Semilinear parabolic equation in 𝐑 N associated with critical Sobolev exponent
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 877-900.

We consider the semilinear parabolic equation u t -Δu=|u| p-1 u on the whole space 𝐑 N , N⩾3, where the exponent p=(N+2)/(N-2) is associated with the Sobolev imbedding H 1 (𝐑 N )⊂L p+1 (𝐑 N ). First, we study the decay and blow-up of the solution by means of the potential-well and forward self-similar transformation. Then, we discuss blow-up in infinite time and classify the orbit.

DOI : 10.1016/j.anihpc.2010.01.002
Classification : 35K55
Mots-clés : Parabolic equation, Critical Sobolev exponent, Cauchy problem, Stable and unstable sets, Self-similarity 
@article{AIHPC_2010__27_3_877_0,
     author = {Ikehata, Ryo and Ishiwata, Michinori and Suzuki, Takashi},
     title = {Semilinear parabolic equation in $ {\mathbf{R}}^{N}$ associated with critical {Sobolev} exponent},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {877--900},
     publisher = {Elsevier},
     volume = {27},
     number = {3},
     year = {2010},
     doi = {10.1016/j.anihpc.2010.01.002},
     mrnumber = {2629884},
     zbl = {1192.35099},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.002/}
}
TY  - JOUR
AU  - Ikehata, Ryo
AU  - Ishiwata, Michinori
AU  - Suzuki, Takashi
TI  - Semilinear parabolic equation in $ {\mathbf{R}}^{N}$ associated with critical Sobolev exponent
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 877
EP  - 900
VL  - 27
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.002/
DO  - 10.1016/j.anihpc.2010.01.002
LA  - en
ID  - AIHPC_2010__27_3_877_0
ER  - 
%0 Journal Article
%A Ikehata, Ryo
%A Ishiwata, Michinori
%A Suzuki, Takashi
%T Semilinear parabolic equation in $ {\mathbf{R}}^{N}$ associated with critical Sobolev exponent
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 877-900
%V 27
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.002/
%R 10.1016/j.anihpc.2010.01.002
%G en
%F AIHPC_2010__27_3_877_0
Ikehata, Ryo; Ishiwata, Michinori; Suzuki, Takashi. Semilinear parabolic equation in $ {\mathbf{R}}^{N}$ associated with critical Sobolev exponent. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 877-900. doi : 10.1016/j.anihpc.2010.01.002. http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.002/

[1] L.A. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271-297 | MR | Zbl

[2] T. Cazenave, P.L. Lions, Solutions globales d'equations de la shaleur semi lineaires, Comm. Partial Differential Equations 9 (1984), 955-978 | MR | Zbl

[3] W. Chen, C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615-622 | MR | Zbl

[4] E.B. Davis, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge (1989) | MR | Zbl

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

[6] M. Escobedo, E. Zuazua, Large time behavior for convection–diffusion equations in R N , J. Funct. Anal. 100 (1991), 119-161 | MR | Zbl

[7] V. Georgiev, Semilinear Hyperbolic Equations, MSJ Mem. vol. 7, Math. Soc. Japan (2000) | MR | Zbl

[8] R. Ikehata, The Palais–Smale condition for the energy of some semilinear parabolic equations, Hiroshima Math. J. 30 (2000), 117-127 | MR | Zbl

[9] R. Ikehata, T. Suzuki, Semilinear parabolic equations involving critical Sobolev exponent: Local and asymptotic behavior of solutions, Differential Integral Equations 13 (2000), 437-477 | MR | Zbl

[10] K. Ishige, T. Kawakami, Asymptotic behavior of solutions for some semilinear heat equations in 𝐑 N , preprint | MR

[11] M. Ishiwata, Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent, Discrete Contin. Dyn. Syst. (2005), 443-452 | MR | Zbl

[12] M. Ishiwata, On the asymptotic behavior of radial positive solutions for semilinear parabolic problem involving critical Sobolev exponent, in preparation

[13] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York (1976) | MR

[14] O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré 4 (1987), 423-452 | EuDML | Numdam | MR | Zbl

[15] T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity, Ann. Inst. H. Poincaré 13 (1996), 1-15 | EuDML | Numdam | MR | Zbl

[16] T. Kawanago, Existence and behavior of solutions for u t =Δu m +u ℓ , Adv. Math. Sci. Appl. 7 (1997), 367-400 | MR | Zbl

[17] V. Komornik, Exact Controllability and Stabilization, Multiplier Method, Masson, Paris (1994) | MR | Zbl

[18] M. Otani, Existence and asymptotic stability of strong solutions of nonlinear evolution equations with a difference term of subdifferentials, Colloq. Math. Soc. Janos Bolyai, Qualitative Theory of Differential Equations, vol. 30, North-Holland, Amsterdam (1980) | Zbl

[19] M. Otani, L ∞ -energy method and its applications, nonlinear partial differential equations and their applications, GAKUTO Internat. Ser. Math. Sci. Appl. 20 (2004), 505-516 | MR | Zbl

[20] L.E. Payne, D.H. Sattinger, Saddle points and unstability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), 273-303 | MR | Zbl

[21] P. Poláčik, private communication

[22] D.H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 30 (1968), 148-172 | MR | Zbl

[23] T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. J. 57 no. 7 (2008), 3365-3396 | MR | Zbl

[24] H. Tanabe, Equations of Evolution, Pitman, London (1979) | MR

Cité par Sources :