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Blow-up for degenerate and singular nonlinear parabolic systems with nonlocal source.

1 Introduction

In this paper, we consider degenerate and singular nonlinear reaction-diffusion equations with nonlocal source of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where [u.sub.0](x), [v.sub.0](x) [member of] [C.sup.2+[alpha]]([0, a]) for some [alpha] [member of] (0, 1) are nonnegative nontrivial functions. [u.sub.0](0) = [u.sub.0](a) = [v.sub.0](0) = [v.sub.0](a) = 0, [u.sub.0] and [v.sub.0] satisfy compatibility conditions, T > 0, a > 0, [r.sub.1], [r.sub.2] [member of] [0, 1), [absolute value of [p.sub.1]] + [r.sub.1] [not equal to] 0, [absolute value of [p.sub.2]] + [r.sub.2] [not equal to] 0.

Let D = (0, a) and [[OMEGA].sub.t] = D x (0, t]. [bar.D] and [[bar.[OMEGA]].sub.t] are their closures, respectively. Since [absolute value of [p.sub.1]] + [r.sub.1] [not equal to] 0, [absolute value of [p.sub.2]] + [r.sub.2] [not equal to] 0, the coefficients of [u.sub.t], [u.sub.x], [u.sub.xx] and [v.sub.t], [v.sub.x], [v.sub.xx] may tend to 0 or [infinity] as x tends to 0, and thus we can regard the equations as degenerate and singular.

Floater [11] and Chan and Liu [6] investigated the blow-up properties of the problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

The motivation for studying problem (1.2) comes form Ockendon's model (see [15]) for the flow in a channel of a fluid whose viscosity depends on temperature

[x.sub.ut] = [u.sub.xx] + [e.sup.u],

where u represents the temperature of the fluid. Floater in [11] approximated [e.sup.u] by [u.sup.p] and considered equation (1.2). In [4], Chan and Chan considered the problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

For q = 0, it is the heat equation; the problem (1.3) and (1.2) (cf. [18, p. 10]) may be used to describe the temperature u(x, t) of a homogeneous and isotropic rod having a constant cross-sectional area with respect to x, and a thermal conductivity K independent of x; inside the rod, there is a nonlinear source producing heat (due to an exothermic reaction) at K f(u) per unit volume per unit time; the object has an initial distribution of temperature [u.sub.0](x), and the temperature at each of its ends is kept at zero. For q = 1, the problem (1.3) may be used to describe the temperature u of the channel flow of a fluid with a temperature-dependent viscosity in the boundary layer (cf. [5,15]); here, x and t denote the coordinates perpendicular and parallel to the channel wall, respectively; hence, [t.sub.b] corresponds to the downstream position where u blows up at some x. In a heat conduction

problem with t denoting the time, the term [x.sup.q] corresponds to the reciprocal of the diffusivity (cf. [2, p. 9]); thus for q > 0, the amount of heat required to raise the temperature of the object approaches to zero as x tends to zero; also for a fixed x [member of] D, [x.sup.q] is a decreasing function of q; physically, decreasing x or increasing q has the effect of shifting the blow-up point towards x = 0.

In [8], Chen and Xie discussed the degenerate and singular semilinear parabolic equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

They established the local existence and uniqueness of a classical solution. Under appropriate hypotheses, they obtained some sufficient conditions for the global existence and blow-up of a positive solution.

In [9], Chen et al. consider the following degenerate nonlinear reaction diffusion equation with nonlocal source

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

They established the local existence and uniqueness of a classical solution. Under appropriate hypotheses, they also got some sufficient conditions for the global existence and blow-up of a positive solution. Furthermore, under certain conditions, it is proved that the blow-up set of the solution is the whole domain.

Very recently, Jun Zhou et al. [18] generalized the results of [9] and investigated the blow-up properties of the following parabolic system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

Under certain conditions, Jun Zhou et al. proved that the blow-up set of the solution of (1.5) is the whole domain. The existence of a unique classical nonnegative solution is established and sufficient conditions for solution that exist globally or blows up in finite time are obtained.

In [14], J. Li et al. considered the effect of the singularity, degeneracy and localized reaction on the behavior of the solution of following problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

and show that the blow-up set of the solution of (1.6) is the whole domain.

Motivated by the results of the papers [8, 9, 18], we slightly modify the method developed by Jun Zhou et al. [18] and Y. Chen et al. [9] and extend the results of [9, 18] to a degenerate and singular parabolic system (1.1). The difficulties are the construction of the corresponding upper solution of (1.1). It is different from [6, 7, 11, 18] that under certain conditions the blow-up set of the solution of (1.1) is the whole domain. But this is consistent with the conclusions in [1, 16, 17].

Before stating our main results, we make some assumptions on the initial data [u.sub.0](x), [v.sub.0](x) and f(s), g(s) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This paper is organized as follows. In the next section, we show the existence of a unique classical solution. In Section 3, we give some criteria for the solution (u(x, t), v(x, t)) to blow-up in finite time and discuss the blow-up set.

2 Local Existence

In this section, we start with the definition of an upper solution of system (1.1).

Definition 2.1. A pair of nonnegative functions ([bar.u](x, t), [bar.v](x, t)) is called an upper solution of (1.1) if ([bar.u](x, t), [bar.v](x, t)) [member of] [(C([0, a] x [0, T)).sup.2] is such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

Similarly, ([u.bar](x, t), [v.bar](x, t)) [member of] [(C([0, a] x [0, T)).sup.2] is called a lower solution if it satisfies all the reversed inequalities in (2.1).

In order to prove the existence of a unique positive solution to (1.1), we must construct the following comparison principle.

Lemma 2.2. Let [b.sub.1](x, t) and [b.sub.2](x, t) be continuous nonnegative functions defined on [0, a] x [0, r] for any r [member of] (0, T), and let (u(x, t), v(x, t)) [member of] (C([[bar.[OMEGA]].sub.r]) [intersection] [C.sup.2,1][([[OMEGA].sub.r]).sup.2] satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then u(x, t) [greater than or equal to] 0 and v(x, t) [greater than or equal to] 0 on [0, a] x [0, T).

Proof. Jun Zhou et al. proved this lemma in [18], so we omit it.

Lemma 2.3. Let (u, v) be the nonnegative solution of (1.1). Let us assume that a pair of nonnegative functions (w(x, t), z(x, t)) [member of] (C([[OMEGA].sub.r]) [intersection] [C.sup.2,1][([[OMEGA].sub.r])).sup.2] is such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

Then (w(x, t), z(x, t)) [greater than or equal to] ([less than or equal to])(u(x, t), v(x, t)) on [0, a] x [0, T).

Proof. We only consider the case "[greater than or equal to]" (as for the other case "[less than or equal to]" the proof is similar). Let [[psi].sub.1](x, t) = w(x, t) - u(x, t) and [[psi].sub.2](x, t) = z(x, t) - v(x, t). Subtracting (1.1) from (2.2) and using the mean value theorem, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[eta].sub.1] and [[eta].sub.2] are some intermediate values between (w, u) and (z, v) satisfying f'([[eta].sub.1]), g'([[eta].sub.2]) [greater than or equal to] 0. Then Lemma 2.2 ensures that ([[psi].sub.1](x, t), [[psi].sub.2](x, t)) [greater than or equal to] (0, 0), that is, (w(x, t), z(x, t)) [greater than or equal to] (u(x, t), v(x, t)) on [0, a] x [0, T).

Obviously, (0, 0) is a lower solution of (1.1), and we need to construct an upper solution. We modify the proof of Jun Zhou et al. [18, Lemma 2.2] to show the following result.

Lemma 2.4. There exists a positive constant [t.sub.0]([t.sub.0] < T) such that the problem (1.1) has an upper solution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and let [K.sub.0] be a positive constant such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Introduce the positive constants [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [K.sub.10] [member of] (0, 1 - [r.sub.1]/2 - [r.sub.1]) and [K.sub.20] [member of] (0, 1 - [r.sub.2]/2 - [r.sub.2]) be positive constants such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [K.sub.1](t), [K.sub.2](t) be the positive solutions of the initial value problems

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

Since [K.sub.1](t) and [K.sub.2](t) are increasing functions, we can choose [t.sub.0] > 0 such that

[K.sub.1](t) [less than or equal to] 2[K.sub.0], [K.sub.2](t) [less than or equal to] 2[K.sub.0] for all t [member of] [0, [t.sub.0]].

Set

[h.sub.1](x,t) = [f.sup.-1] ([K.sub.1(t)][psi](x) + f(0)), [h.sub.2](x,t) = [g.sup.-1] (K.sub.2](t)[psi](x) + g(0)).

Then [h.sub.1](x, t) [greater than or equal to] 0 and [h.sub.2](x, t) [greater than or equal to] 0 on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We show that ([h.sub.1](x, t), [h.sub.2](x, t)) is an upper solution of (1.1) in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. To do this, let us construct two functions [J.sub.1] and [J.sub.2] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For (x, t) [member of] (0, a[K.sub.10]) x (0, [t.sub.0]] [union] (a(1 - [K.sub.10]), a) x (0, [t.sub.0]], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For (x, t) [member of] (0, a[K.sub.20]) x (0, [t.sub.0]] [union] (a(1 - [K.sub.20]), a) x (0, [t.sub.0]], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For (x, t) [member of] [a[K.sub.10], a(1 - [K.sub.10])] x (0, [t.sub.0]] by (2.3), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For (x, t) [member of] [a[K.sub.20], a(1 - [K.sub.20])] x (0, [t.sub.0]] by (2.4), we can get [J.sub.2] [greater than or equal to] 0 with the same argument as that for [J.sub.1]. Thus, [J.sub.1](x, t) [greater than or equal to] 0, [J.sub.2](x, t) [greater than or equal to] 0 in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since f'(s) > 0 and g'(s) > 0 in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So ([h.sub.1](x, t), [h.sub.2](x, t)) is an upper solution of (1.1). The proof is complete.

To show the existence of the classical solution (u(x, t), v(x, t)) of (1.1), let us introduce a cutoff function [rho](x). By Dunford and Schwartz [10, p. 1640], there exists a nondecreasing [rho](x) [member of] [C.sup.3](R) such that [rho](x) = 0 if x [less than or equal to] 0 and [rho](x) = 1 if x [greater than or equal to] 1. Let 0 < [delta] < min {1 - [r.sub.1]/2 - [r.sub.1] a, 1 - [r.sub.2]/2 [r.sub.2] a},

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [u.sub.0[delta]](x) = [[rho].sub.[delta]](x)[u.sub.0](x), [v.sub.0[delta]](x) = [[rho].sub.[delta]](x)[v.sub.0](x). We note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [rho] is nondecreasing, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [D.sub.[delta]] = ([delta], a), let [w.sub.[delta]] = [D.sub.[delta]] x (0, [t.sub.0]], let [[bar.D].sub.[delta]] and [[bar.w].sub.[delta]] be their respective closures, and let [S.sub.[delta]] = {0, a} x (0, [t.sub.0]]. We consider the following regularized problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

By using Schauder's fixed point theorem, we have the following theorem.

Theorem 2.5. The problem (2.5) admits a unique nonnegative solution

([u.sub.[delta]], [v.sub.[delta]]) [member of] [([C.sup.2+[alpha],1+[alpha]/2([[bar.w].sub.[delta]])).sup.2].

Moreover, 0 [less than or equal to] [u.sub.[delta]] [less than or equal to] [h.sub.1](x, t), 0 [less than or equal to] [v.sub.[delta]] [less than or equal to] [h.sub.2](x, t), (x, t) [member of] [[bar.w].sub.[delta]], where [h.sub.1](x, t) and [h.sub.2](x, t) are given by Lemma 2.4.

Proof. By using Lemma 2.3, there exists at most one nonnegative solution ([u.sub.[delta]], [v.sub.[delta]]). To prove existence, we use the Schauder fixed point theorem. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We note that [X.sub.1] and [X.sub.2] are closed convex subsets of the Banach space [C.sup.[alpha],[alpha]/2]([[bar.w].sub.[delta]]). In order to obtain the conclusion, we define another set X = [X.sub.1] x [X.sub.2]. Obviously ([C.sup.[alpha],[alpha]/2][([[bar.w].sub.[delta]])).sup.2] is a Banach space with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and X is a closed convex subset of the Banach space [([C.sup.[alpha],[alpha]/2]([[bar.w].sub.[delta]])).sup.2]. For any ([v.sub.1], [u.sub.1]) [member of] ([X.sub.1] x [X.sub.2]), let us consider the following linearized uniformly parabolic problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

By construction, (0, 0) and ([h.sub.1](x, t), [h.sub.2](x, t)) are lower and upper solutions of problem (2.6). We also note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from Ladde et al. [12, Theorem 4.2.2 on p. 143] that problem (2.6) has a unique solution ([W.sub.[delta]] (x, t; [v.sub.1], [u.sub.1]). [Z.sub.[delta]] (x, t; [v.sub.1], [u.sub.1])) [member of] [([C.sup.2+[alpha],1+[alpha]/2([[bar.w].sub.[delta]])).sup.2] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, we can define a mapping T from X into [([C.sup.2]+[alpha],1+[alpha]/2]([[bar.w].sub.[delta]])).sup.2] such that

T([v.sub.1](x, t), [u.sub.1](x, t)) = ([W.sub.[delta]](x, t; [v.sub.1], [u.sub.1]),[Z.sub.[delta]](x, t; [v.sub.1], [u.sub.1])),

where ([W.sub.[delta]](x, t; [v.sub.1], [u.sub.1]), [Z.sub.[delta]](x, t; [v.sub.1], [u.sub.1])) denotes the unique solution of (2.6) corresponding to ([v.sub.1](x, t), [u.sub.1](x, t)) [member of] X. To use the Schauder fixed point theorem, we need to verify that T maps X into itself and that T is continuous and compact. In fact, TX [subset] X, and the embedding operator from the Banach space [([C.sup.2]+[alpha],1+[alpha]/2]([[bar.w].sub.[delta]])).sup.2] to the Banach space [([C.sup.[alpha][alpha]/2]([[bar.w].sub.[delta]])).sup.2] is compact. Therefore, T is compact. To show that T is continuous, let us consider sequence [v.sub.1n](x, t) which converges to [v.sub.1](x, t) uniformly and [u.sub.1n](x, t) which converges to [u.sub.1](x, t) uniformly in the norm [[parallel]x[parallel].sub.[alpha],[alpha]/2]. We know that [v.sub.1](x, t) [member of] [X.sub.1] and [u.sub.1](x, t) [member of] [X.sub.2]. So we get a sequence {([v.sub.1n](x, t), [u.sub.1n](x, t))} [member of] X, which converges to ([v.sub.1](x, t), [u.sub.1](x, t)) uniformly in the norm [[parallel]x, x[parallel].sub.[alpha],[alpha]/2]. Let ([W.sub.[delta]n](x, t), [Z.sub.[delta]n](x, t)) and ([W.sub.[delta]](x, t), [Z.sub.[delta]](x, t)) be the solutions of (2.6) corresponding to ([v.sub.1n](x, t), [u.sub.1n](x, t)) and ([v.sub.1](x, t), [u.sub.1](x, t)), respectively. Without loss of generality, let us assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let W(x, t) = [W.sub.[delta]n](x, t) - [W.sub.[delta]](x, t),Z(x, t) = [Z.sub.[delta]n](x, t) - [Z.sub.[delta]](x, t). Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From Ladyzenskaja et al. [13, Theorem 4.5.2 on p. 320], there exist positive constants [C.sub.1] (independent of g, [v.sub.1n] and [v.sub.1]) and [C.sub.2] (independent of f, [u.sub.1n] and [u.sub.1]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and we note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [tau] [member of] (0, 1). Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This shows that the mapping T is continuous. By the Schauder fixed point theorem, the proof is complete.

Now we can prove the following local existence result.

Theorem 2.6. There exists some [t.sub.0] < T such that problem (1.1) has a unique nonnegative solution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By Theorem 2.5, the problem (2.5) has a unique nonnegative solution ([u.sub.[delta]], [v.sub.[delta]]) [member of] ([C.sup.2+[alpha],1+[alpha]/2][([[bar.w].sub.[delta]])).sup.2]. It follows from Lemma 2.3 that ([u.sub.[delta]1], [v.sub.[delta]1]) [less than or equal to] ([u.sub.[delta]2], [v.sub.[delta]2]) if [delta]1 > [delta]2. Therefore, l[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and define (u(0, t), v(0, t)) = (0, 0), t [member of] [0, [t.sub.0]]. We show that (u(x, t), v(x, t)) is a classical solution of (1.1) in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For any [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], there exist three domains

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that ([x.sub.1], [t.sub.1]) [member of] Q' [subset] Q" [subset] Q"' [subset] (0, a) x (0, [t.sub.0]] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By the conditions of f and g, we know that [h.sub.1](x, t) and [h.sub.2](x, t) are finite on [bar.Q]'". For any constant q > 1 and some positive constants [K.sub.3] and [K.sub.4], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [a.sup.*.sub.1] = [a'".sub.1] if [q.sub.1] [greater than or equal to] 0, [a.sup.*.sub.1] = [a'".sub.2] if [q.sub.1] < 0, and [a.sup.*.sub.2] = [a'".sub.1] if [q.sub.2] [greater than or equal to] 0, [a.sup.*.sub.2] = [a'".sub.2] if [q.sub.2] < 0. By the local [L.sup.p]-estimate of Ladyzenskaja et al. [13, pp. 341-342, 352], ([u.sub.[delta]], [v.sub.[delta]]) [member of] [([W.sup.2,1.sub.q] (Q")).sup.2]. By the embedding theorem in [15, pp. 61, 80], [W.sup.2,1.sub.q] (Q") [??] [H.sup.[alpha],[alpha]/2](Q") if we choose q > 2/(1 - [alpha]). Then, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for some positive constant [K.sub.5], and we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some positive [K.sub.6], which is independent of [delta], where [tau] [member of] (0, 1). By Ladyzenskaja et al. [16, pp. 351-352], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some positive constant [K.sub.7] independent of [delta]. This implies that [u.sub.[delta]], [u.sub.[delta]t], [u.sub.[delta]x], [u.sub.[delta]xx] and [v.sub.[delta]], [v.sub.[delta]t], [v.sub.[delta]x], [v.sub.[delta]xx] are equicontinuous in Q'. By the Ascoli-Arzela theorem, we know that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some [alpha]' [member of] (0, [alpha]) and some positive constant [K.sub.8] independent of [delta], and that the derivatives of u and v are uniform limits of the corresponding partial derivatives of [u.sub.[delta]] and [v.sub.[delta]], respectively. Hence (u(x, t), v(x, t)) satisfy (1.1), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from 0 [less than or equal to] u(x, t) [less than or equal to] [h.sub.1](x, t), 0 [less than or equal to] v(x, t) [less than or equal to] [h.sub.2](x, t) and [h.sub.1](x, t) [right arrow] 0, [h.sub.2](x, t) [right arrow] 0 as x [right arrow] 0 or x [right arrow] a that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the solution of (1.1) in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This completes the proof.

Theorem 2.7. Let T be the supremum over [t.sub.0] for which there is a unique nonnegative solution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then (1.1) has a unique nonnegative solution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. The proof of this theorem is similar to the proof of [11, Theorem 2.5], so we omit it.

3 Blow-up of Solutions

In this section, we give some global blow-up result of the solution of (1.1). In order to obtain the blow-up result, we assume that [p.sub.1] [greater than or equal to] [r.sub.1] - 1, [p.sub.2] [greater than or equal to] [r.sub.2] - 1 and f(s), g(s) satisfy

f(s) + g(t) [greater than or equal to] [eta] min{f(s + t), g(s + t)} [equivalent to] h(s + t), (3.1)

for some positive constant [eta].

Remark 3.1. Since [(s + t).sup.p] [less than or equal to] [2.sup.p-1]([s.sup.p] + [t.sup.p]), power functions satisfy the property (3.1).

Now, we consider the eigenvalue problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

Using the argument by Jun Zhou et al. [18], we can make [[psi].sub.1](x) satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

Analogously, we consider the eigenvalue problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

As above, we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

Let [C.sub.1] = [[integral].sup.a.sub.0] [[psi].sub.1](x)dx, [C.sub.2] = [[integral].sup.a.sub.0] [[psi].sub.2](x)dx and [lambda] = max{[[lambda].sub.1], [[lambda].sub.2]}, C = min{[C.sub.1], [C.sub.2]}.

Then we have the following result.

Theorem 3.2. Let (u(x, t), v(x, t)) be the solution of problem (1.1). Then the solution of (1.1) blows up in finite time.

Proof. We set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By (1.1), (3.2) and (3.4), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using Jensen's inequality and (3.3), (3.5), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

and

V'(t) [greater than or equal to] - [lambda]V (t) + Caf (U(t)/a). (3.7)

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In fact, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By h"(y) [greater than or equal to] 0 (by f"(s) [greater than or equal to] 0, g"(s) [greater than or equal to] 0), we have that h'(y) is nondecreasing if y > 0. Using L'Hospital's principle, we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Assume by contradiction that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then there exists [y.sub.0] [greater than or equal to] [s.sub.0] such that h(y) [less than or equal to] 3/2 N y, and we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we know that there exists [s.sub.1] [greater than or equal to] [s.sub.0] such that h(y)/y [greater than or equal to] 2[lambda]/C if y [greater than or equal to] [s.sub.1]. Let ([u.sub.0](x), [v.sub.0](x)) be sufficiently larger such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, by (3.6), (3.7), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and integrating this inequality over t from 0 to T, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This completes the proof.

Now, we discuss the global blow-up under the following hypothesis.

Case 1: [p.sub.1] > 0, [r.sub.1] = 0 or [p.sub.2] > 0, [r.sub.2] = 0.

Chan et al. [3,7] showed that Green's function G(x, [xi], t - [tau]) associated with the operator L = [x.sup.p]([partial derivative]/[partial derivative]t) - [[partial derivative].sup.2]/[[partial derivative].sup.2]x with the first boundary condition exists. For ease of reference, we state their results in the following lemma.

Lemma 3.3. (i) For t > [tau], G(x, [xi], t - [tau]) is continuous for (x, t, [xi], [tau]) [member of] ([0, a] x (0, T]) x ((0, a] x [0, T)).

(ii) For each fixed ([xi], [tau]) [member of] (0, a] x [0, T), [G.sub.t](x, [xi], t - [tau]) [member of] C([0, a] x ([tau], T]).

(iii) In {(x, t, [xi], [tau]) : x and [xi] are in (0, a), T [greater than or equal to] t > [tau] [greater than or equal to] 0}, G(x, [xi], t - [tau]) is positive.

Lemma 3.4. For fixed [x.sub.0] [member of] (0, a], given any x [member of] (0, a) and any finite time T, there exist positive constants [C.sub.1] (depending on x and T) and [C.sub.2] (depending on T) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we give the global blow-up result.

Theorem 3.5. Under the assumption of Case 1, if the solution of (1.1) blows up at the point [x.sub.0] [member of] (0, a), then the blow-up set of the solution of (1.1) is [0, a].

Proof. Obviously, the system (1.1) is completely coupled. Therefore, u and v blow up simultaneously if the solution (u, v) blows up in finite time. Without loss of generality, we assume [p.sub.1] > 0, [r.sub.1] = 0, and u(x, t) blows up in finite time T. By Green's second identity we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

for any (x, t) [member of] (0, a) x (0, T). Since u(x, t) blows up at x = [x.sub.0], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By (3.8) and Lemma 3.4, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.9)

On the other hand, for any x [member of] (0, a), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from the above inequality and (3.9) that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For any [??] [member of] {0, a}, we can always find a sequence {([x.sub.n, [t.sub.n])} such that ([x.sub.n], [t.sub.n]) [right arrow] ([??], T)(n [right arrow] 1) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, the blow-up set is [0, a], and this completes the proof.

Case 2: [p.sub.1] = 0, 0 [less than or equal to] [r.sub.1] < 1 or [p.sub.2] = 0, 0 [less than or equal to] [r.sub.2] < 1.

We assert that the blow-up set is the whole domain under certain assumptions.

Theorem 3.6. Under the assumption of Case 2 and if there exists M [member of] (0, + [infinity]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if the solution of (1.1) blows up at the point [x.sub.0] [member of] (0, a), then the blow-up set of the solution of (1.1) is [0, a].

Proof. The proof is similar to the proof presented in [8, 18], so we omit it.

Acknowledgements

This project was supported by the National Natural Science Foundation of China (Grant No. 10871060). and by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 08KJB110005)

Received September 10, 2007; Accepted June 22, 2008 Communicated by Kumbakonam Rajagopal

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[3] C. Y. Chan and W. Y. Chan. Existence of classical solutions for degenerate semilinear parabolic problems. Appl. Math. Comput., 101(2-3):125-149, 1999.

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[6] C. Y. Chan and H. T. Liu. Global existence of solutions for degenerate semilinear parabolic problems. Nonlinear Anal., 34(4):617-628, 1998.

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[8] You Peng Chen and Chun Hong Xie. Blow-up for degenerate, singular, semilinear parabolic equations with nonlocal source. Acta Math. Sinica (Chin. Ser.), 47(1):41-50, 2004.

[9] Youpeng Chen, Qilin Liu, and Chunhong Xie. Blow-up for degenerate parabolic equations with nonlocal source. Proc. Amer. Math. Soc., 132(1):135-145 (electronic), 2004.

[10] Nelson Dunford and Jacob T. Schwartz. Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space. With the assistance of William G. Bade and Robert G. Bartle. Interscience Publishers John Wiley & Sons New York-London, 1963.

[11] M. S. Floater. Blow-up at the boundary for degenerate semilinear parabolic equations. Arch. Rational Mech. Anal., 114(1):57-77, 1991.

[12] G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala. Monotone iterative techniques for nonlinear differential equations. Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics, 27. Pitman (Advanced Publishing Program), Boston, MA, 1985.

[13] O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'ceva. Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1967.

[14] Juan Li, Zejian Cui, and Chunlai Mu. Global existence and blow-up for degenerate and singular parabolic system with localized sources. Appl. Math. Comput., 199(1):292-300, 2008.

[15] J. R. Ockendon. Channel flow with temperature-dependent viscosity and internal viscous dissipation. J. Fluid Mech., 93:737-746, 1979.

[16] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhaiklov. Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii. "Nauka", Moscow, 1987.

[17] Philippe Souplet. Blow-up in nonlocal reaction-diffusion equations. SIAM J. Math. Anal., 29(6):1301-1334 (electronic), 1998.

[18] Jun Zhou, Chunlai Mu, and Zhongping Li. Blowup for degenerate and singular parabolic system with nonlocal source. Bound. Value Probl., Art. ID 21830, 1-19, 2006.

Congming Peng * and Zuodong Yang ([dagger])

Nanjing Normal University Institute of Mathematics School of Mathematics and Computer Science Jiangsu Nanjing 210097, China zdyang_jin@263.net

* Second Address: Math Dept, Tianshui Normal University, Gansu Tianshui 741001, China

([dagger]) Second Address: College of Zhongbei, Nanjing Normal University, Jiangsu Nanjing 210046, China
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