Electronic Journal of Qualitative Theory of Differential Equations

This paper deals with the global existence and the global nonexistence of a doubly nonlinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux. The authors first establish a weak comparison principle, then by constructing various upper and lower solutions, some appropriate conditions for global existence and global nonexistence of solutions are determined respectively.


Introduction
In this paper, we consider the following problem: (u n1 ) t = ∆ m1 u + u α1 v p1 , (v n2 ) t = ∆ m2 v + u p2 v β1 , x ∈ Ω, t > 0, (1.1) where , Ω is a bounded domain in R N with smooth boundary ∂Ω, m i > 1, n i , α i , β i > 0, p i , q i ≥ 0, i = 1, 2. ν denotes the outer unit normal on the boundary, u 0 (x), v 0 (x) ∈ C 1 ( Ω) are positive and satisfy the compatibility conditions.Parobolic equations like Eq.(1.1) appear in population dynamics, chemical reactions, heat transfer like, for instance, the description of turbulent filtration in porous media, the theory of non-Newtonian fluids perturbed by nonlinear terms and forced by rather irregular period in time excitations, the flow of a gas through a porous medium in a turbulent regime or the spread of biological (see [1,2,3] and the references given therein).In particular, Eq.(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, Eq.(1.1) are called the non-Newtonian polytropic filtration equations (see [4]- [8] and the references therein).We refer to [9] for further information on these phenomena.Recently a connection has been revealed with soil science, specifically with flows in reservoirs exhibiting fractured media (see [10]).
In [12], Li et al. considered the following system with nonlinear boundary conditions They obtained necessary and sufficient conditions on the global existence of all positive (weak) solutions.
In [13], Song and Zheng studied the following quasilinear parabolic system with multi-coupled nonlinearities They obtained the necessary and sufficient conditions to the global existence of solutions for 0 < m, n < 1.They also considered the case of m, n ≥ 1 and 0 < m < 1, n ≥ 1.However, they only gave some sufficient conditions to the global existence and blowup of solutions.Motivated by the references cited above, we study the influence of nonlinear reaction terms and nonlinear boundary flux on the existence and nonexistence of global solutions of (1.1) − (1.3).Due to the nonlinear diffusion terms and doubly degeneration for u = 0, |∇u| = 0 or v = 0, |∇v| = 0, we have EJQTDE, 2012 No. 1, p. 2 some new difficulties to be overcome.Noticing that the system (1.1) includes the Newtonian filtration system (p = 2) and the non-Newtonian filtration system (m = 1) formally, so the method for it should be synthetic.In fact, we can use the methods for the above two systems to deal with it.Then we investigate the global existence or blow-up properties of weak solutions to the problem (1.1) depending on the relations among the parameters m 1 , m 2 , n 1 , n 2 , p 1 , p 2 , q 1 , q 2 , α 1 , α 2 , β 1 , β 2 .Note that (1.1) has nonlinear and nonlocal sources u α1 v p1 , u p2 v β1 and nonlinear boundary sources u α2 v q1 , u q2 v β2 , which make the behavior of the solution different from that for that of homogeneous Neumann or Dirichlet boundary value problems.However, it is difficult to use the same methods as that in [13] to get the desired result.To overcome these difficulties, we used some modification of the technique in [12] so that we can handle the nonlinearities.Then, we use some functions to control the nonlocal sources and prove, with the technique in [12], that the control for the nonlocal sources is suitable.Finally we also need to consider the effect of these nonlinear terms in the proof of the global existence(blow-up) property of solutions to (1.1).
Our main results are stated as follows.
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).
Definition 2.1 Let T > 0 and In particular, (u(x, t), v(x, t)) is called a weak solution of (1.1) − (1.3) if it is both a weak upper and a lower solution.For every Next we give some preliminary propositions and a fact.Proposition 2.1 (Comparison principle).Assume that u 0 , v 0 are positive C 1 (Ω) functions and (u, v) is any weak solution of (1.1)-(1.3) in Q T .Also assume that (u, v) ≥ (δ, δ) > 0 and (u, v) are a lower and an upper solution of (1.1)−(1.3) in Q T , respectively, with nonlinear boundary flux (λu α2 v q1 , λu q2 v β2 ) and (λu α2 v q1 , λu q2 v β2 ), and with nonlinear reaction terms (u α1 v p1 , u p2 v β1 ) and Proof.For small σ > 0, letting ψ σ (z) = min{1, max{z/σ, 0}}, z ∈ R, and setting ψ 1 = ψ σ (u − u), according to the definition of solutions and lower solutions, we have As in [14], by letting σ → 0, we get where , it follows from the continuity of u, v, u and v that there exists a τ > 0 sufficiently small such that It follows that Similarly, we have Now, (2.8) and (2.9) combined with the Gronwall's Lemma show that (u, v) Obviously, (δ, δ) is a lower solution of (1.1) Using this fact, as in the above proof we can proof that (u, v) For convenience, we denote δ = min{min Ω u 0 (x), min Ω v 0 (x)} > 0 and 0 < λ < 1 < λ, which are fixed constants. Let with the first eigenvalue λ k normalized by (Ω) C 1 (Ω) and ∂ϕ k (x)/∂ν < 0 on ∂Ω (see [15]- [17]).Thus there exist some positive constants We have also and some positive constant ε k .For the fixed ε k , there exists a positive constant Then the solutions of (1.1) − (1.3) blow up in finite time.
For (3 • ) or (4 • ) or (5 • ), since the solution of the system in [12] is a lower solution of (1.1) − (1.3), in view of the blow up results of [12], under the condition of Proposition 2.2, the solution of (1.1) − (1.3) blows up in finite time. 2 The following Proposition 3 − 5 can be proved in the similar procedure.
At the end of this section, we describe a simple fact without proof.Fact 1 Suppose that positive constants A, B, C, D satisfy AB < CD, then for any two positive constants a, b, there exist two positive constants l 1 , l 2 such that al 3 Proof of the Theorem 1.1 In this section we will divide the proof of Theorem 1.1 into following lemmas.
For (x, t) ∈ Ω × R + , by direct computation, we have EJQTDE, 2012 No. 1, p. 8 Similarly, Moreover, By setting , on the boundary, we have Since the conditions of this lemma, there exist positive constants l 1 , l 2 satisfying Thus, (u, v) is an upper solution of (1.1) − (1.3), which means that the solutions of (1.1) , then all positive solutions of problem (1.1) − (1.3) blow up in finite time.
Proof.We prove this lemma by dividing into following two subcases: On the other hand, on the boundry, we have Moreover, it is easy to see that u(x, 0) ≤ δ ≤ u 0 (x), v(x, 0) ≤ δ ≤ v 0 (x), so (u, v) is a subsolution of (1.1) − (1.3), which blows up in finite time.Subcase (ii).For Consider the problem We know from the Subcase (i) that (w, z) blows up in finite time, so the solutions of (1.1) − (1.3) blow up in finite time.
Proof.We can prove this lemma in the similar way as that of lemma 3.3.2 We get the proof of Theorem 1.1 by combining Proposition 2 and Lemma 3.1-3.4.

Proof of the Theorem 1.2
In this section we will divide the proof of Theorem 1.2 into following lemmas.
Similarly, we can get Moreover, on the boundary, we have Since the conditions of the lemma, there exist a positive constant l 1 , l 2 large such that EJQTDE, 2012 No. 1, p. 12 and Thus, (u, v) is a global upper solution of (1.1) − (1.3).The global existence of solution to (1.1) − (1.3) follows from the comparison principle.
, then all positive solutions of problem (1.1)-(1.3)blow up in finite time.