A NEW SECOND CRITICAL EXPONENT AND LIFE SPAN FOR A QUASILINEAR DEGENERATE PARABOLIC EQUATION WITH WEIGHTED NONLOCAL SOURCES

. In this paper, we consider positive solutions of a Cauchy problem for the following quasilinear degenerate parabolic equation with weighted non- local sources: where N ≥ 1, p > 2, q , r ≥ 1, s ≥ 0, and r + s > 1. We classify global and non-global solutions of the equation in the coexistence region by ﬁnding a new second critical exponent via the slow decay asymptotic behavior of an initial value at spatial inﬁnity, and the life span of non-global solution is studied.

where N ≥ 1, p > 2, q, r ≥ 1, s ≥ 0, and r + s > 1. We classify global and non-global solutions of the equation in the coexistence region by finding a new second critical exponent via the slow decay asymptotic behavior of an initial value at spatial infinity, and the life span of non-global solution is studied.
1. Introduction. It is well known that the positive solution of a Cauchy problem for the semilinear parabolic equation possesses the critical Fujita exponent s c = 1 + 2 N (cf. [3]). Fujita [3] also showed that the positive solution of the Cauchy problem (1) blows up at finite time for any nontrivial initial data, whenever 1 < s < s c ; while there are global solutions for small initial data and non-global solutions for large initial data, if s > s c . Furthermore, Hayakawa [7] and Weissler [14] proved that the critical case s = s c belongs to blowup case. From then on, considerable attention has been paid to the study on the critical Fujita exponent for the parabolic equation with local or nonlocal sources. For a parabolic equation with local source, Galaktionov [5] considered the positive solution of a Cauchy problem for the following quasilinear degenerate parabolic equation: where p > 2, and obtained the critical Fujita exponent s c = p−1+ p N . Afterwards, Qi [13] pointed out that the critical case s = s c belongs to blow-up case. Concerning a nonlocal source problem, Galaktionov and Levine [6] investigated the positive 1698 LINGWEI MA AND ZHONG BO FANG solution of a Cauchy problem for the following semilinear parabolic equation with weighted nonlocal source: where q, r ≥ 1, s ≥ 0, and r+s > 1, and they derived the critical Fujita exponent by the parameter r to classify solutions of the equation. When the nonnegative weight function K(x) belongs to L 1 (R N ), the critical Fujita exponent r c = 1 + 2 N − s; while if the nonnegative weight function K(x) does not belong to L 1 (R N ) and for N s 2 < 1, which is included in blow-up case. Moreover, they also considered the p-Laplace equation with weight nonlocal sources when K(x) ∈ L 1 (R N ) and found the critical Fujita exponent r c = p − 1 − s + p−2 q + p N for s < p−2 q + p N , which is included in blow-up case. Afterwards, Afanas'eva and Tedeev [1] studied the positive solution of a Cauchy problem for the following doubly degenerate parabolic equation with weighted nonlocal sources: They obtained the critical Fujita exponent s c = p + l − 1 + p+1 N − (m + N (q − 1)) r−1 N q , with respect to the parameter s, but this Fujita exponent does not belong to blow-up case.
Note that for the critical Fujita exponent, the region satisfying s > s c or r > r c is a coexistence region of global and non-global solutions for the Cauchy problem. In order to identify global and non-global solutions in the coexistence region, Lee and Ni [8] introduced a new second critical exponent α * = 2 s−1 for problem (1) with s > s c = 1 + 2 N by virtue of the slow decay behavior of the initial data at spatial infinity. More precisely, for problem (1) with initial data u 0 (x) = λϕ(x) and s > s c = 1 + 2 N , there exist constants µ, Λ, Λ 0 > 0 such that the solution blows up in finite time, whenever lim inf |x|→∞ |x| α * ϕ(x) > µ > 0 and λ > Λ, or exists globally, if lim sup |x|→∞ |x| α ϕ(x) < ∞ with α ≥ α * and λ < Λ 0 . Afterwards, Mu et al.
[10] considered problem (2) with s > s c = p − 1 + p N , and they derived a new second critical exponent α * = p s+1−p and a life span of non-global solution. Moreover, concerning the second critical exponent for the Cauchy problem of porous medium equation or doubly degenerate parabolic equation with local sources, one can refer to [9,11]. On the nonlocal problem, recently, Yang et al. [15] studied problem (3) with r > r c , K(x) ∈ L ∞ (R N ) ∩ C(R N ), and K(x) ∼ |x| −m for |x| large, and they found a new second critical exponent α * = 2q+(r−1)(N −m)+ q(r+s−1) .
To the best of our knowledge, much less effort has been devoted to the second critical exponent and life span for the Cauchy problem of a quasilinear degenerate parabolic equation with weighted nonlocal sources. At a glance, our main difficulties lie in the treatment of p-Laplace operator and weighted nonlocal source, and the selection of test function. Motivated by the above works, we investigate the parabolic p-Laplace equation with weighted nonlocal source subject to the initial condition where N ≥ 1, p > 2, q, r ≥ 1, s ≥ 0, r + s > 1, and the initial data u 0 (x) is a nonnegative and continuous function. Meanwhile, the nonnegative weight function Problem (4)-(5) arises in the theory of quasiregular and quasiconformal mappings, which can describe non-Newtonian flux in the mechanics of fluid, population of biological species, and so on (cf. [12,4]). The essential purpose of this paper is to seek the effect of the slow decay behavior of initial data at spatial infinity for the positive solution of problem (4)-(5) and to derive a life span of non-global solution.
Firstly, recall that the critical Fujita exponent r c to problem (4)-(5), given by Galaktionov and Levine [6], is such that Throughout the rest of this paper, C b (R N ) denotes the space of all bounded continuous functions in R N and let Moreover, u(x, t) denotes the solution of problem (4)-(5) with r > r c and u 0 (x) = λϕ(x). Then we can establish a new second critical exponent α * = pq+(r−1)(N −m)+ q(r+s+1−p) .
The main results of this paper are as follows: and r > r c , we have ϕ(x) ∈ Π α and 0 < α < α * or α ≥ α * with λ large enough. Then the solution u(x, t) of problem (4)-(5) blows up at finite time.
Theorem 1.2. Suppose that for p > 2 and r > r c , we have ϕ(x) ∈ Π α and α ≥ α * . Then there exist positive constants λ 0 and C such that the solution u(x, t) of problem (4)-(5) exists globally, if λ ∈ (0, λ 0 ), and u(x, t)satisfies the inequality Then the life span of u(x, t) satisfies The rest of this paper is organized as follows: In Section 2, by virtue of the test function method, we prove Theorem 1.1. Theorem 1.2 is proved in Section 3 by establishing a global supersolution for problem (4)- (5). Finally, we obtain a life span of the non-global solution for problem (4)-(5) in Section 4.

2.
Non-global solution. In this section, by using the test function method, we derive a sufficient condition for which the solution of (4)-(5) blows up at finite time. We give a proof of Theorem 1.1 below: Proof. Since ϕ(x) ∈ Π α and K(x) ∼ |x| −m for |x| large enough, there exist positive constants R 0 , c 1 , and c 2 such that ϕ(x) ≥ c 1 |x| −α and K(x) ≥ c 2 |x| −m for |x| ≥ R 0 . Now, we define the following test function: Then it can be easily seen that and Next, we introduce the auxiliary function where 0 < σ < 1 p . Firstly, differentiating Θ(t), and using (6) and Green's formula, we get Applying Young's inequality to the second term on the right-hand side of (8), we have the inequality Since 0 < σ < 1 p , it follows from (8) and (9) that Then employing Hölder's inequality and (7) to the last term on the right-hand side of (10), we obtain the inequality by virtue of r > r c = p − 1 − s + p−2 q + p N and p > max 2, N [(r−1)q+2] N +q , where p+σ−2 σ+s ∈ (0, 1). Thus, substituting (11) into (10), one can derive the inequality In what follows, we claim that where c is a positive constant. Throughout this paper, we assume that c denotes a positive constant that is independent of x, t, λ, and , for convenience.
Proof. Firstly, we consider the following Cauchy problem: where M is a positive constant given in (26). It is known that the existence and uniqueness of the solution of (22)-(23) have been well established and the radially symmetric self-similar solution was given, see [16], where η = |x| t β , β = 1 α(p−2)+p , and the positive function f (η) is the solution of the following problem: The solution of problem (25)-(26) is decreasing, we can refer to [2]. Since ϕ(x) ∈ Π α , there exists a positive constant c 3 such that ϕ(x) ≤ c 3 (1+|x|) −α for all x ∈ R N . Hence, we can choose c 3 such that lim η→∞ η α f (η) = M > c 3 , and so, there is a positive constant R 1 such that By virtue of (27), it is not difficult to verify that there exists t 0 ∈ (0, 1) such that Next, we claim that Since K(x) ∈ L 1 (R N ), K(x) ∼ |x| −m for |x| large enough, and f (η) is decreasing, when m > N , we have the inequalities , which can be led to m > N − αq. We then obtain by a simple calculation.
Let h(t) be the solution of the following ordinary differential equation: where θ > 1. The local existence and uniqueness of the solution h(t) for (33) follow from the standard theory of initial value problem on ordinary differential equation. Afterwards, we claim that there exists λ 0 > 0 such that h(t) is bounded in [0, +∞) for all λ ∈ [0, λ 0 ). Integrating (33) over [0, t], one can have We then have h(t) ≤ h λ for all t ∈ [0, +∞) and λ ∈ [0, λ 0 ). Now, we construct the following global solution: where U (x, t + t 0 ) is the solution of (22)-(23) and h(t) solves (33) with θ = αβ(r + . Since f (η) is decreasing, it follows from (29) that where Moreover, utilizing (28) to the initial data, we havē Hence, combining (36) with (37), one can see thatū(x, t) is a global supersolution of problem (4)- (5). Furthermore, we can easily show that the decay estimate of the solution u(x, t) for (4)- (5) is for all t > 0. The proof is completed.
4. Life span. In this section, we will give a life span of the non-global solution u(x, t) for problem (4)-(5) by giving a proof of Theorem 1.3.
Proof. Firstly, we have already obtained an upper bound of the blow-up time for u(x, t) in the measure of Θ(t), given in the proof of Theorem 1.1, and the upper bound is given as Then, it follows from ϕ L ∞ (R N ) = lim |x|→∞ ϕ = ϕ ∞ that there exists R 2 > 0 such that |ϕ − ϕ ∞ | < ε for |x| > R 2 and any ε > 0. Meanwhile, by the definition of test function φ (x), we must have for R 0 > R 2 . Thus, from the arbitrariness of ε, let ε → 0 can yield that where c 5 = 2σ r+s+σ−1 σ c . On the other hand, in order to get a lower bound of the blow-up time for the non-global solution u(x, t), we construct a supersolution of (4)- (5). Consider the following ordinary differential equation: g (t) = 1 c4 g r+s (t), t > 0 g(0) = λϕ ∞ .
By a direct calculation, one can see that the solution g(t) of (39) is given by Now, applying c 4 ≤ R N K(x)dx − r−1 q and the comparison principle, it can be easily shown that g(t) is a supersolution of problem (4)- (5). We then obtain a lower bound of the blow-up time, i.e., The proof is completed.