zbMATH — the first resource for mathematics

Linear behaviour of solutions of a superlinear heat equation. (English) Zbl 1263.35137
Summary: We give sufficient conditions which guarantee that solutions of a superlinear heat equation decay to zero at the same rate as the solutions of the linear heat equation with the same initial data. This improves a previous result of T.-Y. Lee and W.-M. Ni [Trans. Am. Math. Soc. 333, No. 1, 365–378 (1992; Zbl 0785.35011)] by showing that this behaviour holds for a significantly larger, and rather tightly defined class of solutions.

35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
[1] Dohmen, C.; Hirose, M., Structure of positive radial solutions to the haraux – weissler equation, Nonlinear anal., 33, 51-69, (1998) · Zbl 0934.34028
[2] Fila, M.; King, J.; Winkler, M.; Yanagida, E., Optimal lower bound of the grow-up rate for a supercritical parabolic equation, J. differential equations, 228, 339-356, (2006) · Zbl 1102.35017
[3] Fila, M.; Winkler, M.; Yanagida, E., Grow-up rate of solutions for a supercritical semilinear diffusion equation, J. differential equations, 205, 365-389, (2004) · Zbl 1072.35042
[4] Fila, M.; Winkler, M.; Yanagida, E., Convergence rate for a parabolic equation with supercritical nonlinearity, J. dynam. differential equations, 17, 249-269, (2005) · Zbl 1087.35049
[5] M. Fila, M. Winkler, E. Yanagida, Slow convergence to zero for a parabolic equation with supercritical nonlinearity, Math. Ann., in press · Zbl 1263.35138
[6] Fujita, H., On the blowing up of solutions of the Cauchy problem for \(u_t = \operatorname{\Delta} u + u^{1 + \alpha}\), J. fac. sci. univ. Tokyo sect. I, 13, 109-124, (1966) · Zbl 0163.34002
[7] Galaktionov, V.; Vázquez, J.L., Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. pure appl. math., 50, 1-67, (1997) · Zbl 0874.35057
[8] Gui, C.; Ni, W.-M.; Wang, X., Further study on a nonlinear heat equation, J. differential equations, 169, 588-613, (2001) · Zbl 0974.35056
[9] Haraux, A.; Weissler, F.B., Nonuniqueness for a semilinear initial value problem, Indiana univ. math. J., 31, 167-189, (1982) · Zbl 0465.35049
[10] Lee, T.-Y.; Ni, W.-M., Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. amer. math. soc., 333, 365-378, (1992) · Zbl 0785.35011
[11] Poláčik, P.; Quittner, P.; Souplet, P., Singularity and decay estimates in superlinear problems via Liouville-type theorems. part II: parabolic equations, Indiana univ. math. J., 56, 879-908, (2007) · Zbl 1122.35051
[12] Poláčik, P.; Yanagida, E., A Liouville property and quasiconvergence for a semilinear heat equation, J. differential equations, 208, 194-214, (2005) · Zbl 1068.35044
[13] Souplet, P.; Weissler, F., Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state, Ann. inst. H. Poincaré anal. non linéaire, 20, 213-235, (2003) · Zbl 1029.35106
[14] Suzuki, R., Asymptotic behavior of solutions of quasilinear parabolic equations with slowly decaying initial data, Adv. math. sci. appl., 9, 291-317, (1999) · Zbl 0930.35089
[15] Wang, X., On the Cauchy problem for reaction – diffusion equations, Trans. amer. math. soc., 337, 549-590, (1993) · Zbl 0815.35048
[16] Yanagida, E., Uniqueness of rapidly decaying solutions to the haraux – weissler equation, J. differential equations, 127, 561-570, (1996) · Zbl 0856.34058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.