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Upper and lower blow-up rate estimates of a semilinear heat equation with a nonlinear boundary condition

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pp. 163–175Vol. 26Issue 2March 2023DOI: 10.47974/JIM-1302XML
Received:
05 Mar 2021
Published Online:
01 Mar 2023
Article type:
A
Language:
EN
Article no.:
JIM-1302
Pages:
163–175

Abstract

In this paper, we consider the blow-up solutions of a semi-linear heat equation with a Neumann boundary condition, where the nonlinear terms, appear in the differential equation and in the boundary conditions, are of exponential types. We study the effect of the nonlinear terms on the last blow-up behavior. Precisely, the upper (lower) blow-up rate estimates are established by using integral equation method and maximum principle techniques. The results show that the presentence of the reaction and boundary terms has important effects on the upper (lower) blow-up rate estimates forms.

Keywords

Subject Classifications

(2010) 35B44

References

[1] P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhuser Advanced Texts, Birkhuser, Basel, (2007).
[2] M. A. Rasheed, H. A. S. Al-Dujaly, T. J. Aldhlki, Blow-up rate estimates for a system of reaction-diffusion equations with gradient terms, International Journal of Mathematics and Mathematical Sciences, 2019(1) 1-7 ; (2019).
[3] M. A. Rasheed and L. J. Barghooth, Blow-up set and upper rate estimate for a semilinear heat equation, Journal of Physics: Conf. Series 1294(2019) 032013.
[4] M. A. Rasheed, Deriving the upper blow-up rate estimate for a parabolic problem, Iraqi Journal of Science, special issue, The first conference of Mathematics, (2020), 200-203.
[5] M. A. Rasheed and M. Chlebik, blow-up rate estimates and blow-up set for a system of two heat equations with coupled nonlinear Neumann boundary conditions, Iraqi journal of Science, 61(1) , 147-152, (2020).
[6] M. A. Rasheed, R. A. Hameed, S. K. Obeid, A. F. Jameel, On numerical blow-up solutions of semilinear heat equations, Iraqi Journal of Science, 61(8), 2077-2086, (2020).
[7] M. A. Rasheed, On blow-up solutions of a parabolic system coupled in both equations and boundary conditions, Baghdad Science Journal, 18(2), 315-321, (2021).
[8] M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow-up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comenian. 60, 35-103, (1991).
[9] J. D. Rossi, The blow-up rate for a semilinear parabolic equation with a nonlinear boundary condition, Acta Math. Univ. Comenian. 67, 343-350, (1998).
[10] Z. Lin and M. Wang, The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditions, Z. Angew. Math. Phys. 50, 361-374, (1999).
[11] S. N. Zheng, F. J. Li and B. C. Liu, Asymptotic behavior for a reaction-diffusion equation with inner absorption and boundary flux, Appl. Math. Lett. 19, 942-948, (2006).
[12] K. Deng, The blow-up behavior of the heat equation with Neumann boundary conditions, J. Math. Anal. Appl. 188, 641-650, (1994).
[13] O. A. Ladyzenskaja, V.A.Solonnikov and N.N.Uralceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, American Mathematical Society, 23, (1968).
[14] C. V. Pao., Nonlinear Parabolic and Elliptic Equations, New York and London: Plenum Press, (1992).
[15] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., (1964).

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