On Blow-up Solutions of A Parabolic System Coupled in Both Equations and Boundary Conditions

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Maan A. Rasheed


This paper is concerned with the blow-up solutions of a system of two reaction-diffusion equations coupled in both equations and boundary conditions. In order to understand how the reaction terms and the boundary terms affect the blow-up properties, the lower and upper blow-up rate estimates are derived. Moreover, the blow-up set under some restricted assumptions is studied.


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Rasheed MA. On Blow-up Solutions of A Parabolic System Coupled in Both Equations and Boundary Conditions. Baghdad Sci.J [Internet]. [cited 2021Jan.20];18(2):0315. Available from: https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/3980


Rasheed M A. On Blow-up Solutions of Parabolic Problems, Ph.D. thesis, School of physical and Mathematical sciences, University of Sussex, UK; 2012.

Liu W, Zhuang H. Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms. Nonlinear Differ. Equ. Appl. 2017; 24(6): doi.org/10.1007/s00030-017-0491-5

Rasheed M A , Al-Dujaly H A S, Aldhlki T J. Blow-Up Rate Estimates for a System of Reaction-Diffusion Equations with Gradient Terms. Int. J. Math. Math. Sci. 2019; 2019(1); 1-7.

Han Y. Blow-up at infinity of solutions to a semilinear heat equation with logarithmic nonlinearity. J. Math. Anal. Appl. 2019; 474(1): 513–517.

Cho CH. A numerical algorithm for blow-up problems revisited. NUMER ALGORITHMS. 2017 Jul 1;75(3):675-97.

Polyanin A D, Shingareva I K. Nonlinear problems with blow-up solutions: Numerical integration based on differential and nonlocal transformations, and differential constraints. Appl Math Comput. 2018; 336: 107–137.

Polyanin A D , Shingareva I K. Nonlinear blow-up problems for systems of ODEs and PDEs: Non-local transformations, numerical and exact solutions. Int J Nonlin Mech. 2019; 111: 28–41.

Fu SC , Guo J S. Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions. J. Math. Anal. Appl. 2002; 29: 458- 475.

Zheng S N, Li F J. Critical exponent for a reaction-diffusion model with absorption and coupled boundary flux. Proc. Edinb. Math. Soc. 2005; 48: 241-252.

Xu S. Non-simultaneous blow-up of a reaction-diffusion system with inner absorption and coupled via nonlinear boundary flux. Bound Value Probl. 2015; 2015(1). doi:10.1186/s13661-015-0483-5

Ding J, & Hu H. Blow-up solutions for nonlinear reaction diffusion equations under Neumann boundary conditions. Appl Anal. 2016; 96(4): 549-562.

Gladkov A. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Commun Pur Appl Anal. 2017; 16(6): 2053–2068.

Liu B, Dong M., Li F. Asymptotic properties of blow-up solutions in reaction–diffusion equations with nonlocal boundary flux. Z. Angew. Math. Phys. 2018; 69(2): doi:10.1007/s00033-018-0920-2.

Lin Z, Xie C. The blow-up rate for a system of heat equations with Neumann boundary conditions. Acta Math. Sinica. 1999; 15: 549-554.

Deng K. Blow-up rates for parabolic systems, Z. Angew. Math. Phys. 1996; 47:132-143.

Ladyzenskaja OA, Solonnikov VA,Uralceva NN. Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs. JAMS. 23, 1968.

Pao C V. Nonlinear Parabolic and Elliptic Equations. New York and London: Plenum Press, 1992.

Hu B, Yin HM. The profile near blow-up time for solution of the heat equation with a non-linear boundary condition. Trans. Amer. Math. Soc. 1994; 346: 117-135.

Friedman A. Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964.

Quittner P, Souplet Ph. Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhuser Advanced Texts, Birkhuser, Basel. 2007.