Symmetry and stability of asymptotic profiles for fast diffusion equations in annuli
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 6, pp. 1155-1173.

This paper is concerned with stability analysis of asymptotic profiles for (possibly sign-changing) solutions vanishing in finite time of the Cauchy–Dirichlet problems for fast diffusion equations in annuli. It is proved that the unique positive radial profile is not asymptotically stable, and moreover, it is unstable for the two-dimensional annulus. Furthermore, the method of stability analysis presented here will be also applied to exhibit symmetry breaking of least energy solutions.

DOI : https://doi.org/10.1016/j.anihpc.2013.08.006
Classification : 35K67,  35J61,  35B40,  35B35,  35B06
Mots clés : Fast diffusion equation, Semilinear elliptic equation, Asymptotic profile, Stability analysis, Symmetry breaking
@article{AIHPC_2014__31_6_1155_0,
     author = {Akagi, Goro and Kajikiya, Ryuji},
     title = {Symmetry and stability of asymptotic profiles for fast diffusion equations in annuli},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1155--1173},
     publisher = {Elsevier},
     volume = {31},
     number = {6},
     year = {2014},
     doi = {10.1016/j.anihpc.2013.08.006},
     zbl = {1332.35154},
     mrnumber = {3280064},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.006/}
}
TY  - JOUR
AU  - Akagi, Goro
AU  - Kajikiya, Ryuji
TI  - Symmetry and stability of asymptotic profiles for fast diffusion equations in annuli
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2014
DA  - 2014///
SP  - 1155
EP  - 1173
VL  - 31
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.006/
UR  - https://zbmath.org/?q=an%3A1332.35154
UR  - https://www.ams.org/mathscinet-getitem?mr=3280064
UR  - https://doi.org/10.1016/j.anihpc.2013.08.006
DO  - 10.1016/j.anihpc.2013.08.006
LA  - en
ID  - AIHPC_2014__31_6_1155_0
ER  - 
Akagi, Goro; Kajikiya, Ryuji. Symmetry and stability of asymptotic profiles for fast diffusion equations in annuli. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 6, pp. 1155-1173. doi : 10.1016/j.anihpc.2013.08.006. http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.006/

[1] G. Akagi, R. Kajikiya, Stability analysis of asymptotic profiles for sign-changing solutions to fast diffusion equations, Manuscr. Math. 141 (2013), 559 -587 | MR 3062598 | Zbl 1282.35050

[2] J.G. Berryman, C.J. Holland, Nonlinear diffusion problem arising in plasma physics, Phys. Rev. Lett. 40 (1978), 1720 -1722 | MR 495716

[3] J.G. Berryman, C.J. Holland, Stability of the separable solution for fast diffusion, Arch. Ration. Mech. Anal. 74 (1980), 379 -388 | MR 588035 | Zbl 0458.35046

[4] J.G. Berryman, C.J. Holland, Asymptotic behavior of the nonlinear diffusion equation n t =(n -1 n x ) x , J. Math. Phys. 23 (1982), 983 -987 | MR 659997 | Zbl 0487.35007

[5] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, J.L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal. 191 (2009), 347 -385 | MR 2481073 | Zbl 1178.35214

[6] M. Bonforte, J. Dolbeault, G. Grillo, J.L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proc. Natl. Acad. Sci. USA 107 (2010), 16459 -16464 | MR 2726546 | Zbl 1256.35026

[7] M. Bonforte, G. Grillo, J.L. Vázquez, Behaviour near extinction for the fast diffusion equation on bounded domains, J. Math. Pures Appl. 97 (2012), 1 -38 | MR 2863762 | Zbl 1241.35013

[8] M. Bonforte, G. Grillo, J.L. Vázquez, Fast diffusion flow on manifolds of nonpositive curvature, J. Evol. Equ. 8 (2008), 99 -128 | MR 2383484 | Zbl 1139.35065

[9] M. Bonforte, J.L. Vázquez, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations, Adv. Math. 223 (2010), 529 -578 | MR 2565541 | Zbl 1184.35083

[10] J. Byeon, Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli, J. Differ. Equ. 136 (1997), 136 -165 | MR 1443327 | Zbl 0878.35043

[11] C.V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differ. Equ. 54 (1984), 429 -437 | MR 760381 | Zbl 0569.35033

[12] E.N. Dancer, On the number of positive solutions of some weakly nonlinear equations on annular regions, Math. Z. 206 (1991), 551 -562 | EuDML 174243 | MR 1100839 | Zbl 0705.35043

[13] E. Feiresl, F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimension, J. Dyn. Differ. Equ. 12 (2000), 647 -673 | MR 1800136 | Zbl 0977.35069

[14] B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys. 68 (1979), 209 -243 | MR 544879 | Zbl 0425.35020

[15] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford (1979) | MR 568909 | Zbl 0423.10001

[16] Y.C. Kwong, Asymptotic behavior of a plasma type equation with finite extinction, Arch. Ration. Mech. Anal. 104 (1988), 277 -294 | MR 1017292 | Zbl 0703.35019

[17] Y.Y. Li, Existence of many positive solutions of semilinear elliptic equations in annulus, J. Differ. Equ. 83 (1990), 348 -367 | MR 1033192 | Zbl 0748.35013

[18] C.S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in 2 , Manuscr. Math. 84 (1994), 13 -19 | EuDML 155974 | MR 1283323 | Zbl 0807.35043

[19] W.-M. Ni, Uniqueness of solutions of nonlinear Dirichlet problems, J. Differ. Equ. 50 (1983), 289 -304 | MR 719451 | Zbl 0476.35033

[20] P. Rosenau, Fast and superfast diffusion processes, Phys. Rev. Lett. 74 (1995), 1056 -1059

[21] G. Savaré, V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations, Nonlinear Anal. 22 (1994), 1553 -1565 | MR 1285092 | Zbl 0821.35081

Cité par Sources :