Self-similar solutions for a semilinear heat equation with critical Sobolev exponent

Indiana University Mathematics Journal

Volumn 57, Issue 3, 2008, Pages 1283-1315

  Naito, Yuki ^a  

^a KOBE UNIVERSITY   (Japan)

Author keywords

Critical Sobolev exponent; ODE approach; Self similar solution; Semilinear heat equation; Variational method

Indexed keywords

EID: 49549085530     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2008.57.3279     Document Type: Article

References (19)

1. H. BREZIS and T. KATO, Remarks on the Schrodinger operator with singular complex potentials, J. Math. Pures Appl. 58 (1979), 137-151.
2. H. BREZIS and E. LIEB, A relation between pointwise convergence of functions and convergence of functionah, Proc. Amer. Math. Soc. 88 (1983), 486-490.
3. H. BREZIS and L. NIRENBERG, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477.
4. T. CAZENAVE and F. B. WEISSLER, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z. 228 (1998), 83-120.
5. M. ESCOBEDO and O. KAVIAN, Variational problems related to self-similar solutions for the beat equation, Nonlinear Anal. TMA . 11 (1987), 1103-1133.
6. V.A. GALAKTIONOV and J.L. VAZQUEZ, Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), 1-67.
7. A. HARAUX and F. B. WEISSLER, Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982), 167-189.
8. O. KAVIAN, Remarks on the large time behavior of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 423-452.
9. T. KAWANAGO, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 1-15.
10. H. KOZONO and M. YAMAZAKI, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations 19 (1994), 959-1014.
11. Y. NAITO, Non-uniqueness of solutions to the Cauchy problem for semilinear heat equations with singular initial data, Math. Ann. 329 (2004), 161-196.
12. An ODE approach to the multiplicity of self-similar solutions for semi-linear heat equations, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 807-835.
13. A variational approach to self-similar solutions for semilinear heat equations, Asymptotic Analysis and Singularity, Advanced Studies in Pure Mathematics, vol. 47-2, 2007, pp. 675-688.
14. Y. NAITO and T. SUZUKI, Radial symmetry of self-similar solutions for semilinear heat equations, J. Differential Equations 163 (2000), 407-428.
15. R. S. PALAIS, The principle of symmetric criticality, Commun. Math. Phys. 69 (1979), 19-30.
16. L. A. PELETIER, D. TERMAN, and F. B. WEISSLER, On the equation Δu + 1/2 x- ℕu + f (u) = 0, Arch. Rational Mech. Anal. 94 (1986), 83-99.
17. P. SOUPLET and F. B. WEISSLER, Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state, Ann. Inst. H. Poincaré Anal. Non Lineaire 20 (2003), 213-235.
18. M. STRUWE, Variational Methods, Springer-Verlag, Berlin, 1996.
19. M. WILLEM, Minimax Theorems, Birkhauser, 1996.