# Article

 Title: On blow-up and asymptotic behavior of solutions to a nonlinear parabolic equation of second order with nonlinear boundary conditions (English) Author: Boni, Théodore K. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 40 Issue: 3 Year: 1999 Pages: 457-475 . Category: math . Summary: We obtain some sufficient conditions under which solutions to a nonlinear parabolic equation of second order with nonlinear boundary conditions tend to zero or blow up in a finite time. We also give the asymptotic behavior of solutions which tend to zero as $t\rightarrow\infty$. Finally, we obtain the asymptotic behavior near the blow-up time of certain blow-up solutions and describe their blow-up set. (English) Keyword: blow-up Keyword: global existence Keyword: asymptotic behavior Keyword: maximum principle MSC: 35B40 MSC: 35K55 MSC: 35K60 idZBL: Zbl 1011.35078 idMR: MR1732489 . Date available: 2009-01-08T18:54:11Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119102 . Reference: [1] Boni T.K.: Sur l'explosion et le comportement asymptotique de la solution d'une équation parabolique semi-linéaire du second ordre.C.R. Acad. Paris, t. 326, Série I, 1 (1998), 317-322. Zbl 0913.35069, MR 1648453 Reference: [2] Chipot M., Fila M., Quittner P.: Stationary solutions, blow-up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions.Acta Math. Univ. Comenianae, Vol. LX, 1 (1991), 35-103. Zbl 0743.35038, MR 1120596 Reference: [3] Egorov Yu.V., Kondratiev V.A.: On blow-up solutions for parabolic equations of second order.in `Differential Equations, Asymptotic Analysis and Mathematical Physics', Berlin, Academie Verlag, 1997, pp.77-84. Zbl 0879.35081, MR 1456179 Reference: [4] Friedman A., McLeod B.: Blow-up of positive solutions of semilinear heat equations.Indiana Univ. Math. J. 34 (1985), 425-447. Zbl 0576.35068, MR 0783924 Reference: [5] Protter M.H., Weinberger H.F.: Maximum Principles in Differential Equations.Prentice Hall, Englewood Cliffs, NJ, 1967. Zbl 0549.35002, MR 0219861 Reference: [6] Rossi J.D.: The blow-up rate for a semilinear parabolic equation with a nonlinear boundary condition.Acta Math. Univ. Comenianae, Vol. LXVII, 2 (1998), 343-350. Zbl 0924.35017, MR 1739446 Reference: [7] Walter W.: Differential-und Integral-Ungleichungen.Springer, Berlin, 1964. Zbl 0119.12205, MR 0172076 .

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