1. Introduction
In this paper, we consider a slow diffusive p-Laplace equation:
(1)ut-div(|∇u|p-2∇u)=|u|q-1u-⨍Ω|u|q-1u dx, (x,t)∈Ω×(0,T),∂u∂n=0, (x,t)∈∂Ω×(0,T),u(x,0)=u0(x), x∈Ω
with ⨍Ωu0dx=0, where Ω is a bounded smooth domain Ω⊂ℝN, p>2, q>p-1, and u0∈L∞(Ω)∩W1,p(Ω), u0≢0, and denote ⨍Ωf dx=(1/|Ω|)∫Ωf dx. It is easy to check that ∫Ωu dx=0; that is, the mass of u is conserved.

The problem (1) with p=2 can be used to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved [1, 2]. If p>2, the problem (1) is the degenerate parabolic equation and appears to be relevant in the theory of non-Newtonian fluids (see [3]). Here, we are mainly interested in the case p>2, namely, the degenerate one. When p=2, (1) becomes the heat equation which has been deeply studied in [4, 5]. When 1<p<2, (1) is singular, which can be handled similar to that of [6].

As an important feature of many evolutionary equations, the properties of blow-up solution have been the subject of intensive study during the last decades. Among those investigations in this area, it was Fujita [7] who first established the so-called theory of critical blow-up exponents for the heat equation with reaction sources in 1966, which can be, of course, regarded as the elegant description for either blow-up or global existence of solutions. From then on, there has been increasing interest in the study of critical Fujita exponents for different kinds of evolutionary equations; see [8, 9] for a survey of the literature. In recent years, special attention has been paid to the blow-up property to nonlinear degenerate or singular diffusion equations with different nonlinear sources, including the inner sources, boundary flux, or multiple sources; see, for example, [3, 10, 11].

In some situations, we have to deal with changing sign solutions. For instance, the changing sign solutions were considered in [1] for the nonlocal and quadratic equation
(2)ut=Δu+u2-⨍Ωu2dx
with the Neumann boundary condition. The study in [5] for
(3)ut=Δu+|u|p-⨍Ω|u|pdx,
a natural generalization of (2), proposed with 1<p≤2 a global existence result (for small initial data) and a new blow-up criterion (based on the partial maximum principle and a Gamma-convergence argument). The authors also conjectured that the solutions blow up when p>2, which was then proved with a positive answer [4]. The changing sign solutions to the reaction-diffusion equation
(4)ut=Δu+f(u,k(t))
were discussed in [2], with such as f(u,k(t))=|u|p-1u-k(t). The blow-up of solutions was obtained even under the source with ⨍Ωf dx=0. The semilinear parabolic equation [12]
(5)ut=Δu+|u|p-1u-⨍Ω|u|p-1u dx
with a homogeneous Neumann’s boundary condition is studied. A blow-up result for the changing sign solution with positive initial energy is established. In [6], a fast diffusive p-Laplace equation with the nonlocal source
(6)ut-div(|∇u|p-2∇u)=|u|q-⨍Ω|u|qdx, (x,t)∈Ω×(0,T),∂u∂n=0, (x,t)∈∂Ω×(0,T),u(x,0)=u0(x), x∈Ω,
was considered. The authors showed that a critical blow-up criterion was determined for the changing sign weak solutions, depending on the size of q and the sign of the natural energy associated. The relationship between the finite time blow-up and the nonpositivity of initial energy was discussed, based on an energy technique.

Notice that (1) is degenerate if p>2 at points where ∇u=0; therefore, there is no classical solution in general. For this, a weak solution for problem (1) is defined as follows.

Definition 1.
A function u∈L∞(Ω×(0,T))∩Lp(0,T,W1,p(Ω)) with ut∈L2(Ω×(0,T)) is called a weak solution of (1) if
(7)∫0t∫Ω[u∂φ∂s-|∇u|p-2∇u·∇φ+(|u|q-⨍Ω|u|q)φ]dx ds =∫Ωu(x,t)φ(x,t)dx-∫Ωu0(x)φ(x,0)dx
holds for all φ∈C1(Ω¯×[0,T]).

The local existence of the weak solutions can be obtained via the standard procedure of regularized approximations [10]. Throughout the paper, we always assume that the weak solution is appropriately smooth for convenience of arguments, instead of considering the corresponding regularized problems.

This paper is organized as follows. In Section 2, we show that the solutions to (1) blow up with nonpositive initial energy. In Section 3, under some assumptions of initial data, we prove that the solutions with bounded initial energy also blow up in finite time.

2. Nonpositive Initial Energy Case
The technique used here is the same as in [4]; define the energy functional by
(8)E(t)=1p∫Ω|∇u|pdx-1q+1∫Ω|u|q+1dx.
and denote
(9)M(t)=12∫Ωu2(x,t)dx, H(t)=∫0tM(s)ds.

Theorem 2.
Assume that p>2, q>p-1, and u0∈L∞(Ω)∩W1,p(Ω), u0≢0, and let the initial energy
(10)E(0)=1p∫Ω|∇u0|pdx-1q+1∫Ω|u0|q+1dx
be nonpositive. Then, there exists T0 with 0<T0<∞, such that
(11)limt→T0M(t)=+∞.

We need three lemmas for the functionals E(t), M(t), and H(t), respectively.

Lemma 3.
The energy E(t) is a nonincreasing function and
(12)E(t)=E(0)-∫0t∫Ω(ut)2dx ds.

Proof.
A direct computation using (1) and by parts yields
(13)ddtE(t)=∫Ω(|∇u|p-2∇u·∇ut-|u|q-1uut)dx=∫Ω(-div(|∇u|p-2∇u)-|u|q-1u)utdx=∫Ω(-ut-⨍Ω|u|q-1u dx)utdx=-∫Ω(ut)2dx.
Integrate from 0 to t to get (12).

Lemma 4.
Assume that p>2, q>p-1, and E(0)≤0. Then, M(t) satisfies the following inequality:
(14)M′(t)≥(q+1)∫0t∫Ω(ut)2dx ds.

Proof.
An easy computation using (1) and the fact ∫Ωu dx=0 and by parts shows that
(15)M′(t)=∫Ωuutdx=∫Ωu(div(|∇u|p-2∇u)+|u|q-1u-⨍Ω|u|q-1u dx)=-∫Ω|∇u|pdx+∫Ω|u|q+1dx=-(q+1)E(t)+q+1-pp∫Ω|∇u|pdx.
The last equality implies
(16)M′(t)≥-(q+1)E(t)=-(q+1)E(0)+(q+1)∫0t∫Ω(ut)2dx ds≥(q+1)∫0t∫Ω(ut)2dx ds,
because of (12) of Lemma 3 and the assumption E(0)≤0.

Lemma 5.
Assume that p>2, q>p-1, and E(0)≤0. Then, H(t) satisfies
(17)q+12(H′(t)-H′(0))2≤H(t)H′′(t).

Proof.
Note the definition of M(t) and H(t), and a simple calculation shows that
(18)H′(t)-H′(0) =M(t)-M(0) =∫0tM′(s)ds=∫0t∫Ωuutdx ds ≤(∫0t∫Ωu2dx ds)1/2(∫0t∫Ω(ut)2dx ds)1/2 ≤(2q+1)1/2(H(t))1/2(M′(t))1/2 =(2q+1)1/2(H(t))1/2(H′′(t))1/2.
Furthermore,
(19)H′(t)-H′(0)=∫0tM′(s)ds≥(q+1)t∫0t∫Ω(ut)2dx ds≥0.
Therefore,
(20)q+12(H′(t)-H′(0))2≤H(t)H′′(t).

Proof of Theorem <xref ref-type="statement" rid="thm1">2</xref>.
Assume for contradiction that the solution u exists for all t>0. We claim that
(21)∫0t0∫Ω(ut)2dx ds>0
for any t0>0. Otherwise, there exists t0>0 such that
(22)∫0t0∫Ω(ut)2dx ds=0,
and hence ut=0 for a.e. (x,t)∈Ω×(0,t0]. Therefore, noticing E(t)≤0 by Lemma 3, we have from (15) that
(23)∫Ω|∇u|pdx=0
for a.e. t∈(0,t0]. Using the Poincaré inequality with ∫Ωu dx=0, we have u=0 for a.e. (x,t)∈Ω×(0,t0]. This contradicts u0≢0.

Integrating (14) from t0 to t, we have
(24)M(t)≥M(t0)+(q+1)∫t0t∫0τ∫Ω(ut)2dx ds dτ,
which implies that
(25)limt→∞H′(t)=limt→∞M(t)=+∞.
Thus, there exists t*≥t0 such that for all t≥t*(26)3q+54(H′(t))2≤(q+1)[H′(t)-H′(0)]2.
Thus, combining (17), we further have
(27)3q+54(H′(t))2≤2H(t)H′′(t)
for all t≥t*. Now, we consider the function G(t)=(H(t))-((q-1)/4). Combining with the above inequality and a simple calculation shows that
(28)G′′(t)=q-14(H(t))(-q-7)/4 ×(q+34(H′(t))2-H(t)H′′(t))≤-(q-1)232(H(t))(-q-7)/4(H′(t))2≤0
for all t≥t*. However, since
(29)limt→∞H(t)=limt→∞M(t)=∞,
we also have
(30)limt→∞G(t)=0,
which is a contradiction.

3. Bounded Initial Energy Case
Define
(31)W(Ω)={u∈W1,p(Ω)∣∫Ωu dx=0}
with the norm ∥u∥=(∫Ω|∇u|pdx)1/p. Let B be the optimal constant of the embedding inequality
(32)∥u∥q+1≤B∥∇u∥p,
where p-1<q≤(Np/(N-p)+)-1. Set
(33)α1=B-(q+1)/(q-p+1),E1=(1p-1q+1)B-p(q+1)/(q-p+1)>0.

Theorem 6.
Assume that p>2, p-1<q≤(Np/(N-p)+)-1. Let the initial data u0 satisfying E(0)≤E1 and ∥∇u0∥p>α1. Then, there exists T1 with 0<T1<∞, such that
(34)limt→T1M(t)=+∞.

First, we prove the following two Lemmas, similar to the idea in [13].

Lemma 7.
Assume that u is a solution of the system (1). If E(0)<E1 and ∥∇u0∥p>α1. Then, there exists a positive constant α2>α1, such that
(35)∥∇u∥p≥α2, for any t≥0,(36)∥u∥q+1≥Bα2, for any t≥0.

Proof.
Let ∥∇u∥p=α and by (32), we have
(37)E(t)=1p∫Ω|∇u|pdx-1q+1∫Ω|u|q+1dx≥1p∥∇u∥pp-1q+1Bq+1∥∇u∥pq+1=1pαp-1q+1Bq+1αq+1.
For convenience, we define
(38)g(α)=1pαp-1q+1Bq+1αq+1.
It is easy to find that g increases if 0<α<α1 and decreases if α>α1. Moreover, g(α)→-∞ as α→∞ and g(α1)=E1. Due to E(0)<E1, there exists α2>α1 such that g(α2)=E(0). Let ∥∇u0∥p=α0; thus α0>α1. Then by (37) and (38), we have g(α0)≤E(0)=g(α2), which implies that α0≥α2. For contradiction to establish (35), we assume that there exists t0>0 such that
(39)α1<∥∇u(·,t0)∥p<α2.
It follows from (37) and (38) that
(40)E(t0)≥g(∥∇u(·,t0)∥p)>g(α2)=E(0),
which is in contradiction with Lemma 3. Hence, (35) is established.

Next to prove (36),
(41)E(t)=1p∫Ω|∇u|pdx-1q+1∫Ω|u|q+1dx≤E(0),
which implies that
(42)1q+1∫Ω|u|q+1dx≥1p∫Ω|∇u|pdx-E(0)≥1pα2p-g(α2).
Therefore, (36) is concluded.

Define
(43)F(t)=E1-E(t), for any t≥0.
Then, we have the following.

Lemma 8.
Assume that u is a solution of the system (1). If E(0)<E1 and ∥∇u0∥p>α1. Then for all t≥0,
(44)0<F(0)≤F(t)≤1q+1∫Ω|u|q+1dx.

Proof.
By Lemma 3, we know that F′(t)≥0. Thus,
(45)F(t)≥F(0)=E1-E(0)>0.
According to (35) of Lemma 7, a simple computation shows that
(46)F(t)=E1-1p∫Ω|∇u|pdx+1q+1∫Ω|u|q+1dx≤E1-1pB-p(q+1)/(q-p+1)+1q+1∫Ω|u|q+1dx=-1q+1B-p(q+1)/(q-p+1)+1q+1∫Ω|u|q+1dx≤1q+1∫Ω|u|q+1dx,
which guarantees the conclusion of the lemma.

At the end, let us finish the proof of Theorem 6.

Proof of Theorem <xref ref-type="statement" rid="thm2">6</xref>.
According to (15), we have
(47)M′(t)=-∫Ω|∇u|pdx+∫Ω|u|q+1dx=∫Ω|u|q+1dx-pE(t)-pq+1∫Ω|u|q+1dx=q+1-pq+1∫Ω|u|q+1dx-pE1+pF(t).
By using (33) and (36), we obtain
(48)pE1=(1-pq+1)B-p(q+1)/(q+1-p)=α1q+1α2q+1q+1-pq+1Bq+1α2q+1≤α1q+1α2q+1q-p+1q+1∫Ω|u|q+1dx.
Combining (47) and (48), we get
(49)M′(t)≥(1-α1q+1α2q+1)q+1-pq+1∫Ω|u|q+1dx+pF(t)≥(1-α1q+1α2q+1)q+1-pq+1|Ω|(1-q)/2M(q+1)/2.
Since q>p-1>1, M(t) blows up at a finite time. The proof of Theorem 6 is complete.

Remark 9 (behavior of the energy<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M157"><mml:mi> </mml:mi><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>).
Similar to Theorem 1.3 of [5], it is easy to be proved. Let p>2, p-1<q≤(Np/(N-p)+)-1, and let u be a weak solution of (1). If there exists a constant C0>0 and a time T0′>0, such that the solution u exists on [0,T0′) and satisfies E(t)≥-C0 on [0,T0′), then F(t) is bounded on [0,T0′). Thus, the above result and Theorem 6 reveal that even though the initial energy could be chosen as positive, the energy E(t) needs to become negative at a certain time and then goes to -∞. Otherwise, E(t) has a lower bound on [0,+∞); thus F(t) is bounded on [0,+∞). It is in contradiction with Theorem 6.