Global Existence and Blow-Up Solutions and Blow-Up Estimates for Some Evolution Systems with p-Laplacian with Nonlocal Sources

This paper deals with p-Laplacian systems ut − div(|∇u|p−2∇u) = ∫ Ωv α(x, t)dx, x ∈Ω, t > 0, vt − div(|∇v|q−2∇v) = ∫ Ωu β(x, t)dx, x ∈ Ω, t > 0, with null Dirichlet boundary conditions in a smooth bounded domain Ω ⊂ RN , where p,q ≥ 2, α,β ≥ 1. We first get the nonexistence result for related elliptic systems of nonincreasing positive solutions. Secondly by using this nonexistence result, blow up estimates for above p-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained under Ω = BR = {x ∈ RN : |x| < R} (R > 0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exist globally or blow up in finite time.

As well as the nonexistence of positive solutions of the related elliptic systems, (1.2) Equations (1.1) are the classical reaction-diffusion system of Fujita-type for p = q = 2.If p = 2, q = 2, (1.1) appears in the theory of non-Newtonian fluids [1,2] and in nonlinear filtration theory [3].In the non-Newtonian fluids theory, the pair (p, q) is a characteristic quantity of the medium.Media with (p, q) > (2,2) are called dilatant fluids and those with (p, q) < (2,2) are called pseudoplastics.If (p, q) = (2,2), they are Newtonian fluids.
In the past two decades, many physical phenomena were formulated into nonlocal mathematical models (see [4][5][6][7][8][9] and the references therein) and studied by many authors.Degenerate parabolic equations involving a nonlocal source, which arise in a population model that communicates through chemical means, were studied in [10,11].
As a matter of course, (1.1) with p = q = 2 give semilinear parabolic equations and have been studied by many authors.Over the last few years, much effort has been devoted to the study of blow-up properties for nonlocal semilinear parabolic equations of the type v t = v + g(t) (see [12][13][14]).Conditions on blowing up, blow-up set, blow-up rate, and asymptotic behavior of solutions are obtained, see [4,5].The problem concerning (1.1) includes the existence and multiplicity of global solutions, blowing-up, blow-up rates and blow-up sets, uniqueness and nonuniqueness, and so forth.For (1.2), there are problems such as existence and nonexistence, uniqueness and nonuniqueness, and so on.On the contrary, it seems that little is known about the result for quasilinear reaction-diffusion system (non-Newtonian filtration systems) and quasilinear elliptic system (e.g., [15][16][17][18]).For the scalar problem, a few authors (see [8,19]) investigated the following equation: with initial and boundary conditions.Roughly speaking, their results are (1) the solution u exists globally if q < p − 1, and (2) u blows up in finite time if q > p − 1 and u 0 (x) is sufficiently large.The authors in [7] studied the following equation: with null Dirichlet conditions and obtained that the solution either exists globally or blows up in finite time.Under appropriate hypotheses, they have local theory of the solution and obtain that the solution either exists globally or blows up in finite time.
The authors in [9] deal with the following reaction-diffusion system: with initial and boundary conditions.They proved that there exists a unique classical solution and the solution either exists globally or blows up in finite time.Furthermore, they obtain the blow-up set and asymptotic behavior provided that the solution blows up in finite time.
For p-Laplacian systems, Yang and Lu in [15] studied the following equations: (1.6) They derive some estimates near the blow-up point for positive solutions and nonexistence of positive solutions of the relate elliptic systems.
The main purpose of this paper is to derive some estimates near the blow-up point and investigate the global existence and blow-up of solutions for problem (1.1).
The outline of the paper is as follows.In the next section, we investigate the global nonexistence for elliptic system (1.2).Section 3 is devoted to blow-up estimate for system (1.1).In Section 4, we give the local existence and uniqueness of system (1.1).In Section 5, we give the blow-up property of solutions to (1.1).
After finishing this paper, we learn from a recent paper by Li [20] that he obtained the results of global existence and blow-up of solutions for (1.1).As we will show in Sections 4 and 5, our proof for the results of global existence and blow-up of solutions given here is simpler than [20].
To prove Theorem 2.1, system (1.2) can be written in radial coordinates as ) By the similar argument of [15, Lemma 2], we can prove the following lemmas.
(2.13)This is impossible, however, since from Lemma 2.3, estimate implies that This contradiction concludes the proof of the theorem.
If N = 2, p = 2, proceeding similarly as above implies that for r 0 ≤ s ≤ t.Letting t → +∞ in the inequality, we obtain a contraction.Finally, if N > max{p, q} ≥ 2 holds, we know from Theorem 2.1 that system (3.14) has no positive solutions.We conclude that (3.11) is true.It follows from (3.11) that there exists t 1 ∈ (0,T) such that for any t ∈ (t 1 ,T), we have Integrating (3.18) on (t,s) ⊆ (t 1 ,T) and then letting s → T, we obtain By using condition (vi) in (3.19), we have u(x,t) ≤ u(0,t) ≤ c 1 (T − t) −δ1 for any (x,t) ∈ Q T \Q t1 . (3.20) In the same way, we have the blow-up estimate for v.The proof is complete.
Remark 3.2.From the condition in Theorem 3.1, we fell that the condition (vii) is rather strong.We guess that the condition (vii) may be removed and a better result can be obtained: Further discussion on this problem will be made.

Local existence and uniqueness
In this section, we study the global existence of (1.1) under appropriate hypotheses.From the point of physics, we need only to consider the nonnegative solutions.Moreover, if we assume u 0 (x),v 0 (x) ≥ 0, by Lemma 4.5 (proved later), we can show that (u(x,t),v(x,t)) ≥ 0 a.e. in Ω × (0,T).Since (1.1) are the degenerate parabolic equations for |∇u| = 0, |∇v| = 0, one cannot expect the existence of classical solution of (1.1).As it is now well known that degenerate equations need not posses classical solutions, most of studies of p-Laplacian equations concerned with weak solutions (see [7,9]).We begin by giving a precise denition of a weak solution for problem (1.1).Let Q T = Ω × (0,T), T > 0, hold for all 0 < t 1 < t 2 < T, where ψ i (x,t) ∈ Ψ (i = 1,2).A weak solution of (1.1) is a vector function which is both a subsolution and a supersolution of (1.1).For every T < ∞, if (u,v) is a solution of (1.1), we say (u,v) is global.By a modification of the method given in [7], we obtain the following results.
Proof of this lemma is similar as in [7] only need a little modification, we omit it here.
To the limit function (u(x,t),v(x,t)) of the sequence (u n (x,t),v n (x,t)), we divide our proof into four steps.
Step 1.There exist a small T 0 > 0 and a constant M > 0, independent of n, such that To this end, we consider the ordinary differential equation: where p = max{α,β}.It is obvious that there exists T 0 > 0, such that (4.5) has a bounded solution We draw the conclusion.

Global existence and blow-up
In this section, we will discuss the global existence and blow-up in finite time of the solution for system (1.1).Our approach in a combination principle and super-and subtechniques which are similar as in [7].Firstly, we suppose p, q > 2.
Then the solution of system (1.1) exists globally.
Then the solution of system (1.1) blows up in finite time.
Proof of Theorem 5.2.(i) Without lose of generality, we can suppose that 0 ∈ Ω.We get our conclusion by a small modification of the results of [8,Section 4].
(ii) To prove u(x,t) and v(x,t) blow-up in finite time, according to sub-and supersolution, we need only to find blowing up subsolutions.The proof is similar, as here we use an argument as done in [5,7].
Grant no.04KJB110062), and the Science Foundation of Nanjing Normal University (Grant no.2003SXXXGQ2B37).
Definition 4.1.A pair of function (u(x,t),v(x,t)) is called a sub-(or super-) solutions of (1.1) on Q T if and only if .1) Z. Cui and Z.Yang 9 2∇v n ∇v nt dx dt