Critical Exponents for a Doubly Degenerate Parabolic System Coupled via Nonlinear Boundary Flux

Abstract

Abstract. The paper deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical Fujita curve is conjectured with the aid of some new results.

A nonlinear diffusion system coupled via nonlinear boundary flux

“…The aim of this paper is to give a simple criteria of the classification of global existence and nonexistence of solutions to system (1.1)-(1.3) by using a combination of various kinds self-similar sub-or super-solutions and the basic properties of the so-called M-matrix for general powers m i , indices p ij and number k 1, which complicate the interaction among various components u i . Paradoxically, our proof is more simple than of [22,32] in the sense we do not need some specific computations of parameters in the construction of self-similar sub-or supersolutions, even though we are dealing with an abstract system without specific number k.…”

“…In particular, many papers have been devoted to study critical exponents of (1.1)- (1.3) in the slow diffusion case (see [8,23,33,36,38]). Recently, many authors transfer their attention to the fast diffusion case (see [6,8,20,21,31]), and many important results about critical exponents have been obtained. The concept of critical Fujita exponents was proposed by Fujita in the 1960s during discussion of the heat conduction equation with a nonlinear source (see [7]).…”

“…Theorem 1.1 suggests that the global existence or nonexistence is completely characterized by whether the matrix I − P is M-matrix or not, in case that the algebraic system (1.13) has a solution k with k i > 0 for some i. The assumption on k, which holds naturally if one investigates the systems studied in [17,18,[20][21][22][23]32], is rather technical. On the other hand, if there exists i such that q ii > p i +2 m i +2p i +1 , then I − P is not an M-matrix by Remark 1.1.…”

“…Note that there are no such results for the problem posed on a bounded domain (see [11]). [17,18,[20][21][22][23]32] are covered by the above theorems when taking special parameters. Remark 1.5.…”

“…Eqs. (1.1) include both the porous medium operator (with p i = 0, k = 1 or k = 2) and the gradient-diffusivity the p-Laplacian operator (m i = 1, k = 1 or k = 2) as special cases, which have been the subject of intensive study (see [4,6,9,8,12,14,[20][21][22]26,34,36] and references therein).…”