1998), 745-758. | MR 1635420 | Zbl 0909.35068

[18] M. Marcus - L. Véron, Initial trace of positive solutions of some nonlinear parabolic equations, Comm. Partial Differential Equations 24 (1999), 1445-1499. | MR 1697494 | Zbl 1059.35054

[19] J. L. Vázquez - E. Zuazua, Complexity of large time behaviour of evolution equations with bounded data, Chinese Ann. Math., Ser. B, 23 (2002), 293-310. | MR 1924144 | Zbl 1002.35020

[20] C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Rational Mech. Anal. 138 (1997), 279-306. | MR 1465095 | Zbl 0882.35061

[21] F. B. Weissler, Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations, Arch. Rational Mech. Anal. 91 (1986), 247-266. | MR 806004 | Zbl 0604.34034