Abstract
We study the behaviour of solutions of the Cauchy problem for a supercritical semilinear parabolic equation which converge to zero from above as t → ∞. We show that any algebraic decay rate slower than the self-similar one occurs for some initial data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz M. and Stegun I. (1972). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Wiley, New York
Fila M., Winkler M. and Yanagida E. (2004). Grow-up rate of solutions for a supercritical semilinear diffusion equation. J. Differ. Equ. 205: 365–389
Fila M., Winkler M. and Yanagida E. (2005). Convergence rate for a parabolic equation with supercritical nonlinearity. J. Dyn. Differ. Equ. 17: 249–269
Fila M., King J., Winkler M. and Yanagida E. (2006). Optimal lower bound of the grow-up rate for a supercritical parabolic equation. J. Differ. Equ. 228: 339–356
Fujita H. (1966). On the blowing up of solutions of the Cauchy problem for u t = Δu + u 1+α. J. Fac. Sci. Univ. Tokyo Sect. I 13: 109–124
Galaktionov V. and Vázquez J.L. (1997). Continuation of blow-up solutions of nonlinear heat equations in several space dimensions. Commun. Pure Appl. Math. 50: 1–67
Gui C., Ni W.-M. and Wang X. (1992). On the stability and instability of positive steady states of a semilinear heat equation in \({\mathbb{R}}^n\) Commun. Pure Appl. Math. 45: 1153–1181
Gui C., Ni W.-M. and Wang X. (2001). Further study on a nonlinear heat equation. J. Differ. Equ. 169: 588–613
Lee T.Y. and Ni W.M. (1992). Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem. Trans. Am. Math. Soc. 333: 365–378
Li Y. (1992). Asymptotic behavior of positive solutions of equation Δu + K(x)u p = 0 in R n. J. Differ. Equ. 95: 304–330
Poláčik P., Quittner P. and Souplet P. (2007). Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: Parabolic equations. Indiana Univ. Math. J. 56: 879–908
Poláčik P. and Yanagida E. (2003). On bounded and unbounded global solutions of a supercritical semilinear heat equation. Math. Ann. 327: 745–771
Poláčik P. and Yanagida E. (2005). A Liouville property and quasiconvergence for a semilinear heat equation. J. Differ. Equ. 208: 194–214
Quittner, P., Souplet, P.: Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States. Birkhäuser Advanced Texts (2007)
Souplet P. and Weissler F. (2003). Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state. Ann. Inst. H. Poincaré Analyse non linéaire 20: 213–235
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fila, M., Winkler, M. & Yanagida, E. Slow convergence to zero for a parabolic equation with a supercritical nonlinearity. Math. Ann. 340, 477–496 (2008). https://doi.org/10.1007/s00208-007-0148-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-007-0148-5