Universal solutions of a nonlinear heat equation on ℝ N

Thierry Cazenave; Flávio Dickstein; Fred B. Weissler

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 1, page 77-117
  • ISSN: 0391-173X

Abstract

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In this paper, we study the relationship between the long time behavior of a solution u ( t , x ) of the nonlinear heat equation u t - Δ u + | u | α u = 0 on ℝ N (where α > 0 ) and the asymptotic behavior as | x | → ∞ of its initial value u 0 . In particular, we show that if the sequence of dilations λ n 2 / α u 0 ( λ n · ) converges weakly to z ( · ) as λ n → ∞ , then the rescaled solution t 1 / α u ( t , · t ) converges uniformly on ℝ N to 𝒰 ( 1 ) z along the subsequence t n = λ n 2 , where 𝒰 ( t ) 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 ∞ 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 α < 2 / N .

How to cite

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Cazenave, Thierry, Dickstein, Flávio, and Weissler, Fred B.. "Universal solutions of a nonlinear heat equation on $\mathbb {R}^N$." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.1 (2003): 77-117. <http://eudml.org/doc/84500>.

@article{Cazenave2003,
abstract = {In this paper, we study the relationship between the long time behavior of a solution $u(t,x)$ of the nonlinear heat equation $u_t-\Delta u+|u|^\alpha u=0$ on $\mathbb \{R\}^N$ (where $\alpha &gt;0$) and the asymptotic behavior as $|x|\rightarrow \infty $ of its initial value $u_0$. In particular, we show that if the sequence of dilations $\lambda _n^\{2/\alpha \} u_0(\lambda _n\cdot )$ converges weakly to $z(\cdot )$ as $\lambda _n\rightarrow \infty $, then the rescaled solution $t^\{1/\alpha \} u(t,\cdot \sqrt\{t\})$ converges uniformly on $\mathbb \{R\}^N $ to $\{\mathcal \{U\}\}(1)z$ along the subsequence $t_n=\lambda _n^2$, where $\{\mathcal \{U\}\}(t)$ 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 $\alpha &lt;2/N$.},
author = {Cazenave, Thierry, Dickstein, Flávio, Weissler, Fred B.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {77-117},
publisher = {Scuola normale superiore},
title = {Universal solutions of a nonlinear heat equation on $\mathbb \{R\}^N$},
url = {http://eudml.org/doc/84500},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Cazenave, Thierry
AU - Dickstein, Flávio
AU - Weissler, Fred B.
TI - Universal solutions of a nonlinear heat equation on $\mathbb {R}^N$
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 1
SP - 77
EP - 117
AB - In this paper, we study the relationship between the long time behavior of a solution $u(t,x)$ of the nonlinear heat equation $u_t-\Delta u+|u|^\alpha u=0$ on $\mathbb {R}^N$ (where $\alpha &gt;0$) and the asymptotic behavior as $|x|\rightarrow \infty $ of its initial value $u_0$. In particular, we show that if the sequence of dilations $\lambda _n^{2/\alpha } u_0(\lambda _n\cdot )$ converges weakly to $z(\cdot )$ as $\lambda _n\rightarrow \infty $, then the rescaled solution $t^{1/\alpha } u(t,\cdot \sqrt{t})$ converges uniformly on $\mathbb {R}^N $ to ${\mathcal {U}}(1)z$ along the subsequence $t_n=\lambda _n^2$, where ${\mathcal {U}}(t)$ 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 $\alpha &lt;2/N$.
LA - eng
UR - http://eudml.org/doc/84500
ER -

References

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  1. [1] C. Bardos – L. Tartar, Sur l’unicité rétrograde des équations paraboliques et quelques questions voisines, Arch. Rational Mech. Anal. 50 (1973), 10-25. Zbl0258.35039MR338517
  2. [2] H. Brezis – A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), 73-97. Zbl0527.35043MR700049
  3. [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. Zbl0627.35046MR853963
  4. [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. Zbl0996.35031MR1855457
  5. [5] T. Cazenave – F. Dickstein – F. B. Weissler, Universal solutions of the heat equation, preprint 2001. 
  6. [6] T. Cazenave – F. Dickstein – F. B. Weissler, Chaotic behavior of solutions of the Navier-Stokes system in ℝ N , in preparation. Zbl1208.35101
  7. [7] T. Cazenave – F. B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z. 228 (1998), 83-120. Zbl0916.35109MR1617975
  8. [8] R. L. Devaney, “Overview: Dynamics of Simple Maps, in Chaos and Fractals”, Proc. Symp. Appl. Math. 39, Providence, 1989. MR1010233
  9. [9] M. Escobedo – O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal. 11 (1987), 1103-1133. Zbl0639.35038MR913672
  10. [10] M. Escobedo – O. Kavian, Asymptotic behavior of positive solutions of a nonlinear heat equation, Houston J. Math. 13 (1987), 39-50. Zbl0666.35046MR959221
  11. [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. Zbl0838.35015MR1335757
  12. [12] J-M. Ghidaglia, Some backward uniqueness results, Nonlinear Anal. 10 (1986), 777-790. Zbl0622.35029MR851146
  13. [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. Zbl0529.35041MR748242
  14. [14] A. Haraux, “Systèmes dynamiques dissipatifs et applications”, R.M.A. 17, P. G. Ciarlet et J.-L. Lions (eds.), Masson, Paris, 1991. Zbl0726.58001MR1084372
  15. [15] L. Herraiz, Asymptotic behaviour of solutions of some semilinear parabolic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), 49-105. Zbl0918.35025MR1668560
  16. [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. Zbl0598.35050MR837255
  17. [17] M. Kwak, A semilinear heat equation with singular initial data, Proc. Royal Soc. Edinburgh Sect. A 128 (1998), 745-758. Zbl0909.35068MR1635420
  18. [18] M. Marcus – L. Véron, Initial trace of positive solutions of some nonlinear parabolic equations, Comm. Partial Differential Equations 24 (1999), 1445-1499. Zbl1059.35054MR1697494
  19. [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. Zbl1002.35020MR1924144
  20. [20] C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Rational Mech. Anal. 138 (1997), 279-306. Zbl0882.35061MR1465095
  21. [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. Zbl0604.34034MR806004

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