# Global existence and finite time blowup for a nonlocal parabolic system.

1 Introduction and main resultsIn this paper, we investigate the global existence and finite time blowup of nonnegative solutions for the following parabolic system with nonlocal sources

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.1)

where [OMEGA] be a bounded domain in [R.sup.N] (N [greater than or equal to] 1) with smooth boundary [partial derivative][OMEGA], and [u.sub.0] (x), [v.sub.0] (x) are nonnegative bounded functions in [OMEGA], constants [alpha], [beta] [greater than or equal to] 1, p, q > 0, where [[parallel]*[parallel].sup.a.sub.a] = [[integral].sub.[OMEGA]] [[absolute value of x].sup.a]dx.

Equations (1.1) constitute a simple example of a reaction diffusion system exhibiting a nontrivial coupling on the unknowns u(x, t), v(x, t). Such as heat propagations in a two-component combustible mixture [1], chemical processes [2], interaction of two biological groups without self-limiting [3], etc.

In the past several decades, a number of works have been contributed to the study of the following weakly coupled reaction-diffusion system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.2)

(see [1],[4]-[6] and references therein), especially its special cases [p.sub.1] = [q.sub.2] = 0 (variational) or [q.sub.1] = [p.sub.2] = 0 (uncoupled single equation). For the case [p.sub.1] = [q.sub.2] = 0, Escobedo analyzed the boundedness and blow-up of solutions [7]; Caristi obtained the blow-up estimates of solutions [4]. Wu Yuan studied the uniqueness of generalized solutions with degenerate diffusion [6]. It is well known that there have been much more results for the uncoupled single equation case [q.sub.1] = [p.sub.2] = 0, including necessary and sufficient conditions for finite blow-up [8], estimates of blow-up time [9], blow-up rates [10] and blow-up behavior [11].

The general form of (1.2) was systematically studied by Escobedo [5]. They gave a complete analysis on the critical blow-up and the global existence numbers for the Cauchy problem of (1.2), where the introduced parameters [alpha] and [beta] satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

played important roles in their framework.

Meanwhile, the system (1.2) was also studied by Wang in [12] and Zheng in [13] with different methods. Some interesting results concerning the global existence and blow-up conditions of the solutions are established.

Lately, Li et al. in[14] and Zhang et al. in[15] studied the following system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

respectively. They obtained some results on the global solutions, the blow-up solutions and the blow-up rates.

Our present work is motivated by [5] and [12]-[15] mentioned above. The main purpose is to extend Escobedo's method in [5] to system (1.1) and establish the critical exponent which concern with the global existence and finite time blowup of solutions.

For a solution (u(x, t), v(x, t)) of (1.1), we define

[T.sup.*] = [T.sup.*](n, v) = sup{T > 0: (n, v) is bounded and satisfies (1.1)}.

Note that if [T.sup.*] < + [infinity], then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

in this case, we say that the solution (n, v) blows up in finite time.

Throughout the remainder of this paper, we denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, let us state our main results, the two theorems concern the global existence and blow-up conditions of the solutions to system (1.1).

Theorem 1. If one of the following conditions holds:

(1) p, q < 1 and pq < (1 - p)(1 - q);

(2) p, q < 1, pq > (1 - p) (1 - q) and the initial data [u.sub.0](x), [v.sub.0](x) are sufficiently small;

(3) p, q < 1, pq = (1 - p)(1 - q) and the domain ([absolute value of [OMEGA]]) is sufficiently small. Then every nonnegative solution of system (1.1) exists globally.

Theorem 2. If one of the following conditions holds:

(1) p, q < 1, pq > (1 - p)(1 - q), and the initial data n0 (x), v0 (x) are sufficiently large;

(2) p, q < 1,pq = (1 - p)(1 - q), the domain contains a sufficiently large ball, and initial data [u.sub.0](x), [v.sub.0](x) are sufficiently large. Then the nonnegative solution of system (1.1) blows up in a finite time.

This paper is organized as follows. In the next Section, we establish the local existence and give some auxiliary lemmas. In Section 3, which concerns global existence, we prove Theorem 1. Theorem 2 which deals with the blow-up phenomenon is proved in Section 4.

2 Local existence and comparison principle

At first, we give the maximum principle and the comparison principle for the nonlocal parabolic system. Let 0 < T < + [infinity], we set [Q.sub.T] = [OMEGA] x (0, T), [[bar.Q].sub.T] = [bar.[OMEGA]] x [0, T], [S.sub.T] = [[partial derivative].sub.[OMEGA]] x (0, T) and define the following class of test functions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 1. A pair of vector function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined on [[bar.Q].sub.T], for some T > 0, is called a sub-solution of (1.1), if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and all the following hold:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(3) For every t [member of] [0, T] and any [[psi].sub.1], [[psi].sub.2] [member of] [PSI],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A super-solution ([bar.u] (x, t), [bar.u](x, t)) can be defined in a similar way.

A weak solution (n, v) of (1.1) is a vector function which is both a sub-solution and a super-solution of (1.1). For every T < + [infinity], if (n, v) is a weak solution of (1.1), we say the (n, v) is global.

Lemma 1. (Maximnm principle) Snppose that [[omega].sub.1] (x, t), [[omega].sub.2](x, t) [member of] [C.sup.2,1] ([Q.sub.T]) [intersection] C([[bar.Q].sub.T]) and satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.2)

where [f.sub.i] (x, t) and [c.sub.ij] (x, t) are the nonnegative functions in [Q.sub.T]. Then [[omega].sub.i](x, t) [greater than or equal to] 0 on [[bar.Q].sub.T].

Proof. The technique for proving the maximum principle for parabolic equation is quite standard. Here we shall sketch the argument for the convenience of the reader.

Suppose the strict inequality of (2.2) hold, then we assert that [[omega].sub.i](x, t) > 0 (i = 1,2) on [[bar.Q].sub.T]. According to [[omega].sub.i](x, 0) > 0(i = 1,2), by continuity, there exist [delta] > 0 such that [[omega].sub.i](x, t) > 0 for all x [member of] [OMEGA],0 [less than or equal to] t [less than or equal to] [delta]. Let

A = {[delta] < T : [[omega].sub.i](x, t) > 0, (x, t) [member of] [OMEGA] x [0,[delta]],i = 1,2}

and [bar.t] = sup A, then 0 < [bar.t] [less than or equal to] T.

If [bar.t] < T, there holds [[omega].sub.i](x, t) [greater than or equal to] 0 in [OMEGA] x (0, [bar.t]], and at least one of [[omega].sub.1], [[omega].sub.2] vanishes at ([bar.x], [bar.t]) for some [bar.x] [member of] [bar.[OMEGA]]. Furthermore, by the boundary conditions we know that [bar.x] [member of] [OMEGA]. Without loss of generality, we suppose [[omega].sub.1] ([bar.x], [bar.t]) = 0. In view of the boundary conditions, we know [[omega].sub.1] > 0 on [partial derivative][OMEGA] x (0, [bar.t]]. So [[omega].sub.1] takes the nonnegative minimum on [[bar.Q].sub.[bar.t]] at ([bar.x],[bar.t]). Then, at ([bar.x],[bar.t]) we find that

[M.sub.1][omega] = [[omega].sub.1t] - [DELTA][[omega].sub.1] - [f.sub.1] [[integral].sub.[OMEGA]] ([c.sub.11] [[omega].sub.1] + [c.sub.12][[omega].sub.2])dx [less than or equal to] 0.

This is a contradiction from (2.2). Hence [bar.t] = T, that is [[omega].sub.i](x, t) > 0(i = 1,2) on [[bar.Q].sub.T].

Now, we consider the general case. Take constant [gamma] satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where e is any fixed positive constant. In view of (2.2), we can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it follows that [[omega].sub.i](x, t) [greater than or equal to] 0(i = 1,2) on [[bar.Q].sub.T]. Thus the proof is completed.

Based on the above lemma, we obtain the following comparison principle.

Lemma 2. (Comparison principle) Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and ([bar.u], [bar.v]) be a nonnegative sub-solution and a nonnegative super-solution of system (1.1), respectively. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] < ([bar.u], [bar.v]) on [Q.sub.T] if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and either

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

hold.

Proof. Subtracting the integral inequalities of (2.1) for ([bar.u], [bar.v]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and using the mean value theorem, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and ([bar.u], [bar.v]) on [Q.sub.T] are bounded, it follows from [alpha] [greater than or equal to] 1 that [H.sub.1] (x,s) is a bounded, nonnegative function. Similarly, [D.sub.1] (s) is bounded if p/[alpha] [greater than or equal to] 1. Now if p/[alpha] < 1, we have [D.sub.1](s) [less than or equal to] [[rho].sup.p/[alpha]-1] by the assumptions. Thus, appropriate test function [[psi].sub.1] may be chosen exactly as in [16] to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.4)

where [[omega].sub.+] = max{[omega], 0} and [k.sub.1] > 0 is a bounded constant. Similarly, we can prove

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

for some bounded constant [k.sub.2] > 0. Now, (2.4)-(2.5) combined with the Gronwall's lemma show [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [less than or equal to] ([bar.u], [bar.v]) since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus the proof is completed.

In order to prove the local existence of solution, for k = 1,2, ..., we consider the following corresponding regularized system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.6)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [u.sub.0i](x), [v.sub.0i](x) are smooth approximation of [u.sub.0](x), [v.sub.0](x) with supp [u.sub.0i] [subset] [OMEGA] and supp [v.sub.0i] [subset] [OMEGA], respectively. It is known that the system (2.6) has a unique classical solution ([u.sup.i.sub.k], [v.sup.i.sub.k]) [member of] C([bar.[OMEGA]] x [0, [T.sub.i](k))) [intersection] [C.sup.2,1] ([OMEGA] x (0, [T.sub.i](k))) for 0 < [T.sub.i](k) [less than or equal to] [infinity] by the classical theory for parabolic equation, where [T.sub.i](k) is the maximal existence time. By a direct computation and the classical maximum principle, we have [u.sup.i.sub.k], [v.sup.i.sub.k] [greater than or equal to] 1/k. Hence ([u.sup.i.sub.k], [v.sup.i.sub.k]) satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

with the corresponding initial and boundary conditions. At the same time, passing to the limit i [right arrow] [infinity], it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and ([u.sub.k], [v.sub.k]) is a weak solution of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

with the corresponding initial and boundary conditions on [Q.sub.T(k)], where T(k) = [lim.sub.i[right arrow][infinity]] [T.sub.i](k) is the maximal existence time. Here a weak solution of (2.8) is defined in a manner similar to that for (1.1), only the integral equalities for u and v, (2.1) may be replaced with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.10) respectively.

Since [u.sup.i.sub.k], [v.sup.i.sub.k] [greater than or equal to] 1/k, applying Lemma 1, we have the following lemma.

Lemma 3. Assume that [omega](x, t),s(x, t) [member of] C([bar.[OMEGA]] x [0, [T.sub.i](k))) [intersection] [C.sup.2,1] ([OMEGA] x (0, [T.sub.i](k))) is a sub- (or super-) solution of (2.7). Then ([omega], s) [less than or equal to] ([greater than or equal to])([u.sup.i.sub.k], [v.sup.i.sub.k]) on [bar.[OMEGA]] x [0, [T.sub.i](k)).

According to Lemma 3, we have

Lemma 4. If [k.sub.1] > [k.sub.2], then we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on [bar.[OMEGA]] x [0, [T.sub.i]([k.sub.2])) and [T.sub.i]([k.sub.1]) [greater than or equal to] [T.sub.i]([k.sub.2]).

Then, from Lemma 4, passing to the limit i [right arrow] [infinity], it happens that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and T([k.sub.1]) [greater than or equal to] T([k.sub.2]) if [k.sub.1] > [k.sub.2].

Therefore, the limit [T.sup.*] = [lim.sub.k[right arrow][infinity]] T(k) exists and, as well, the point-wise limit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

exist for any (x, t) [member of] [bar.[OMEGA]] x [0, [T.sup.*]). Furthermore, as the convergence of the sequence is monotone, passing to the limit k [right arrow] [infinity] in the identities (2.9) and (2.10) is justified by monotone and dominated convergence theorems for any [[psi].sub.1], [[psi].sub.2] [member of] [PSI] and t [member of] [0, [T.sup.*]). Thus, the following theorem is established.

Theorem 3. (Local existence and continuation) Assume [u.sub.0], [v.sub.0] [greater than or equal to] 0, [u.sub.0], [v.sub.0] [member of] [L.sup.[infinity]] ([OMEGA]), there is a [T.sup.*] = [T.sup.*] ([u.sub.0], [v.sub.0]) > 0 such that there exists a nonnegative weak solution (u(x, t), v(x, t)) of (1.1) for each T < [T.sup.*]. Furthermore, either [T.sup.*] = + [infinity] or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3 Global existence

In this section, we will prove Theorem 1. According to Lemma 2, we only need to construct bounded, positive super-solutions for any T > 0.

Let [phi](x) be the unique positive solution of the following linear elliptic problem

-[DELTA][phi](x) = 1, x [member of] [OMEGA]; [psi](x) = 0, x [member of] [partial derivative][OMEGA].

Denote C = [max.sub.x[member of][OMEGA]] [phi](x), then 0 [less than or equal to] [phi](x) [less than or equal to] C. Now, we define the functions [bar.u], [bar.v] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.1)

where constants [l.sub.1], [l.sub.2] < 1, and k > 0 will be fixed later. Clearly, for any T > 0, ([bar.u], [bar.v]) is a bounded function and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, a series of computations yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.3)

Similarly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.4)

(1) If p, q < 1 and pq < (1 - p)(1 - q), then there exist constant 0 < [[epsilon].sub.1] < 1 such that

(1 - [[epsilon].sub.1])(1 - p) > (1 + [[epsilon].sub.1])q, (1 - [[epsilon].sub.1])(1 - q) > (1 + [[epsilon].sub.1])p. (3.5)

Hence, AL = [([[epsilon].sub.1], [[epsilon].sub.1]).sup.T] yields

(1 - p)[l.sub.1] - p[l.sub.2] = [[epsilon].sub.1], -q[l.sub.1] + (1 - q)[l.sub.2] = [[epsilon].sub.1]. (3.6)

That is,

[l.sub.1] = [[epsilon].sub.1] (1 - q) + [[epsilon].sub.1]p/(1 - p)(1 - q) - pq' [l.sub.2] = [[epsilon].sub.1](1 - p) + [[epsilon].sub.1]q/(1 - p)(1 - q) - pq. (3.7)

Moreover, 0 < [l.sub.1], [l.sub.2] < 1. Therefore, we can choose k sufficiently large such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.9)

Now, it follows from (3.2)-(3.9) that ([bar.u], [bar.v]) is a positive super-solution of (1.1). (2) Next, if p, q < 1 and pq > (1 - p) (1 - q), then there exist constant 0 < [[epsilon].sub.2] < 1 such that

(1 + [[epsilon].sub.2])(1 - q) < (1 - [[epsilon].sub.2])p, (1 + [[epsilon].sub.2])(1 - p) < (1 - [[epsilon].sub.2])q. (3.10)

Hence, AL = [(-[[epsilon].sub.2], -[[epsilon].sub.2]).sup.T] yields

(1 - p)[l.sub.1] - p[l.sub.2] = -[[epsilon].sub.2], -q[l.sub.1] + (1 - q)[l.sub.2] = - [[epsilon].sub.2]. (3.11)

Namely,

[l.sub.1] = -[[epsilon].sub.2](1 - q) + [[epsilon].sub.2]p/(1 - p)(1 - q) - pq, [l.sub.2] = - [[epsilon].sub.2](1 - p) + [[epsilon].sub.2]q/(1 - p)(1 - q) - pq. (3.12)

Furthermore, 0 < [l.sub.1], [l.sub.2] < 1. Therefore, we can choose k sufficiently small such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.13)

And provided [u.sub.0](x), [v.sub.0](x) are also sufficiently small to satisfy (3.9). Then, from (3.2)-(3.4) and (3.9), (3.10)-(3.13), we know that ([bar.u], [bar.v]) is a positive super-solution of (1.1).

(3) Finally, if p, q < 1 and pq = (1 - p)(1 - q), then we may choose positive constants [l.sub.1], [l.sub.2] < 1 such that

p/1 - p = [l.sub.1]/[l.sub.2] = 1 - q/q (3.14)

That is [l.sub.1] = p([l.sub.1] + [l.sub.2]), [l.sub.2] = q([l.sub.1] + [l.sub.2]). Without loss of generality, we may assume that [OMEGA] [subset] [subset] B, where B is a sufficiently large ball. And denote [[phi].sub.B](x) is the unique positive solution of the following linear elliptic problem

-[DELTA][phi](x) = 1, x [member of] B; [phi](x) = 0, x [member of] [partial derivative]B.

Let [C.sub.0] = [max.sub.x[member of]B] [phi]B (x), then C < C0. Therefore, as long as [OMEGA] is sufficiently small and such that

[absolute value of [OMEGA]] < min{[([l.sub.1]/[C.sub.0] + 1).sup.[alpha]/p], [([l.sub.2]/[C.sub.0] + 1).sup.[beta]/q] (3.15)

Furthermore, choose k large enough to satisfy (3.9). Then, it follows from (3.2)(3.4) and (3.14)-(3.15) that ([bar.u], [bar.v]) is a positive super-solution of (1.1).

Thus, according to the Lemma 2, the proof of Theorem 1 is completed.

4 Blow-up results

In this section, we prove Theorem 2, to this end, we only need to construct blowing-up positive sub-solutions.

Denote by [[lambda].sub.1] > 0 and [[phi].sub.1] (x) the first eigenvalue and the corresponding eigenfunction of the following eigenvalue problem

-[DELTA][phi](x) = [lambda][phi](x), x [member of] [OMEGA]; [phi](x) = 0, x [member of] [partial derivative][OMEGA]. (4.1)

It is well known that [[phi].sub.1](x) may be normalized as [[phi].sub.1](x) > 0 in [OMEGA] and [max.sub.[OMEGA]] [[phi].sub.1](x) = 1.

Now, we define the functions [??](x, t), [??](x, t) as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4.2)

where [l.sub.1], [l.sub.2] > 1, and s(t) is the unique positive solution of the following Cauchy problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.3)

where constants [k.sub.1], [k.sub.2], [r.sub.2] > 0 and [r.sub.1] [greater than or equal to] 1 to be fixed later. Clearly, s(t) [greater than or equal to] [delta] and become unbounded in finite time T([delta]). Next, we will show that ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is a sub-solution of problem (1.1). A series of computations yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4-4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4-5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4-6)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(1) If p, q < 1 and pq > (1 - p)(1 - q), then there exist constant 0 < [[epsilon].sub.3] < 1 such that

(1 + [[epsilon].sub.3])(1 - q) > (1 - [[epsilon].sub.3])p, (1 + [[epsilon].sub.3])(1 - p) > (1 - [[epsilon].sub.3])q. (4-7)

Hence, AL = [(-[[epsilon].sub.3], -[[epsilon].sub.3]).sup.T] yields

(1 - p)[l.sub.1] - [pl.sub.2] = -[[epsilon].sub.3], -[ql.sub.1] + (1 - q)[l.sub.2] = -[[epsilon].sub.3]. (4-8)

Namely,

[l.sub.1] = [[epsilon].sub.3](1 - q) + [[epsilon].sub.3]p/(1 - p)(1 - q) - pq, [l.sub.2] = [[epsilon].sub.3](1 - p) + [[epsilon].sub.3]p/(1 - p)(1 - q) - pq. (4-9)

Moreover, [l.sub.1], [l.sub.2] > 1. Therefore, if we choose

[k.sub.1] = min{[c.sub.1]/[l.sub.1], [c.sub.2]/[l.sub.2]}, [k.sub.2] = max{[[lambda].sub.1][l.sub.1]/[c.sub.1], [[lambda].sub.1][l.sub.2]/[c.sub.2]}, [r.sub.1] = 1, [r.sub.2] = [[epsilon].sub.3]. (4-10)

Then, [k.sub.1], [k.sub.2], [r.sub.2] > 0, [r.sub.1] [greater than or equal to] 1. Thus, assume that [u.sub.0](x), [v.sub.0](x) large enough to satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4-11)

Now, it follows from (4.1)-(4.11) that ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is a positive sub-solution of (1.1), which blows up in finite time since s(t) does.

(2) Next, we consider the case p, q < 1 and pq = (1 - p)(1 - q). Clearly, there exist positive constants [l.sub.1], [l.sub.2] > 1 such that

p([l.sub.1] + [l.sub.2]) - [l.sub.1] = 0, q([l.sub.1] + [l.sub.2]) - [l.sub.2] = 0. (4-12)

Without loss of generality, we may assume that 0 [member of] [OMEGA], and let [B.sub.R](0) be a ball such that [B.sub.R](0) [subset] [subset] [OMEGA]. In the following, we will prove that ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) blows up in finite time in the ball BR. Because of so, ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) does blow up in the larger domain [OMEGA].

Denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the first eigenvalue and the corresponding eigenfunction of the following eigenvalue problem

-[phi]"(r) - [N - 1/r][phi]'(r) = [lambda][phi](r), r [member of] (0, R); [phi]'(0) = 0, [phi](R) = 0.

It is well known that [[phi].sub.R](r) can be normalized as [[phi].sub.R](r) > 0 in [B.sub.R] and [[phi].sub.R](0) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By the scaling property (let [tau] = r/R) of eigenvalues and eigenfunctions we see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[phi].sub.R](r) = [[phi].sub.1](r/R) = [[phi].sub.1]([tau]), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[phi].sub.1]([tau]) are the first eigenvalue and the corresponding normalized eigenfunction of the eigenvalue problem in the unit ball B1(0). Moreover,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, we define the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where s(t) is confined as in (4.3). Then, a similar calculation as that of (4.4)-(4.6) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4-13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (4-14)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and [K.sub.1], [K.sub.2] are constants independent of R. Then, in view of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we may assume that R, that is, the ball [B.sub.R](0), is sufficiently large that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4-15)

We choose

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4-16)

Then, [k.sub.1], [k.sub.2] > 0. On the other hand, assume that [u.sub.0](x), [v.sub.0](x) large enough to satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4-17)

It follows from (4.12)-(4.17) that ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is a positive sub-solution of (1.1) in the ball [B.sub.R](0), which blows up in finite time since s(t) does.

Thus, according to the Lemma 2, the proof of Theorem 2 is completed.

Acknowledgements

The authors are supported by National Natural Science Foundation of China and they would like to express their many thanks to the Editor and Reviewers for their constructive suggestions to improve the previous version of this paper.

References

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Zhengqiu Ling ([dagger]), Zejia Wang

* This work is supported by the NNSF of China(11071100)

([dagger]) Corresponding author.

Received by the editors August 2011--In revised form in May 2012.

Communicated by P. Godin.

2010 Mathematics Subject Classification : 35K51; 35K55; 35B33.

Institute of Mathematics and Information Science, Yulin Normal University, Yulin 537000, Guangxi, PR China

email: lingzq00@tom.com

College of Mathematics and Informational Science, Jianxi Normal University, Nanchang 330022, PR China

email: wangzj1979@gmail.com

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Author: | Ling, Zhengqiu; Wang, Zejia |
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Publication: | Bulletin of the Belgian Mathematical Society - Simon Stevin |

Article Type: | Report |

Date: | Apr 1, 2013 |

Words: | 4615 |

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