Advances in Differential Equations

Critical Fujita exponents for a coupled non-Newtonian filtration system

Sining Zheng and Miaoqing Tian Sining Zheng and Miaoqing Tian

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In 1966, Fujita started the close investigation of the blow-up phenomena arising in some semilinear parabolic problems and found the so-called Fujita exponent. Later, Galaktionov and Levine researched nonlinear parabolic equations involving $p$-Laplace operators with nonlinear boundary conditions. In the present paper we extend their results to a non-Newtonian filtration system coupled via nonlinear boundary conditions. In terms of a characteristic algebraic system introduced for the problem, we obtain a clear and simpler presentation of both the critical Fujita exponent and the blow-up rate for this complicated coupled system containing six nonlinear exponent parameters. The proof for establishing the critical exponents is based on careful constructions of the comparison functions, especially for the blow-up case, by using the Barlenbratt-type solutions. The analysis of the blow-up rate relies on the appropriate scale transformation of the independent and the dependent variables with some differential inequalities satisfied by the $L^\infty$-norm of blowing up solutions. This paper provides a complete result for such a coupled degenerate parabolic system.

Article information

Adv. Differential Equations, Volume 15, Number 3/4 (2010), 381-400.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K65: Degenerate parabolic equations 35B33: Critical exponents 35B40: Asymptotic behavior of solutions


Sining Zheng; Miaoqing Tian, Sining Zheng; Miaoqing Tian. Critical Fujita exponents for a coupled non-Newtonian filtration system. Adv. Differential Equations 15 (2010), no. 3/4, 381--400.

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