Blowup for a Non-Newtonian Polytropic Filtration System Coupled via Nonlinear Boundary Flux
© Zhongping Li et al. 2008
Received: 18 November 2007
Accepted: 1 March 2008
Published: 1 April 2008
We study the global existence and the global nonexistence of a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux. We first establish a weak comparison principle, then discuss the large time behavior of solutions by using modified upper and lower solution methods and constructing various upper and lower solutions. Necessary and sufficient conditions on the global existence of all positive (weak) solutions are obtained.
Parabolic equations like (1.1) appear in population dynamics, chemical reactions, heat transfer, and so on. In particular, (1.1) may be used to describe the nonstationary flows in a porous medium of fluids with a power dependence of the tangential stress on the velocity of displacement under polytropic conditions. In this case, (1.1) are called the non-Newtonian polytropic filtration equations (see [ 1, 2, 3, 4, 5, 6] and the references therein). For the Neuman problem (1.1)–(1.3), the local existence of solutions in time has been established, see survey in .
with . In , they obtained the necessary and sufficient conditions to the global existence of solutions for . In , they considered the case of or and obtained the necessary and sufficient blowup conditions for the special case (the ball centered at the origin in with radius ). However, for the general domain , they only gave some sufficient conditions to the global existence and the blowup of solutions.
The main purpose of this paper is to study the influence of nonlinear power exponents on the existence and nonexistence of global solutions of (1.1)–(1.3). By using upper- and lower-solution methods, we obtain the necessary and sufficient conditions on the existence of global (weak) solutions to (1.1)–(1.3). Our main results are stated as follows.
If we extend the solution to (1.6) to the interval by symmetry, we get a solution to the same problem (1.6) with the condition at , substituted by a condition at , Conversely, symmetric solutions to this latter problem are solutions to the original problem (1.6). The problem (1.1)–(1.3) is the more general -dimensional version of the problem (1.6). Theorems 1.1–1.4 extend the results of the problem (1.6) into multidimensional case and it seems to be a natural extension of Wang .
The rest of this paper is organized as follows. Some preliminaries will be given in Section 2 . Theorems 1.1–1.4 will be proved in Sections 3–5, respectively.
As it is well known that degenerate and singular equations need not possess classical solutions, we give a precise definition of a weak solution to (1.1)–(1.3).
Next we give some preliminary propositions and lemmas.
Proposition 2.2 (comparison principle).
Assume that are positive functions and is any weak solution of (1.1)–(1.3). Also assume that and are a lower and an upper solution of (1.1)–(1.3) in , respectively, with nonlinear boundary flux and , where . Then we have in .
We know from  that blows up in finite time. Since , by the comparison principle, is a lower solution of (1.1)–(1.3) and blows up in finite time if .
The following propositions can be proved in the similar procedure.
At the end of this section, we describe two simple lemmas without proofs.
3. Proof of Theorem 1.1
We get the proof of Theorem 1.1 by combining Proposition 2.3 and Lemmas 3.1 and 3.2.
4. Proof of Theorems 1.2 and 1.3
It has been shown from Proposition 2.4 and Lemmas 4.1 and 4.2 that Theorem 1.2 is true.
In a similar way to the proof of Theorem 1.2, we have Theorem 1.3.
5. Proof of Theorem 1.4
By Proposition 2.6 and Lemmas 5.1 and 5.2, we see that Theorem 1.4 holds.
This work was partially supported by NNSF of China (10771226), was partially supported by Natural Science Foundation Project of CQ CSTC (2007BB0124), and was partially supported by Natural Science Foundation Project of China West Normal University (07B047).
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