Blow-Up in a Slow Diffusive 𝑝-Laplace Equation with the Neumann Boundary Conditions

Abstract

We study a slow diffusive 𝑝-Laplace equation in a bounded domain with the Neumann boundary conditions. A natural energy is associated to the equation. It is shown that the solution blows up in finite time with the nonpositive initial energy, based on an energy technique. Furthermore, under some assumptions of initial data, we prove that the solutions with bounded initial energy also blow up.

1. Introduction

In this paper, we consider a slow diffusive 𝑝-Laplace equation: 𝑢_𝑡−div|∇𝑢|^𝑝−2∇𝑢=|𝑢|^𝑞−1𝑢−_Ω|𝑢|^𝑞−1()(),𝑢𝑑𝑥,𝑥,𝑡∈Ω×0,𝑇𝜕𝑢()(),𝑢()𝜕𝑛=0,𝑥,𝑡∈𝜕Ω×0,𝑇𝑥,0=𝑢₀(𝑥),𝑥∈Ω (1) with ⨍_Ω𝑢₀𝑑𝑥=0, where Ω is a bounded smooth domain Ω⊂ℝ^𝑁, 𝑝>2, 𝑞>𝑝−1, and 𝑢₀∈𝐿^∞(Ω)∩𝑊^1,𝑝(Ω), 𝑢₀≢0, and denote ⨍_Ω𝑓𝑑𝑥=(1/|Ω|)∫_Ω𝑓𝑑𝑥. It is easy to check that ∫_Ω𝑢𝑑𝑥=0; that is, the mass of𝑢is 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 𝑢_𝑡=Δ𝑢+𝑢²−_Ω𝑢²𝑑𝑥 (2) with the Neumann boundary condition. The study in [5] for 𝑢_𝑡=Δ𝑢+|𝑢|^𝑝−_Ω|𝑢|^𝑝𝑑𝑥, (3) 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 𝑢_𝑡=Δ𝑢+𝑓(𝑢,𝑘(𝑡)) (4) 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 ⨍_Ω𝑓𝑑𝑥=0. The semilinear parabolic equation [12] 𝑢_𝑡=Δ𝑢+|𝑢|^𝑝−1𝑢−_Ω|𝑢|^𝑝−1𝑢𝑑𝑥 (5) 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 𝑢_𝑡−div|∇𝑢|^𝑝−2∇𝑢=|𝑢|^𝑞−_Ω|𝑢|^𝑞()(),𝑑𝑥,𝑥,𝑡∈Ω×0,𝑇𝜕𝑢()(),𝑢()𝜕𝑛=0,𝑥,𝑡∈𝜕Ω×0,𝑇𝑥,0=𝑢₀(𝑥),𝑥∈Ω, (6) 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))∩𝐿^𝑝(0,𝑇,𝑊^1,𝑝(Ω)) with 𝑢_𝑡∈𝐿²(Ω×(0,𝑇)) is called a weak solution of (1) if ^𝑡_0Ω𝑢𝜕𝜑−||𝜕𝑠∇𝑢^𝑝−2∇𝑢⋅∇𝜑+|𝑢|^𝑞−_Ω|𝑢|^𝑞𝜑=𝑑𝑥𝑑𝑠_Ω𝑢(𝑥,𝑡)𝜑(𝑥,𝑡)𝑑𝑥−_Ω𝑢₀(𝑥)𝜑(𝑥,0)𝑑𝑥 (7) 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 𝐸(𝑡)=1𝑝_Ω|∇𝑢|^𝑝1𝑑𝑥−𝑞+1_Ω|𝑢|^𝑞+1𝑑𝑥. (8) and denote 𝑀(𝑡)=12_Ω𝑢²(𝑥,𝑡)𝑑𝑥,𝐻(𝑡)=^𝑡₀𝑀(𝑠)𝑑𝑠. (9)

Theorem 2. Assume that 𝑝>2, 𝑞>𝑝−1, and 𝑢₀∈𝐿^∞(Ω)∩𝑊^1,𝑝(Ω), 𝑢₀≢0, and let the initial energy 𝐸(0)=1𝑝_Ω∇𝑢₀^𝑝1𝑑𝑥−𝑞+1_Ω𝑢₀^𝑞+1𝑑𝑥 (10) be nonpositive. Then, there exists 𝑇₀ with 0<𝑇₀<∞, such that lim_𝑡→𝑇0𝑀(𝑡)=+∞. (11)

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 𝐸(𝑡)=𝐸(0)−^𝑡_0Ω𝑢_𝑡²𝑑𝑥𝑑𝑠. (12)

Proof. A direct computation using (1) and by parts yields 𝑑𝐸(𝑡)=𝑑𝑡_Ω|∇𝑢|^𝑝−2∇𝑢⋅∇𝑢_𝑡−|𝑢|^𝑞−1𝑢𝑢_𝑡=𝑑𝑥_Ω−div|∇𝑢|^𝑝−2∇𝑢−|𝑢|^𝑞−1𝑢𝑢_𝑡𝑑𝑥=_Ω−𝑢_𝑡−_Ω|𝑢|^𝑞−1𝑢𝑑𝑥𝑢_𝑡𝑑𝑥=−_Ω𝑢_𝑡²𝑑𝑥. (13) 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: 𝑀^′(𝑡)≥𝑞+1^𝑡_0Ω𝑢_𝑡²𝑑𝑥𝑑𝑠. (14)

Proof. An easy computation using (1) and the fact ∫_Ω𝑢𝑑𝑥=0 and by parts shows that 𝑀^′(𝑡)=_Ω𝑢𝑢_𝑡=𝑑𝑥_Ω𝑢div|∇𝑢|^𝑝−2∇𝑢+|𝑢|^𝑞−1𝑢−_Ω|𝑢|^𝑞−1𝑢𝑑𝑥=−_Ω|∇𝑢|^𝑝𝑑𝑥+_Ω|𝑢|^𝑞+1𝐸(𝑡)+𝑑𝑥=−𝑞+1𝑞+1−𝑝𝑝_Ω|∇𝑢|^𝑝𝑑𝑥. (15) The last equality implies 𝑀^′(𝑡)𝐸(𝑡)𝐸(0)+≥−𝑞+1=−𝑞+1𝑞+1^𝑡_0Ω𝑢_𝑡²≥𝑑𝑥𝑑𝑠𝑞+1^𝑡_0Ω𝑢_𝑡²𝑑𝑥𝑑𝑠, (16) 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 𝑞+12𝐻^′(𝑡)−𝐻^′(0)²≤𝐻(𝑡)𝐻^′′(𝑡). (17)

Proof. Note the definition of 𝑀(𝑡) and 𝐻(𝑡), and a simple calculation shows that 𝐻^′(𝑡)−𝐻^′(0)(𝑡)(0)==𝑀−𝑀_𝑡0𝑀^′(𝑠)𝑑𝑠=^𝑡_0Ω𝑢𝑢_𝑡≤𝑑𝑥𝑑𝑠_𝑡0Ω𝑢²𝑑𝑥𝑑𝑠^1/2𝑡_0Ω𝑢_𝑡²𝑑𝑥𝑑𝑠^1/2≤2𝑞+1^1/2(𝐻(𝑡))^1/2𝑀^′(𝑡)^1/2=2𝑞+1^1/2(𝐻(𝑡))^1/2𝐻^′′(𝑡)^1/2. (18) Furthermore, 𝐻^′(𝑡)−𝐻^′(0)=^𝑡₀𝑀^′(𝑠)≥𝑡𝑑𝑠𝑞+1^𝑡_0Ω𝑢_𝑡²𝑑𝑥𝑑𝑠≥0. (19) Therefore, 𝑞+12𝐻^′(𝑡)−𝐻^′(0)²≤𝐻(𝑡)𝐻^′′(𝑡). (20)

Proof of Theorem 2. Assume for contradiction that the solution 𝑢 exists for all 𝑡>0. We claim that ^𝑡0_0Ω𝑢_𝑡²𝑑𝑥𝑑𝑠>0 (21) for any 𝑡₀>0. Otherwise, there exists 𝑡₀>0 such that ^𝑡0_0Ω𝑢_𝑡²𝑑𝑥𝑑𝑠=0, (22) and hence 𝑢_𝑡=0 for a.e. (𝑥,𝑡)∈Ω×(0,𝑡₀]. Therefore, noticing 𝐸(𝑡)≤0 by Lemma 3, we have from (15) that _Ω|∇𝑢|^𝑝𝑑𝑥=0 (23) for a.e. 𝑡∈(0,𝑡₀]. Using the Poincaré inequality with ∫_Ω𝑢𝑑𝑥=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 𝑀(𝑡)≥𝑀𝑡₀+𝑞+1^𝑡_𝑡0^𝜏_0Ω𝑢_𝑡²𝑑𝑥𝑑𝑠𝑑𝜏, (24) which implies that lim_𝑡→∞𝐻^′(𝑡)=lim_𝑡→∞𝑀(𝑡)=+∞. (25) Thus, there exists 𝑡^∗≥𝑡₀ such that for all 𝑡≥𝑡^∗3𝑞+54𝐻^′(𝑡)²≤𝑞+1𝐻^′(𝑡)−𝐻^′(0)². (26) Thus, combining (17), we further have 3𝑞+54𝐻^′(𝑡)²≤2𝐻(𝑡)𝐻^′′(𝑡) (27) 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 𝐺^′′(𝑡)=𝑞−14(𝐻(𝑡))^{(−𝑞−7)/4}×𝑞+34𝐻^′(𝑡)²−𝐻(𝑡)𝐻^′′(𝑡)≤−𝑞−1²(𝐻(𝑡32))^{(−𝑞−7)/4}𝐻^′(𝑡)²≤0 (28) for all 𝑡≥𝑡^∗. However, since lim_𝑡→∞𝐻(𝑡)=lim_𝑡→∞𝑀(𝑡)=∞, (29) we also have lim_𝑡→∞𝐺(𝑡)=0, (30) which is a contradiction.

3. Bounded Initial Energy Case

Define 𝑊(Ω)=𝑢∈𝑊^1,𝑝(Ω)∣_Ω𝑢𝑑𝑥=0 (31) with the norm ‖𝑢‖=(∫_Ω|∇𝑢|^𝑝𝑑𝑥)^1/𝑝. Let 𝐵 be the optimal constant of the embedding inequality ‖𝑢‖_𝑞+1≤𝐵‖∇𝑢‖_𝑝, (32) where 𝑝−1<𝑞≤(𝑁𝑝/(𝑁−𝑝)₊)−1. Set 𝛼₁=𝐵^{−(𝑞+1)/(𝑞−𝑝+1)},𝐸₁=1𝑝−1𝐵𝑞+1^{−𝑝(𝑞+1)/(𝑞−𝑝+1)}>0. (33)

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 lim_𝑡→𝑇1𝑀(𝑡)=+∞. (34)

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 ‖∇𝑢‖_𝑝≥𝛼₂,𝑓𝑜𝑟𝑎𝑛𝑦𝑡≥0,‖𝑢‖_𝑞+1≥𝐵𝛼₂,𝑓𝑜𝑟𝑎𝑛𝑦𝑡≥0. (35)(36)

Proof. Let ‖∇𝑢‖_𝑝=𝛼 and by (32), we have 𝐸(𝑡)=1𝑝_Ω|∇𝑢|^𝑝1𝑑𝑥−𝑞+1_Ω|𝑢|^𝑞+1≥1𝑑𝑥𝑝‖∇𝑢‖^𝑝_𝑝−1𝐵𝑞+1^𝑞+1‖∇𝑢‖_𝑝^𝑞+1=1𝑝𝛼^𝑝−1𝐵𝑞+1^𝑞+1𝛼^𝑞+1. (37) For convenience, we define 𝑔(𝛼)=1𝑝𝛼^𝑝−1𝐵𝑞+1^𝑞+1𝛼^𝑞+1. (38) 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 𝛼₁<∇𝑢⋅,𝑡_0𝑝<𝛼₂. (39) It follows from (37) and (38) that 𝐸𝑡₀≥𝑔∇𝑢⋅,𝑡_0𝑝>𝑔𝛼₂=𝐸(0), (40) which is in contradiction with Lemma 3. Hence, (35) is established.
Next to prove (36), 𝐸(𝑡)=1𝑝_Ω|∇𝑢|^𝑝1𝑑𝑥−𝑞+1_Ω|𝑢|^𝑞+1𝑑𝑥≤𝐸(0), (41) which implies that 1𝑞+1_Ω|𝑢|^𝑞+11𝑑𝑥≥𝑝_Ω|∇𝑢|^𝑝(0)≥1𝑑𝑥−𝐸𝑝𝛼^𝑝₂−𝑔𝛼₂. (42) Therefore, (36) is concluded.

Define 𝐹(𝑡)=𝐸₁−𝐸(𝑡),forany𝑡≥0. (43) Then, we have the following.

Lemma 8. Assume that 𝑢is a solution of the system (1). If 𝐸(0)<𝐸₁ and ‖∇𝑢₀‖_𝑝>𝛼₁. Then for all 𝑡≥0, (0)(𝑡)≤10<𝐹≤𝐹𝑞+1_Ω|𝑢|^𝑞+1𝑑𝑥. (44)

Proof. By Lemma 3, we know that 𝐹^′(𝑡)≥0. Thus, 𝐹(𝑡)≥𝐹(0)=𝐸₁−𝐸(0)>0. (45) According to (35) of Lemma 7, a simple computation shows that 𝐹(𝑡)=𝐸₁−1𝑝_Ω|∇𝑢|^𝑝1𝑑𝑥+𝑞+1_Ω|𝑢|^𝑞+1𝑑𝑥≤𝐸₁−1𝑝𝐵^{−𝑝(𝑞+1)/(𝑞−𝑝+1)}+1𝑞+1_Ω|𝑢|^𝑞+11𝑑𝑥=−𝐵𝑞+1^{−𝑝(𝑞+1)/(𝑞−𝑝+1)}+1𝑞+1_Ω|𝑢|^𝑞+1≤1𝑑𝑥𝑞+1_Ω|𝑢|^𝑞+1𝑑𝑥, (46) which guarantees the conclusion of the lemma.

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

Proof of Theorem 6. According to (15), we have 𝑀^′(𝑡)=−_Ω|∇𝑢|^𝑝𝑑𝑥+_Ω|𝑢|^𝑞+1=𝑑𝑥_Ω|𝑢|^𝑞+1(𝑡)−𝑝𝑑𝑥−𝑝𝐸𝑞+1_Ω|𝑢|^𝑞+1=𝑑𝑥𝑞+1−𝑝𝑞+1_Ω|𝑢|^𝑞+1𝑑𝑥−𝑝𝐸₁+𝑝𝐹(𝑡). (47) By using (33) and (36), we obtain 𝑝𝐸₁=𝑝1−𝐵𝑞+1^{−𝑝(𝑞+1)/(𝑞+1−𝑝)}=𝛼₁^𝑞+1𝛼₂^𝑞+1𝑞+1−𝑝𝐵𝑞+1^𝑞+1𝛼₂^𝑞+1≤𝛼₁^𝑞+1𝛼₂^𝑞+1𝑞−𝑝+1𝑞+1_Ω|𝑢|^𝑞+1𝑑𝑥. (48) Combining (47) and (48), we get 𝑀^′(𝑡)≥𝛼1−^1𝑞+1𝛼₂^𝑞+1𝑞+1−𝑝𝑞+1_Ω|𝑢|^𝑞+1(𝑡)≥𝛼𝑑𝑥+𝑝𝐹1−^1𝑞+1𝛼₂^𝑞+1𝑞+1−𝑝|Ω|𝑞+1^(1−𝑞)/2𝑀^(𝑞+1)/2. (49) 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 energy𝐸(𝑡)). 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 𝑇^′₀>0, such that the solution 𝑢exists on [0,𝑇^′₀) and satisfies 𝐸(𝑡)≥−𝐶₀ on [0,𝑇^′₀), then 𝐹(𝑡) is bounded on [0,𝑇^′₀). 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.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (Grants nos. 11226179, 11201045) and the Doctor Startup Fund of Dalian Nationalities University (Grant no. 0701-110030).