zbMATH — the first resource for mathematics

A bound for global solutions of semilinear heat equations. (English) Zbl 0595.35057
The author proves that every global (classical) solution of equation \(u_ t=\Delta u+u^ p\) in a bounded domain in \(R^ n\) is bounded by a constant depending only on the sup-norm of the initial data if \(n/2<(p+1)/(p-1).\)
Reviewer: B.Nowak

35K55 Nonlinear parabolic equations
35K25 Higher-order parabolic equations
35B35 Stability in context of PDEs
35K15 Initial value problems for second-order parabolic equations
Full Text: DOI
[1] Cazenave, T., Lions, P. L.: Solutions globales d’équation de la chaleur semilinéaires. Commun. Partial Differ. Equations9, 955-978 (1984) · Zbl 0555.35067
[2] Gidas, B. and Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Commun. Partial Differ. Equations6, 883-901 (1981) · Zbl 0462.35041
[3] Ladyzhenskaya, O. A., Solonnikov, V. A. Ural’ceva, N. N.: Linear and quasilinear equations of parabolic types. Transl. Math. Monogr. vol.23, Providence, R.I.: Am. Math. Soc. 1968
[4] Lions, P. L.: Asymptotic behavior of some nonlinear heat equations. Nonlinear phenomena. Physica5D. 293-306 (1982) · Zbl 1194.35459
[5] Ni, W. M., Sacks, P. E., Tavantzis, J.: On the asymptotic behavior of solutions of certain quasilinear equation of parabolic type. J. Differ. Equations54, 97-120 (1984) · Zbl 0565.35053
[6] Weissler, F.:L p -energy and blow-up for a semilinear heat equation, Proc. AMS Summer Inst. 1983 (to appear)
[7] Weissler, F.: Local existence and nonexistence for semilinear parabolic equations inL p . Indiana Univ. Math. J.29, 79-102 (1980) · Zbl 0443.35034
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.