Critical Fujita exponents for a coupled non-Newtonian filtration system

Abstract

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.