On Existence, Uniform Decay Rates, and Blow-Up for Solutions of a Nonlinear Wave Equation with Dissipative and Source

Abstract

This paper studies the blow-up and existence, and asymptotic behaviors of the solution of a nonlinear hyperbolic equation with dissipative and source terms. By using Galerkin procedure and the perturbed energy method, the local and global existence of solution is established. In addition, by the concave method, the blow-up of solutions can be obtained.

1. Introduction

In this paper, we investigate the following nonlinear wave equation: 𝑢_𝑡^𝜌𝑢_𝑡𝑡+Δ²𝑢+𝑢_𝑡^𝑚𝑢_𝑡+𝛾Δ²𝑢_𝑡=^𝑁_𝑖=1𝜕𝜕𝑥_𝑖𝜎_𝑖𝑢_𝑥𝑖+|𝑢|^𝑝−1𝑢,(𝑥,𝑡)∈Ω×(0,𝑇),𝑢=𝜕𝑢()(],𝑢()𝜕𝑛=0,𝑥,𝑡∈𝜕Ω×0,𝑇𝑥,0=𝑢₀(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥),𝑥∈Ω, (1.1) where Ω is a bounded domain in ℝ^𝑛 with smooth boundary 𝜕Ω, Δ is a Laplace operator, and 𝜕𝑢/𝜕𝑛|_𝜕Ω indicates derivative of 𝑢 in outward normal direction of 𝜕Ω. In addition to, if 𝑛>3,1<𝑝<(𝑛+2)/(𝑛−2) and if 𝑛=1,2, then 𝑝>1. 𝛾≥0 is a constant, 𝜎_𝑖(𝑠)(𝑖=1,…,𝑁) are given in (A1) later.

In 1968, Greenberg et al. [1] first suggested and studied the following equation: 𝑢_𝑡𝑡−𝑢_𝑥𝑥𝑡=𝜎𝑢_𝑥𝑥. (1.2) Under the condition 𝜎^′(𝑠)>0 and higher smooth conditions on 𝜎(𝑠) and initial data, they claimed the global existence of classical solutions for the initial boundary value problem of (1.2).

The multidimensional form of the following: 𝑢_𝑡𝑡−Δ𝑢_𝑡−^𝑁_𝑖=1𝜕𝜕𝑥_𝑖𝑎_𝑖𝑥,𝑡,𝑢_𝑥𝑖=𝑓(𝑥,𝑡) (1.3) was first studied by Clement [2, 3]. Exploiting the monotone operator method, he obtained the global existence of weak solutions for the initial boundary value problem of (1.2).

Our model comes from [4]. In [4], Yang has studied the global existence, asymptotic behavior, and blow-up of solutions for a nonlinear wave equations with dissipative term: 𝑢_𝑡𝑡+Δ²𝑢+𝜆𝑢_𝑡=^𝑁_𝑖=1𝜕𝜕𝑥_𝑖𝜎_𝑖𝑢_𝑥𝑖, (1.4) with the same initial and boundary conditions as that of (1.1). In our model, we add damping and source terms which enhance the difficulty of proving the existence and decay of solution of (1.2).

More related studies of the damped hyperbolic equation with dissipative term or damping term can be found in papers [5–15].

The paper is organized as follows. In Section 2, we present some notations, and results needed later and main results. Section 3 contains the statement and the proofs of the decay of solutions. Section 4 gives the statement and the proofs of the blow-up of solutions.

2. Preliminaries

We first introduce the following abbreviations: 𝑄_𝑇=Ω×(0,𝑇),𝐿_𝑝=𝐿_𝑝(Ω),𝑊^𝑚,𝑝=𝑊^𝑚,𝑝(Ω),𝐻^𝑚=𝑊^𝑚,2,𝐻^𝑚₀=𝑊₀^𝑚,2,‖⋅‖₂=‖⋅‖_𝐿2,‖⋅‖_1,2=‖⋅‖_𝐻¹₀. (2.1) Let (⋅,⋅) denote the 𝐿₂-inner product. We denote the dual of 𝑊₀^1,𝑝 by 𝑊^−1,𝑝′, with 𝑝^′=𝑝/(𝑝−1), and ‖⋅‖_−1,𝑝′=‖⋅‖_{𝑊′−1,𝑝}. Now we make the following assumptions:(A1)𝜎_𝑖∈𝐶¹(𝑅),𝜎_𝑖(𝑠)𝑠>0 and 𝐶₁|𝑠|^𝑟≤𝜎_𝑖(𝑠)≤𝐶₂|𝑠|^𝑟, (2.2)for some 𝑟≥1, if 𝑛=1,2, else if 𝑛≥3, then 𝑟≤𝑛/(𝑛−2), so that ‖∇𝑢‖_2𝑟≤𝐵₁‖Δ𝑢‖₂, 𝑖=1,2,…,𝑁, where 𝐵₁ is the optimal embedding constant.(A2) The initial data 𝑢₀∈𝐻²₀,𝑢₁∈𝐻¹₀. (2.3)(A3)𝑚<𝜌+1; If 𝑛>3, then 0<𝜌<𝑚<4/(𝑛−2) and if 𝑛=1,2, then 0<𝜌<𝑚. (A4) If 𝑛>3, 1<𝑝<(𝑛+2)/(𝑛−2) and if 𝑛=1,2, then 𝑝>1. Without loss of generality, we assume that 𝑟<𝑝.

And(B1)𝜎_𝑖∈𝐶¹(𝑅),𝜎_𝑖(𝑠)𝑠<0, and 𝐶₁|𝑠|^𝑟≤𝜎_𝑖(𝑠)≤𝐶₂|𝑠|^𝑟, (2.4)for some 𝑟≥1, if 𝑛=1,2, else if 𝑛≥3, then 𝑟≤𝑛/(𝑛−2), so that ‖∇𝑢‖_2𝑟≤𝐵₁‖Δ𝑢‖₂, 𝑖=1,2,…,𝑁. where 𝐵₁ is the optimal embedding constant, and 𝐶₂<(𝑟+1)/(𝑝+1).(B2) If 𝑛>3, then 𝑚<𝑟−1<𝑝−1<4/(𝑛−2) and if 𝑛=1,2, then 0<𝑚<𝑟−1<𝑝−1.(B3)𝜌+1<𝑟, and if 𝑛>3, then 𝜌<4/(𝑛−2) and if 𝑛=1,2, then 0<𝜌.

Throughout this paper, we use the embedding 𝐻²₀(Ω)↪𝐿^𝑞(Ω) which implies ‖𝑢‖_𝑞≤𝐵₂‖Δ𝑢‖₂ when 2≤𝑞≤2𝑛𝑛−2if𝑛≥3,𝑞≥1if𝑛=1,2, (2.5) where 𝐵₂ is an optimal embedding constant. We introduce the following functionals: 𝐴_𝑖(𝑠)=^𝑠₀𝜎_𝑖𝜂𝐸(𝑡)=1𝑑𝜂,𝑢𝜌+2^{𝑡𝜌+2𝜌+2}+12‖Δ𝑢‖²₂+^𝑁_𝑖=1Ω𝐴_𝑖𝑢_𝑥𝑖‖𝑢‖𝑑𝑥−^𝑝+1𝑝+1,𝐽(𝑡)=1𝑝+12‖Δ𝑢‖²₂−𝐶₂‖‖𝑟+1∇𝑢^𝑟+1_𝑟+1−‖𝑢‖^𝑝+1_𝑝+1,𝐼(𝑡)=‖‖𝑝+1Δ𝑢²₂−𝐶₂‖∇𝑢‖^𝑟+1_𝑟+1−‖𝑢‖^𝑝+1_𝑝+1. (2.6)(2.7)(2.8)(2.9) Because 𝑟<𝑝, we have 𝐽(𝑡)=𝐼(𝑡)+𝑟+1𝑟−12()‖‖𝑟+1Δ𝑢²₂+𝑝−𝑟()‖𝑢‖𝑝+1𝑟+1^𝑝+1𝑝+1. (2.10)

Theorem 2.1. Assume that (A1)–(A4) hold, then problem (1.1) has a unique solution 𝑢 satisfying 𝑢∈𝐿_∞[0,𝑇];𝐻²₀∩𝑊^1,∞[0,𝑇];𝐿_𝜌+2;𝑢_𝑡∈𝐿₂[0,𝑇];𝐻²₀∩𝐿^∞[0,𝑇];𝐿_𝜌+2, (2.11) where 𝑇<1.

Theorem 2.2. Assume that (A1)–(A4) hold, 𝑢 is the local solution of the problem (1.1). And 𝐶₀2(𝑟+1)𝐸(0)𝑟−1^(𝑟−1)/2+2(𝑟+1)𝐸(0)𝑟−1^(𝑝−1)/2𝐼(0)<1,>0, (2.12) where 𝐶₀=max{𝐵₁𝐶₂,𝐵₂}, then 𝑢 is a global bounded solution, moreover, 𝐸(𝑡)≤𝑀1+𝑡1+𝑡^1/(𝑚+2)+𝑡^{(𝑚−𝜌)/(𝑚+2)}+𝑡^1/2,∀𝑡∈[0,∞),lim_𝑡→∞‖Δ𝑢‖²₂=lim_𝑡→∞𝑢_𝑡^𝜌+2_𝜌+2=0, (2.13)(2.14) where 𝑀>0 is a constant.

Theorem 2.3. Assume that (A1)–(A4) and 𝑝𝐵₂2()𝑝+1𝑟+1𝑟−1^(𝑝+1)/2(𝐸(0))^(𝑝−1)/2<1 (2.15) hold, 𝑢 is the local solution of the problem (2.7). Consider (2.12) are satisfied, then 𝐸(𝑡)≤𝐾𝑒^−𝜅𝑡,∀𝑡∈[0,∞), (2.16) where 𝐾,𝜅>0 are constants.

Remark 2.4. When 𝛾>0, we will use perturbed energy method, which is different to the method of the proof of Theorem 2.2, to prove Theorem 2.3.

Theorem 2.5. Assume that (B1)–(B3), (A4) hold, and there exist some (𝑢₀,𝑢₁)∈𝐻²(Ω)×𝐿^𝜌+2(Ω), then the solution of the problem (1.1) blows-up at the limited time 𝑇^∗>0.

3. Decay of Solutions

In this section, we prove Theorems 2.1–2.3. First, we give the following Lemma.

Lemma 3.1 (see [2]). Let Ω be any bounded domain in 𝐑^𝑁, {𝜔_𝑘}^∞_𝑘=1 be an orthogonal basis in 𝐿₂(Ω). Then for any 𝜀>0, there exists a positive number 𝑁_𝜀 such that ‖𝑢‖≤^𝑁𝜀_𝑘=1𝑢,𝜔_𝑘^21/2+𝜀‖𝑢‖_1,𝑝, (3.1) for all 𝑢∈𝑊₀^1,𝑝(2≤𝑝<∞).

Proof of Theorem 2.1. We look for approximate solutions 𝑢^𝑛(𝑡) of problem (1.1) of the form 𝑢^𝑛(𝑡)∶=^𝑛_𝑗=1𝑇_𝑗𝑛(𝑡)𝜔_𝑗, (3.2) where {𝜔_𝑗}^𝑛_𝑗=1 is an orthogonal basis in 𝐻²₀, and also in 𝐿₂, and the coefficients {𝑇_𝑗𝑛}^𝑛_𝑗=1 satisfy 𝑇_𝑗𝑛(𝑡)=(𝑢^𝑛(𝑡),𝜔_𝑗) with 𝑢^𝑛_𝑡(𝑡)^𝜌𝑢^𝑛_𝑡𝑡(𝑡),𝜔_𝑗+Δ²𝑢^𝑛(𝑡),𝜔_𝑗+𝑢^𝑛_𝑡^𝑚𝑢^𝑛_𝑡(𝑡),𝜔_𝑗+𝛾Δ²𝑢^𝑛_𝑡(𝑡),𝜔_𝑗=_𝑖𝜕𝜕𝑥_𝑖𝜎_𝑖𝑢^𝑛_𝑥𝑖(𝑡),𝜔_𝑗+𝑢^𝑛(𝑡)^𝑝−1𝑢^𝑛(𝑡),𝜔_𝑗𝑢,𝑡>0,𝑗=1,…,𝑛,_𝑛(0)=𝑢^𝑛₀,𝑢^𝑛_𝑡(0)=𝑢^𝑛₁. (3.3) Since 𝐶^∞₀ is dense in 𝐻²₀ and 𝐿₂, we choose 𝑢^𝑛₀,𝑢^𝑛₁∈𝐶^∞₀ such that 𝑢^𝑛₀→𝑢₀in𝐻²₀,𝑢^𝑛₁→𝑢₁in𝐻¹₀as𝑛→∞. (3.4) The above system of o.d.e. has a local solution 𝑢^𝑛(𝑡) defined in some interval [0,𝑇_𝑛). The following will prove that the 𝑇_𝑛 can be substituted by some 𝑇>0.
Multiply (3.3) by 𝑇_𝑗𝑛′(𝑡) and summing up about 𝑗, we get 𝑑1𝑑𝑡𝑢𝜌+2^{𝑛𝑡𝜌+2𝜌+2}+12Δ𝑢^𝑛2₂+𝑢^𝑛_𝑡^𝑚+2_𝑚+2+𝛾Δ𝑢^𝑛_𝑡²₂=^𝑁_𝑖=1𝜎^𝑛_𝑖𝑢_𝑥𝑖,𝑢^𝑛_𝑥𝑖𝑡+𝑢^𝑛𝑝−1𝑢^𝑛,𝑢^𝑛_𝑡. (3.5) A simple integration of (3.5) over (0,𝑡) leads to 1𝑢𝜌+2^{𝑛𝑡𝜌+2𝜌+2}+12Δ𝑢^𝑛2₂+^𝑡₀𝑢^𝑛_𝑠^𝑚+2_𝑚+2+𝛾𝑢^𝑛_𝑠²₂𝑑𝑠=𝐶₃+^𝑁_𝑖=1^𝑡₀𝜎^𝑛_𝑖𝑢_𝑥𝑖,𝑢^𝑛_𝑥𝑖𝑠𝑑𝑠+^𝑡₀𝑢^𝑛𝑝−1𝑢^𝑛,𝑢^𝑛_𝑠𝑑𝑠, (3.6) where 𝐶₃=1/(𝜌+2)‖𝑢^𝑛₁‖^𝜌+2_𝜌+2+(1/2)‖Δ𝑢^𝑛₀‖²₂>0.
We now estimate the last two terms at the right-hand side of (3.6). Using Hӧlder inequality, Young's inequality, and the embedding theorem, we know there exist 𝑞,𝑤>0, satisfying that 1𝑞+1𝑤+𝑛−22𝑛=1, (3.7) where (𝑟−1)𝑞=2𝑛,1𝑛−2𝑤=𝑛+2−()2𝑛𝑟−1)(𝑛−2≥2𝑛𝑛−2,2𝑛𝑝−1𝑞=2𝑛,1𝑛−2𝑤=𝑛+2−()2𝑛𝑝−1𝑛−2≥2𝑛𝑛−2.2𝑛 (3.8) Consider the following: ^𝑁_𝑖=1^𝑡₀𝜎_𝑖𝑢^𝑛_𝑥𝑖,𝑢^𝑛_𝑥𝑖𝑠𝑑𝑠≤𝐶₂^𝑡_0Ω∇𝑢^𝑛𝑟−1∇𝑢^𝑛∇𝑢^𝑛_𝑠𝑑𝑥𝑑𝑠≤𝐶₂^𝑡₀∇𝑢^𝑛𝑟−1_{(𝑟−1)𝑞}∇𝑢^𝑛_𝑤∇𝑢^𝑛_{𝑠2𝑛/(𝑛−2)}𝑑𝑠≤𝐶₂𝐶^𝑡₀Δ𝑢^𝑛₂^𝑟−1Δ𝑢^𝑛₂Δ𝑢^𝑛_𝑠2𝑑𝑠=𝐶₂𝐶^𝑡₀Δ𝑢^𝑛𝑟₂Δ𝑢^𝑛_𝑠2𝑑𝑠≤𝐶₂𝐶12𝜖^𝑡₀Δ𝑢^𝑛₂^2𝑟𝜖𝑑𝑠+2^𝑡₀Δ𝑢^𝑛_𝑠²₂𝑑𝑠≤𝐶₂𝐶12𝜖^𝑡₀Δ𝑢^𝑛₂^2𝑝𝑑𝑠^𝑟/𝑝+12𝜖^𝑡₀1𝑑𝑠^{(𝑝−𝑟)/𝑝}+𝜖2^𝑡₀Δ𝑢^𝑛_𝑠²₂𝑑𝑠≤𝐶₂𝐶1𝑝2𝜖𝑟^𝑡₀Δ𝑢^𝑛₂^2𝑝𝑑𝑠+𝐶₂𝐶12𝜖𝑝−𝑟𝑝+𝜖2𝐶₂𝐶^𝑡₀Δ𝑢^𝑛_𝑠²₂𝑑𝑠,∀𝜖>0, (3.9) assuming that 𝑡<1.
Similarly, ^𝑡₀𝑢^𝑛𝑝−1𝑢^𝑛,𝑢^𝑛_𝑠𝑑𝑠≤^𝑡₀𝑢^𝑛𝑝−1_{(𝑝−1)𝑞}𝑢^𝑛_𝑤𝑢^𝑛_{𝑠2𝑛/(𝑛−2)}𝑑𝑠≤𝐶^𝑡₀Δ𝑢^𝑛₂^𝑝−1Δ𝑢^𝑛₂Δ𝑢^𝑛_𝑠21𝑑𝑠≤𝐶2𝜖^𝑡₀Δ𝑢^𝑛₂^2𝑝𝜖𝑑𝑠+2𝐶^𝑡₀Δ𝑢^𝑛_𝑠²₂𝑑𝑠,∀𝜖>0. (3.10) Using (3.6)–(3.10), we have ^𝑡₀𝑢^𝑛_𝑠^𝑚+2_𝑚+2𝑑𝑠+𝛾^𝑡₀Δ𝑢^𝑛_𝑠²₂1𝑑𝑠+2Δ𝑢^𝑛2₂+1𝑢𝜌+2^{𝑛𝑡𝜌+2𝜌+2}≤𝐶₄+𝐶(𝜀)^𝑡₀Δ𝑢^𝑛₂^2𝑝𝜖𝑑𝑠+2𝐶^𝑡₀Δ𝑢^𝑛_𝑠²₂. (3.11) Choosing 𝜖=(1/𝐶)𝛾 in (3.11), we have ^𝑡₀𝑢^𝑛_𝑠^𝑚+2_𝑚+21𝑑𝑠+2Δ𝑢^𝑛2₂+1𝑢𝜌+2^{𝑛𝑡𝜌+2𝜌+2}+𝛾2^𝑡₀Δ𝑢^𝑛_𝑠²₂𝑑𝑠≤𝐶₄+𝐶^𝑡₀Δ𝑢^𝑛₂^2𝑝𝑑𝑠, (3.12) where 𝐶₄=𝐶₃+((𝐶₂𝐶)/2𝜖)((𝑝−𝑟)/𝑝). Assuming 𝑌_𝑛(𝑡)=2(𝐶₄+𝐶∫^𝑡₀‖Δ𝑢^𝑛‖₂^2𝑝), we have 𝑌^′_𝑛(𝑡)≤2𝐶𝑌^𝑝_𝑛. (3.13) A simple integration of (3.13) over (0,𝑡) leads to 𝑌_𝑛(𝑡)≤𝑌_𝑛^1−𝑝(0)−2𝐶𝑝−1𝑡^{−1/(𝑝−1)}; (3.14) this implies that Δ𝑢^𝑛2₂+2𝑢𝜌+2^{𝑛𝑡𝜌+2𝜌+2}≤𝑌_𝑛^1−𝑝(0)−2𝐶𝑝−1𝑡^{−1/(𝑝−1)}. (3.15) Though 𝑌_𝑛(𝑡) may blow up, there exists 0<𝑇<min{1,𝑇_𝑛} satisfying Δ𝑢^𝑛2₂+𝑢^𝑛_𝑡^𝜌+2_𝜌+2≤𝐶,∀𝑡∈[0,𝑇], (3.16) where 𝐶 is independent of 𝑛.
Moreover, ^𝑡₀𝑢^𝑛_𝑠^𝑚+2_𝑚+2𝑑𝑠+^𝑡₀Δ𝑢^𝑛_𝑠²₂𝑑𝑠≤𝐶. (3.17) By (3.17), ^𝑡₀𝑢^𝑛_𝑠(𝑠)^𝑚𝑢^𝑛_𝑠(𝑠)^{(𝑚+2)/(𝑚+1)}_{(𝑚+2)/(𝑚+1)}𝑑𝑠≤^𝑡₀𝑢^𝑛_𝑠(𝑠)^𝑚+2_𝑚+2[].𝑑𝑠≤𝐶,𝑡∈0,𝑇 (3.18) By (3.16), 𝑢^𝑛(𝑠)^𝑝−1𝑢^𝑛(𝑠)^{(𝑝+1)/𝑝}_{(𝑝+1)/𝑝}≤𝑢^𝑛(𝑠)^𝑝+1_𝑝+1≤Δ𝑢^𝑛(𝑠)₂^𝑝+1≤𝐶. (3.19) From (3.16)–(3.19), we have 𝑢^𝑛∈𝐿_∞[0,𝑇];𝐻²₀;𝑢^𝑛_𝑡∈𝐿₂[0,𝑇];𝐻²₀∩𝐿_∞[0,𝑇];𝐿_𝜌+2. (3.20) So the solution 𝑢^𝑛(𝑡) of problem (3.3) exists on [0,𝑇] for each 𝑛. On the other hand, we can extract a subsequence from 𝑢^𝑛, still denoted by 𝑢^𝑛, such that 𝑢^𝑛→𝑢weak^∗in𝐿_∞[0,𝑇];𝐻²₀𝑢_𝑛𝑡,→𝑢_𝑡weak^∗in𝐿_∞[0,𝑇];𝐿_𝜌+2∩𝐿₂[0,𝑇];𝐻²₀, (3.21)(3.22) as 𝑛→∞. By (3.21), the Sobolev embedding theorem and the continuity of 𝜎_𝑖(𝑠), for 𝑡∈[0,𝑇], ∇𝑢^𝑛(𝑡)→∇𝑢(𝑡)stronglyin𝐿₂𝜎,a.e.onΩ,_𝑖𝑢^𝑛_𝑥𝑖(𝑡)→𝜎_𝑖𝑢_𝑥𝑖(𝑡),a.e.onΩ,𝑖=1,…𝑁, (3.23) as 𝑛→∞. By Lemma 3.1, (3.21)-(3.22), for any 𝜀>0, there exist positive constant 𝑁_1𝜀 and 𝑁_2𝜀 independent of 𝑢^𝑛 and 𝑢^𝑛_𝑡, respectively, such that, as 𝑛→∞, 𝑢^𝑛(𝑡)−𝑢(𝑡)≤^𝑁1𝜀_𝑘=1𝑢^𝑛−𝑢,𝜔_𝑘^21/2+𝜀𝑢^𝑛(𝑡)−𝑢(𝑡)_1,2≤𝑀𝜀,_𝑇0𝑢^𝑛_𝑡(𝑠)−𝑢_𝑡(𝑠)²₂𝑑𝑠≤2^𝑁2𝜀_𝑘=1^𝑇₀𝑢^𝑛_𝑡(𝑠)−𝑢_𝑡(𝑠),𝜔_𝑘²𝑑𝑠+𝜀^2𝑇₀𝑢^𝑛_𝑡(𝑠)−𝑢_𝑡(𝑠)²_1,2𝑑𝑠≤𝑀𝜀². (3.24)
By the arbitrariness of 𝜀 we get 𝑢^𝑛→𝑢stronglyin𝐿_∞[0,𝑇];𝐿₂,a.e.on𝑄_𝑇,𝑢^𝑛_𝑡→𝑢_𝑡stronglyin𝐿₂𝑄_𝑇,a.e.on𝑄_𝑇. (3.25) From the continuity of |𝑢_𝑡|^𝑚𝑢_𝑡 and (3.25) we know that |𝑢^𝑛_𝑡|^𝑚𝑢^𝑛_𝑡→|𝑢_𝑡|^𝑚𝑢_𝑡 a.e. on 𝑄_𝑇. With the same methods used above we easily get |𝑢^𝑛|^𝑝−1𝑢^𝑛→|𝑢|^𝑝−1𝑢 a.e. on 𝑄_𝑇. Integrating (3.3) over (0,𝑡),𝑡<𝑇 gets 1𝑢𝜌+1^𝑛𝑡(𝑡)^𝜌+1,𝜔_𝑗+^𝑡₀Δ𝑢^𝑛(𝑠),Δ𝜔_𝑗𝑑𝑠+^𝑡₀𝑢^𝑛_𝑠^𝑚𝑢^𝑛_𝑠(𝑠),𝜔_𝑗𝑑𝑠+^𝑡₀𝛾Δ𝑢^𝑛_𝑠(𝑠),Δ𝜔_𝑗=𝑑𝑠_𝑖𝑡0𝜕𝜕𝑥_𝑖𝜎_𝑖𝑢^𝑛_𝑥𝑖(𝑠),𝜔_𝑗𝑑𝑠+^𝑡₀𝑢^𝑛(𝑠)^𝑝−1𝑢^𝑛(𝑠),𝜔_𝑗𝑑𝑠,𝑡>0,𝑗=1,…,𝑛. (3.26) Exploiting (3.16)-(3.17), we have ^𝑡₀Δ𝑢^𝑛(𝑠),Δ𝜔_𝑗𝑑𝑠≤𝐶^𝑡₀Δ𝑢^𝑛2₂𝑢𝑑𝑠≤𝐶𝑇,_𝑛𝑡(𝑡)^𝑚𝑢^𝑛_𝑡(𝑡),𝜔_𝑗≤𝑢^𝑛_𝑡^𝑚+1_𝜌+2𝜔_{𝑗((𝜌+2)/(𝑚+1))′}≤𝑢^𝑛_𝑡^𝜌+2_𝜌+2𝜔_𝑗^{((𝜌+2)/(𝑚+1))′}_{((𝜌+2)/(𝑚+1))′}≤𝐶,_𝑡0𝑢^𝑛_𝑠(𝑠)^𝑚𝑢^𝑛_𝑠(𝑠),𝜔_𝑗𝑑𝑠≤^𝑡₀𝑢^𝑛_𝑠^𝑚+2_𝑚+2+𝜔_𝑗^𝑚+2_𝑚+2𝑑𝑠≤𝐶𝑇,_𝑡0𝑢^𝑛(𝑠)^𝑝−1𝑢^𝑛(𝑠),𝜔_𝑗𝑑𝑠≤𝐶𝑇. (3.27)
Let 𝑛→∞ in (3.26) and we deduce from (3.23), (3.27) and the Lebesgue-dominated convergence theorem that 1𝑢𝜌+1^𝑡(𝑡)^𝜌+1,𝜔_𝑗+^𝑡₀Δ𝑢(𝑠),Δ𝜔_𝑗𝑑𝑠+^𝑡₀𝑢_𝑠(𝑡)^𝑚𝑢_𝑠(𝑠),𝜔_𝑗𝑑𝑠+^𝑡₀𝛾Δ𝑢_𝑠(𝑠),Δ𝜔_𝑗=𝑑𝑠_𝑖𝑡0𝜕𝜕𝑥_𝑖𝜎_𝑖𝑢_𝑥𝑖(𝑠),𝜔_𝑗𝑑𝑠+^𝑡₀|𝑢(𝑠)|^𝑝−1𝑢(𝑠),𝜔_𝑗𝑑𝑠,𝑡>0,𝑗=1,…,𝑛. (3.28) This implies 𝑢∈𝐿_∞([0,𝑇];𝐻²₀) is a local weak solution of problem (1.1). The proof of Theorem 2.1 is completed.

Secondly, we prove Theorem 2.2. First we give two lemmas. It is easy to prove what follows.

Lemma 3.2. The modified energy functional satisfies, along solutions of (1.1), 𝐸^′(𝑡)=−𝑢_𝑡^𝑚+2_𝑚+2−𝛾Δ𝑢_𝑡²₂≤0. (3.29)

Lemma 3.3. Assume that (A1)–(A3) hold, satisfying 𝐶₀2(𝑟+1)𝐸(0)𝑟−1^(𝑟−1)/2+2(𝑟+1)𝐸(0)𝑟−1^(𝑝−1)/2𝐼(0)<1,>0. (3.30) Then 𝐼(𝑡)>0.

Proof. Since 𝐼(0)>0, then there exists (by continuity) 𝑇_∗≤𝑇 such that 𝐼(𝑡)>0,forall𝑡∈[0,𝑇_∗), this gives 𝐽(𝑡)=𝐼(𝑡)+𝑟+1𝑟−12()‖‖𝑟+1Δ𝑢²₂+𝑝−𝑟()‖𝑢‖𝑝+1𝑟+1^𝑝+1𝑝+1≥𝑟−12()‖‖𝑟+1Δ𝑢²₂+𝑝−𝑟)‖𝑢‖𝑝+1(𝑟+1^𝑝+1𝑝+1. (3.31) By using (2.8), (2.9), (3.31), and Lemma 3.2, we easily have ‖Δ𝑢‖²₂≤2(𝑟+1)𝐽(𝑡)≤2()𝑟−1𝑟+1𝐸(𝑡)≤2()𝑟−1𝑟+1𝐸(0)𝑟−1,∀𝑡∈0,𝑇_∗. (3.32) We then exploit (2.12), and (3.32) to obtain 𝐶₂‖∇𝑢‖^𝑟+1_𝑟+1≤𝐵₁𝐶₂‖Δ𝑢‖₂^𝑟+1≤𝐵₁𝐶₂‖Δ𝑢‖₂^𝑟−1‖Δ𝑢‖²₂≤𝐵₁𝐶₂2(𝑟+1)𝐸(0)𝑟−1^(𝑟−1)/2‖Δ𝑢‖²₂,‖𝑢‖^𝑝+1_𝑝+1≤𝐵₂‖Δ𝑢‖₂^𝑝+1≤𝐵₂2(𝑟+1)𝐸(0)𝑟−1^(𝑝−1)/2‖Δ𝑢‖²₂,∀𝑡∈0,𝑇_∗. (3.33) Using (3.33), we have 𝐶₂‖∇𝑢‖^𝑟+1_𝑟+1+‖𝑢‖^𝑝+1_𝑝+1≤𝐵₁𝐶₂2(𝑟+1)𝐸(0)𝑟−1^(𝑟−1)/2‖Δ𝑢‖²₂+𝐵₂2(𝑟+1)𝐸(0)𝑟−1^(𝑝−1)/2‖Δ𝑢‖²₂≤𝐶₀2(𝑟+1)𝐸(0)𝑟−1^(𝑟−1)/2+2(𝑟+1)𝐸(0)𝑟−1^(𝑝−1)/2‖Δ𝑢‖²₂<‖Δ𝑢‖²₂,∀𝑡∈0,𝑇_∗, (3.34) where 𝐶₀=max{𝐵₂,𝐵₁𝐶₂}.
Therefore, 𝐼(𝑡)=‖Δ𝑢‖²₂−𝐶₂‖∇𝑢‖^𝑟+1_𝑟+1−‖𝑢‖^𝑝+1_𝑝+1>0, (3.35) for all 𝑡∈[0,𝑇_∗]. By repeating this procedure, and using the fact that lim_{𝑡→𝑇∗}𝐶₀2(𝑟+1)𝐸(𝑡)𝑟−1^(𝑟−1)/2+2(𝑟+1)𝐸(𝑡)𝑟−1^(𝑝−1)/2≤lim_{𝑡→𝑇∗}𝐶₀2(𝑟+1)𝐸(0)𝑟−1^(𝑟−1)/2+2(𝑟+1)𝐸(0)𝑟−1^(𝑝−1)/2<1, (3.36) the proof is completed.

Lemma 3.4. Assume that (A1)–(A4) hold, (2.12) satisfy. Then the solution is global existence. More, exist positive constant 𝑀>0 has ‖Δ𝑢‖²₂+𝑢_𝑡^𝑝+2_𝑝+2≤𝑀;^𝑡₀𝑢_𝑠^𝑚+2_𝑚+2𝑑𝑠≤𝑀;^𝑡₀Δ𝑢_𝑠²₂𝑑𝑠≤𝑀,∀𝑡∈[0,∞). (3.37)

Proof. It suffices to show that ‖Δ𝑢‖²₂+𝑢_𝑡^𝜌+2_𝜌+2 (3.38) is bounded independently of 𝑡. To achieve this, we use (2.9), (3.31), and Lemma 3.2 to get 𝐸(0)=𝐸(𝑡)+^𝑡₀𝑢_𝑠^𝑚+2_𝑚+2𝑑𝑠+𝛾^𝑡₀Δ𝑢_𝑠²₂(𝑡)+1𝑑𝑠≥𝐽𝑢𝜌+2^{𝑡𝜌+2𝜌+2}+^𝑡₀𝑢_𝑠^𝑚+2_𝑚+2𝑑𝑠+𝛾^𝑡₀Δ𝑢_𝑠²₂≥𝑑𝑠𝑟−12()‖‖𝑟+1Δ𝑢²₂+𝑝−𝑟()‖𝑢‖𝑝+1𝑟+1^𝑝+1𝑝+1+1𝑢𝜌+2^{𝑡𝜌+2𝜌+2}+^𝑡₀𝑢_𝑠^𝑚+2_𝑚+2𝑑𝑠+𝛾^𝑡₀Δ𝑢_𝑠²₂≥𝑑𝑠𝑟−12()‖‖𝑟+1Δ𝑢²₂+1𝑢𝜌+2^{𝑡𝜌+2𝜌+2}+^𝑡₀𝑢_𝑠^𝑚+2_𝑚+2𝑑𝑠+𝛾^𝑡₀Δ𝑢_𝑠²₂𝑑𝑠. (3.39) Since 𝐼(𝑡),𝐽(𝑡) are positive. Therefore, ‖Δ𝑢‖²₂+𝑢_𝑡^𝜌+2_𝜌+2≤𝐶𝐸(0). (3.40) Moreover, ^𝑡₀𝑢_𝑠^𝑚+2_𝑚+2𝑑𝑠≤𝐸(0);𝛾^𝑡₀Δ𝑢_𝑠²₂𝑑𝑠≤𝐸(0), (3.41) where 𝐶 is a positive constant, which depends only on 𝑟.

Lemma 3.5. Assume that (A1)–(A4) hold, (2.12) satisfy. Then exist 𝐶>0 has ^𝑡₀𝑢_𝑠^𝜌+2_𝜌+2𝑑𝑠≤𝐶𝑡^{(𝑚−𝜌)/(𝑚+2)};_𝑡0‖Δ𝑢‖²₂𝑑𝑠≤𝐶^𝑡₀𝐼(𝑠)𝑑𝑠. (3.42)(3.43)

Proof. Using Lemma 3.4, we have ^𝑡₀𝑢_𝑠^𝜌+2_𝜌+2𝑑𝑠≤^𝑡₀𝑢_𝑠^𝑚+2_𝜌+2𝑑𝑠^{(𝜌+2)/(𝑚+2)𝑡}₀1𝑑𝑠^{(𝑚−𝜌)/(𝑚+2)}≤𝐶^𝑡₀𝑢_𝑠^𝑚+2_𝑚+2𝑑𝑠^{(𝜌+2)/(𝑚+2)}𝑡^{(𝑚−𝜌)/(𝑚+2)}≤𝐶𝑡^{(𝑚−𝜌)/(𝑚+2)}. (3.44) Using (3.34), we have 𝐶₂‖∇𝑢‖^𝑟+1_𝑟+1+‖𝑢‖^𝑝+1_𝑝+1≤𝐵𝐶₀2(𝑟+1)𝐸(0)𝑟−1^(𝑟−1)/22(𝑟+1)𝐸(0)𝑟−1^(𝑝−1)/2‖Δ𝑢‖²₂∶=𝜂‖Δ𝑢‖²₂. (3.45) So 1−𝜂‖Δ𝑢‖²₂≤𝐼(𝑡); (3.46) the proof is complete.

Lemma 3.6. Assume that (A1)–(A4) hold, (2.12) satisfy. Then there exists a 𝐶>0, having ^𝑡₀𝐼(𝑠)𝑑𝑠≤𝐶𝑡^1/2+𝑡^{(𝑚−𝜌)/(𝑚+2)}+𝑡^1/(𝑚+2)+1. (3.47)

Proof. By multiplying the differential equation in (1.1) by 𝑢 and integrating over Ω, using integration by parts (2.9) and assumption (A1), we obtain 𝑑0=𝑢𝑑𝑡^𝑡𝜌𝑢_𝑡−𝑢𝜌+1,𝑢^{𝑡𝜌+2𝜌+2}+‖‖𝜌+1Δ𝑢²₂+𝑢_𝑡^𝑚𝑢_𝑡Δ,𝑢+𝛾²𝑢_𝑡,𝑢+^𝑁_𝑖=1𝜎_𝑖𝑢_𝑥𝑖,𝑢_𝑥𝑖−‖𝑢‖^𝑝+1_𝑝+1≥𝑑𝑢𝑑𝑡^𝑡𝜌𝑢_𝑡−𝑢𝜌+1,𝑢^{𝑡𝜌+2𝜌+2}+‖‖𝜌+1Δ𝑢²₂+𝑢_𝑡^𝑚𝑢_𝑡,𝑢+𝛾Δ𝑢_𝑡,Δ𝑢−𝐶₂‖∇𝑢‖^𝑟+1_𝑟+1−‖𝑢‖^𝑝+1_𝑝+1=𝑑𝑢𝑑𝑡^𝑡𝜌𝑢_𝑡−𝑢𝜌+1,𝑢^{𝑡𝜌+2𝜌+2}(𝑡)+𝑢𝜌+1+𝐼_𝑡𝑚𝑢_𝑡,𝑢+𝛾Δ𝑢_𝑡,Δ𝑢. (3.48) So ^𝑡₀𝐼(𝑠)𝑑𝑠≤𝑢_𝑡^𝜌+1_𝜌+2‖𝑢‖_𝜌+2+𝑢₁^𝜌+2_𝜌+2𝑢_0𝜌+2+^𝑡₀𝑢_𝑠^𝜌+2_𝜌+2+𝜌+1𝑑𝑠_𝑡0𝑢_𝑠^𝑚𝑢_𝑠,𝑢𝑑𝑠+𝛾^𝑡₀Δ𝑢_𝑠,Δ𝑢𝑑𝑠. (3.49) Using (A3), (3.42)-(3.43), we have 𝑢_𝑡^𝜌+1_𝜌+2‖𝑢‖_𝜌+2≤𝐶𝑢_𝑡^𝜌+2_𝜌+2^{(𝜌+1)/(𝜌+2)}‖Δ𝑢‖₂≤𝐶,_𝑡0Ω𝑢_𝑠^𝑚𝑢_𝑠𝑢𝑑𝑥𝑑𝑠≤^𝑡₀𝑢_𝑠^𝑚+1_𝑚+2‖𝑢‖_𝑚+2𝑑𝑠≤𝐶^𝑡₀𝑢_𝑠^𝑚+1_𝑚+2‖Δ𝑢‖₂𝑑𝑠≤𝐶^𝑡₀𝑢_𝑠^𝑚+1_𝑚+2𝑑𝑠≤𝐶^𝑡₀𝑢_𝑠^𝑚+2_𝑚+2^{(𝑚+1)/(𝑚+2)}𝑡^1/(𝑚+2)≤𝐶𝑡^1/(𝑚+2),𝛾^𝑡₀Δ𝑢_𝑠,Δ𝑢𝑑𝑠≤𝛾^𝑡₀Δ𝑢_𝑠2‖Δ𝑢‖₂𝑑𝑠≤𝐶^𝑡₀Δ𝑢_𝑠²₂𝑑𝑠^1/2𝑡^1/2≤𝐶𝑡^1/2. (3.50) Therefore, ^𝑡₀𝐼(𝑠)𝑑𝑠≤𝑀1+𝑡^1/(𝑚+2)+𝑡^{(𝑚−𝜌)/(𝑚+2)}+𝑡^1/2. (3.51)

Proof of Theorem 2.2. First, 𝑡↦(1+𝑡)𝐸(𝑡) is also absolutely continuous, and we have 𝑑)𝐸(𝑡(𝑡).𝑑𝑡((1+𝑡))≤𝐸 (3.52) A simple integration of (3.52) over (0,𝑡) leads to (1+𝑡)𝐸(𝑡)≤𝐸(0)+^𝑡₀𝐸(𝑠)(0)+1𝑑𝑠≤𝐸𝜌+2^𝑡₀𝑢_𝑠^𝜌+2_𝜌+21𝑑𝑠+2^𝑡₀‖Δ𝑢‖²₂+𝑑𝑠_𝑁0𝑡0Ω𝐴_𝑖𝑢_𝑥𝑖1𝑑𝑥𝑑𝑠−𝑝+1^𝑡₀‖𝑢‖^𝑝+1_𝑝+1(0)+1𝑑𝑠≤𝐸𝜌+2^𝑡₀𝑢_𝑠^𝜌+2_𝜌+21𝑑𝑠+2^𝑡₀‖Δ𝑢‖²₂+𝑑𝑠_𝑁0𝑡0Ω𝐴_𝑖𝑢_𝑥𝑖1𝑑𝑥𝑑𝑠+𝑝+1^𝑡₀‖𝑢‖^𝑝+1_𝑝+1(0)+1𝑑𝑠≤𝐸𝜌+2^𝑡₀𝑢_𝑠^𝜌+2_𝜌+21𝑑𝑠+2^𝑡₀‖Δ𝑢‖²₂+𝐶𝑑𝑠₂𝑟+1^𝑡₀‖∇𝑢‖^𝑟+1_𝑟+11𝑑𝑠+𝑝+1^𝑡₀‖𝑢‖^𝑝+1_𝑝+1𝑑𝑠. (3.53) Using the upper inequality, (3.45), (3.46), and (3.52), we have ()𝐸(𝑡)(0)+11+𝑡≤𝐸𝜌+2^𝑡₀𝑢_𝑠^𝜌+2_𝜌+21𝑑𝑠+2^𝑡₀‖Δ𝑢‖²₂𝑑𝑠+𝐶^𝑡₀‖Δ𝑢‖²₂(0)+1𝑑𝑠≤𝐸𝜌+2^𝑡₀𝑢_𝑠^𝜌+2_𝜌+2𝑑𝑠+𝐶^𝑡₀𝐼(𝑠)𝑑𝑠. (3.54) Apply (3.42), (3.47), we can get (2.13).
Using (3.31) and Lemma 3.3, we get 𝐽(𝑡)≥0,𝐼(𝑡)>0. (2.13) implies lim𝑡→∞𝐸(𝑡)=0. So when 𝑡→∞, we have ‖𝑢_𝑡‖^𝜌+2_𝜌+2→0 and 𝐽(𝑡)→0. It is that (2.14) is satisfied. Theorem 2.2 is complete.
Following we will prove Theorem 2.3. For this purpose we set 𝐿(𝑡)∶=𝐸(𝑡)+𝜀Ψ(𝑡), (3.55) where 𝜀 is a positive constant and Ψ(𝑡)=1𝜌+1_Ω𝑢_𝑡^𝜌𝑢_𝑡𝑢𝑑𝑥. (3.56)

Lemma 3.7. Let 𝜀 be small enough. Then there exist two positive constants 𝛼₁ and 𝛼₂ such that 𝛼₁𝐿(𝑡)≤𝐸(𝑡)≤𝛼₂𝐿(𝑡). (3.57)

Proof. By Lemma 3.4 and Young's inequality, a direct computation gives 𝐿(𝑡)(𝑡)+𝜀≤𝐸𝜌+2_Ω𝑢_𝑡^𝜌+2𝜀𝑑𝑥+𝜌+2_Ω|𝑢|^𝜌+2(𝑡)+𝜀𝑑𝑥≤𝐸𝑢𝜌+2^{𝑡𝜌+2𝜌+2}+𝜀𝐵₂2()𝐸(0)𝜌+2𝑟+1𝑟−1^𝜌/2‖Δ𝑢‖²₂≤𝐸(𝑡)+𝜀𝐸(𝑡)+𝜀𝐵₂2()𝜌+2𝑟+1𝑟−1^(𝜌+2)/2𝐸(0)^𝜌/2𝐸(𝑡)≤1𝛼₁𝐸(𝑡). (3.58) Similarly, we have 𝐿(𝑡)(𝑡)−𝜀≥𝐸𝜌+2_Ω𝑢_𝑡^𝜌+2𝜀𝑑𝑥−𝜌+2_Ω|𝑢|^𝜌+2(𝑡)−𝜀𝑑𝑥≥𝐸𝑢𝜌+2^{𝑡𝜌+2𝜌+2}−𝜀𝐵₂2()𝐸(0)𝜌+2𝑟+1𝑟−1^𝜌/2‖Δ𝑢‖²₂≥𝐸(𝑡)−𝜀𝐸(𝑡)−𝜀𝐵₂2()𝜌+2𝑟+1𝑟−1^(𝜌+2)/2𝐸(0)^𝜌/2𝐸(𝑡)≥1𝛼₂𝐸(𝑡), (3.59) provided that 𝜀 is small enough.

Lemma 3.8. Assume that the conditions of Theorem 2.3 hold, then the function Ψ(𝑡)1∶=𝜌+1_Ω𝑢_𝑡^𝜌𝑢_𝑡𝑢𝑑𝑥 (3.60) satisfies, along the solution of (1.1), Ψ^′(𝑡)1≤−4𝐸(𝑡)+𝜇𝑢_𝑡^𝑚+2_𝑚+2+𝜔‖Δ𝑢‖²₂. (3.61)

Proof. Applying equations of (1.1), we see Ψ^′(𝑡)=1𝑢𝜌+1^{𝑡𝜌+2𝜌+2}+_Ω𝑢_𝑡^𝜌𝑢_𝑡𝑡=1𝑢𝑑𝑥𝑢𝜌+1^{𝑡𝜌+2𝜌+2}+_Ω−Δ²𝑢−𝑢_𝑡^𝑚𝑢_𝑡+^𝑁_𝑖=1𝜕𝜕𝑥_𝑖𝜎_𝑖𝑢_𝑥𝑖−𝛾Δ²𝑢_𝑡+|𝑢|^𝑝−1𝑢=1𝑢𝑑𝑥𝑢𝜌+1^{𝑡𝜌+2𝜌+2}−‖Δ𝑢‖²₂−𝑢_𝑡^𝑚𝑢_𝑡,𝑢−^𝑁_𝑖=1𝜎_𝑖𝑢_𝑥𝑖,𝑢_𝑥𝑖−𝛾Δ𝑢_𝑡,Δ𝑢+‖𝑢‖^𝑝+1_𝑝+11=−𝑢𝜌+2^{𝑡𝜌+2𝜌+2}−‖Δ𝑢‖²₂−^𝑁_𝑖=1Ω𝐴_𝑖𝑢_𝑥𝑖𝑑𝑥+^𝑁_𝑖=1Ω𝐴_𝑖𝑢_𝑥𝑖−𝜎_𝑖𝑢_𝑥𝑖,𝑢_𝑥𝑖−𝑢𝑑𝑥_𝑡𝑚𝑢_𝑡,𝑢−𝛾Δ𝑢_𝑡,Δ𝑢+‖𝑢‖^𝑝+1_𝑝+1+2𝜌+3𝑢𝜌+2𝜌+1^{𝑡𝜌+2𝜌+2}≤−𝐸(𝑡)−𝑢_𝑡^𝑚𝑢_𝑡,𝑢−𝛾Δ𝑢_𝑡,Δ𝑢+2𝜌+3𝑢𝜌+2𝜌+1^{𝑡𝜌+2𝜌+2}+𝑝‖𝑢‖𝑝+1^𝑝+1𝑝+1+^𝑁_𝑖=1Ω𝐴_𝑖𝑢_𝑥𝑖−𝜎_𝑖𝑢_𝑥𝑖𝑢_𝑥𝑖𝑑𝑥. (3.62) Exploiting the assumption (A1) and Young's inequality, we have ^𝑁_𝑖=1Ω𝐴_𝑖𝑢_𝑥𝑖−𝜎_𝑖𝑢_𝑥𝑖𝑢_𝑥𝑖𝑝𝑑𝑥≤0.‖𝑢‖𝑝+1^𝑝+1𝑝+1≤𝑝𝐵₂^𝑝+1‖‖𝑝+1Δ𝑢₂^𝑝+1≤𝑝𝐵₂^𝑝+12()𝑝+1𝑟+1𝑟−1^(𝑝+1)/2(𝐸(0))^(𝑝−1)/2𝐸(𝑡)𝑢_𝑡^𝑚𝑢_𝑡≤1,𝑢(𝑚+2)𝛿^{(𝑚+2)/(𝑚+1)}𝑢_𝑡^𝑚+2_𝑚+2+𝛿^𝑚+2𝑢𝑚+2^{𝑡𝑚+2𝑚+2}≤1(𝑚+2)𝛿^{(𝑚+2)/(𝑚+1)}𝑢_𝑡^𝑚+2_𝑚+2+𝛿𝐵₂^𝑚+22()𝐸(0)𝑚+2𝑟+1𝑟−1^𝑚/2‖Δ𝑢‖²₂≤1(𝑚+2)𝛿^{(𝑚+2)/(𝑚+1)}𝑢_𝑡^𝑚+2_𝑚+2+𝛿𝐵₂^𝑚+22()𝑚+2𝑟+1𝑟−1^(𝑚+2)/2(𝐸(0))^𝑚/2𝐸(𝑡)𝛾,∀𝛿>0Δ𝑢_𝑡,Δ𝑢≤𝛾𝜂⁻¹2Δ𝑢_𝑡²₂+𝛾𝜂2‖Δ𝑢‖²₂,∀𝜂>0. (3.63) Exploiting (3.63) and (3.62), we get Ψ′(𝑡)≤−1−𝑝𝐵₂^𝑝+22()𝑝+1𝑟+1𝑟−1^(𝑝+1)/2(𝐸(0))^(𝑝−1)/2−𝛿𝐵₂^𝑚+22()𝑚+2𝑟+1𝑟−1^(𝑚+2)/2(𝐸(0))^𝑚/22()−𝛾𝜂𝑟+1𝐸(𝑡)+1𝑟−1(𝑚+2)𝛿^{(𝑚+2)/(𝑚+1)}𝑢_𝑡^𝑚+2_𝑚+2+𝛾𝜂⁻¹+2𝜌+3𝐸(0)𝜌+1𝜌+2𝜌+2^{𝜌/(𝜌+2)}𝐵²₂‖Δ𝑢‖²₂. (3.64) Choosing 𝛿 satisfies 𝛿𝐵₂^𝑚+22()𝐸(0)𝑚+2𝑟+1𝑟−1^𝑚/2=141−𝑝𝐵₂^𝑝+22()𝑝+1𝑟+1𝑟−1^(𝑝+1)/2(𝐸(0))^(𝑝−1)/2 (3.65) and 𝜂 satisfies 2()𝛾𝜂𝑟+1=1𝑟−141−𝑝𝐵₂^𝑝+22()𝑝+1𝑟+1𝑟−1^(𝑝+1)/2(𝐸(0))^(𝑝−1)/2, (3.66) at the above; the proof of (3.61) is completed.

Proof of Theorem 2.3. Using (3.61) and Lemma 3.7, we have 𝐿^′(𝑡)=𝐸^′(𝑡)+𝜀Ψ^′(𝑡)≤−𝑢_𝑡^𝑚+2_𝑚+2−𝛾Δ𝑢_𝑡−𝜀4𝐸(𝑡)+𝜀𝜇𝑢_𝑡^𝑚+2_𝑚+2+𝜀𝜔Δ𝑢_𝑡𝜀=−4𝐸(𝑡)−𝛾−𝜀𝜔Δ𝑢_𝑡−1−𝜀𝜇𝑢_𝑡^𝑚+2_𝑚+2≤−𝜀𝛼₁4𝐿(𝑡)−𝛾−𝜀𝜔Δ𝑢_𝑡−1−𝜀𝜇𝑢_𝑡^𝑚+2_𝑚+2. (3.67) Choosing 𝜀 satisfies 𝜀≤min{𝛾/𝜔,1/𝜇}. So we have 𝐿^′(𝑡)≤−𝜀𝛼₁4𝐿(𝑡),∀𝑡≥0. (3.68) A simple integration of (3.68) over (0,𝑡) leads to 𝐿(𝑡)≤𝐿(0)𝑒^{−(𝜀𝛼1/4)𝑡},∀𝑡≥0. (3.69) Exploiting Lemma 3.7 again, we have 𝐸(𝑡)≤𝛼₂𝐿(0)𝑒^{−(𝜀𝛼1/4)𝑡}∶=𝐾𝑒^−𝜅𝑡,∀𝑡≥0, (3.70) where 𝐾,𝜅>0 are constants. The proof of Theorem 2.3 is complete.

4. Blow-Up of Solutions

Proof of Theorem 2.5. Assuming that the solution of (1.1) is global, we have 𝐸^′(𝑡)=−𝑢_𝑡^𝑚+2_𝑚+2−𝛾Δ𝑢_𝑡²₂≤0. (4.1) So ‖Δ𝑢‖²₂≤𝐶₃; (4.2) we set 𝑄(𝑡)∶=−^𝑡₀𝐸(𝑠)𝑑𝑠+(𝑜𝑡+𝜔)_Ω𝑢²₀𝑑𝑥, (4.3) where 𝑜,𝜔 are constants and will be given later.
Consider 𝑄^′(𝑡)=−𝐸(𝑡)+𝑜_Ω𝑢²₀𝑑𝑥≥𝑜_Ω𝑢²₀𝑑𝑥−𝐸(0). (4.4) Choosing 𝑜 satisfies the following condition in (4.4): 𝑜_Ω𝑢²₀𝑑𝑥−𝐸(0)=𝑄^′(0)>0, (4.5) so we have 𝑄^′(𝑡)≥𝑄^′(0)>0,∀𝑡∈[0,𝑇]. (4.6) Moreover, we have 𝑄^′(𝑡)−𝑄^′(0)=𝐸(0)−𝐸(𝑡)=−^𝑡₀𝐸^′(𝑠)𝑑𝑠=^𝑡₀𝑢_𝑠^𝑚+2_𝑚+2+𝛾Δ𝑢_𝑠²₂𝑑𝑠. (4.7) Define 𝐾(𝑡)∶=𝑄^1−𝛾(𝑡)+𝜀𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌𝑢_𝑠𝑢𝑑𝑥𝑑𝑠, (4.8) where 𝜀>0 will be given later, and 0<𝛾≤min𝑟−𝜌−1(),𝜌+2𝑟+1𝑝−𝑚−1(),𝑝+1𝑚+1𝑟−𝑚−1(𝑚+1)(𝑟+1). (4.9) Multiplying (1.1) by 𝑢 and a direct computation yield 𝐾^′(𝑡)=1−𝛾𝑄^−𝛾𝑄^′(𝑡)+𝜀𝜌+1_Ω𝑢₁^𝜌𝑢₁𝑢₀𝜀𝑑𝑥𝑑𝑠+𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌𝑢_𝑠𝑢_𝑠=𝑄𝑑𝑥𝑑𝑠1−𝛾^−𝛾𝑄^′(𝑡)+𝜀𝜌+1_Ω𝑢₁^𝜌𝑢₁𝑢₀𝜀𝑑𝑥𝑑𝑠+𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌+2𝑑𝑥𝑑𝑠+𝜀^𝑡_0Ω𝑢_𝑠^𝜌𝑢_𝑠𝑠=𝑄𝑢𝑑𝑥𝑑𝑠1−𝛾^−𝛾𝑄^′(𝑡)+𝜀𝜌+1_Ω𝑢₁^𝜌𝑢₁𝑢₀𝜀𝑑𝑥𝑑𝑠+𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌+2𝑑𝑥𝑑𝑠−𝜀^𝑡_0ΩΔ²𝑢𝑢𝑑𝑥𝑑𝑠−𝜀^𝑡_0Ω𝑢_𝑠^𝑚𝑢_𝑠𝑢𝑑𝑥𝑑𝑠−𝜀𝛾^𝑡_0ΩΔ²𝑢_𝑠𝑢𝑑𝑥𝑑𝑠+𝜀^𝑁_𝑖=1^𝑡_0Ω𝜕𝜕𝑥_𝑖𝜎_𝑖𝑢_𝑥𝑖𝑢𝑑𝑥𝑑𝑠+𝜀^𝑡_0Ω|𝑢|^𝑝−1=𝑄𝑢𝑢𝑑𝑥𝑑𝑠1−𝛾^−𝛾𝑄^′(𝑡)+𝜀𝜌+1_Ω𝑢₁^𝜌𝑢₁𝑢₀𝜀𝑑𝑥𝑑𝑠+𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌+2𝑑𝑥𝑑𝑠−𝜀^𝑡_0Ω|Δ𝑢|²𝑑𝑥𝑑𝑠−𝜀𝛾^𝑡_0ΩΔ𝑢_𝑠Δ𝑢𝑑𝑥𝑑𝑠−𝜀^𝑡_0Ω𝑢_𝑠^𝑚𝑢_𝑠𝑢𝑑𝑥𝑑𝑠−𝜀^𝑁_𝑖=1^𝑡_0Ω𝜎_𝑖𝑢_𝑥𝑖𝑢_𝑥𝑖𝑑𝑥𝑑𝑠+𝜀^𝑡₀‖𝑢‖^𝑝+1_𝑝+1𝑑𝑠. (4.10) Exploiting (4.7) and (B1), we have 𝛾^𝑡_0ΩΔ𝑢_𝑠𝛾Δ𝑢𝑑𝑥𝑑𝑠≤𝜂^2𝑡₀Δ𝑢_𝑠²₂𝑑𝑠+𝜂²𝛾^𝑡₀‖Δ𝑢‖²₂≤𝛾𝑑𝑠𝜂²𝑄^′(𝑡)−𝑄^′(0)+𝜂²𝛾𝐶₃𝑇,_𝑡0Ω𝑢_𝑠^𝑚𝑢_𝑠𝜍𝑢𝑑𝑥𝑑𝑠≤^𝑚+2𝑚+2^𝑡_0Ω|𝑢|^𝑚+2𝑑𝑥𝑑𝑠+𝑚+1𝜍𝑚+2^{−((𝑚+1)/(𝑚+2))𝑡}_0Ω𝑢_𝑠^𝑚+2_𝑚+2≤𝜍𝑑𝑥𝑑𝑠_𝑚+2𝑚+2^𝑡_0Ω|𝑢|^𝑚+2𝑑𝑥𝑑𝑠+𝑚+1𝜍𝑚+2^{−((𝑚+1)/(𝑚+2))}𝑄^′(𝑡)−𝑄^′(0)−^𝑡_0Ω𝜎_𝑖𝑢_𝑥𝑖𝑢_𝑥𝑖𝑑𝑥𝑑𝑠≥𝐶₄^𝑡₀‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠. (4.11)(4.12)(4.13) Using (4.10)–(4.13), we have 𝐾^′(𝑡)≥1−𝛾𝑄^−𝛾𝑄^′(𝑡)+𝜀𝜌+1_Ω𝑢₁^𝜌𝑢₁𝑢₀𝜀𝑑𝑥+𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌+2𝑑𝑥𝑑𝑠+𝜀𝐶₄^𝑡₀‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠−𝜀^𝑡₀‖Δ𝑢‖²₂𝑑𝑠+𝜀^𝑡₀‖𝑢‖^𝑝+1_𝑝+1𝑑𝑠−𝜀𝜂²𝛾𝑇𝐶₃−𝜀𝑚+1𝜍𝑚+2^{−(𝑚+1)/(𝑚+2)}+𝛾𝜂²𝑄^′(𝑡)𝜍−𝜀^𝑚+2𝑚+2^𝑡_0Ω|𝑢|^𝑚+2𝑑𝑥𝑑𝑠+𝜀𝑚+1𝜍𝑚+2^{−(𝑚+1)/(𝑚+2)}+𝛾𝜂²𝑄^′(0). (4.14) Choosing 𝜍,𝜂 satisfies 𝜍^{−((𝑚+2)/(𝑚+1))}=𝑀₁𝑄^−𝛾(𝑡),1𝜂²=𝑀₂𝑄^−𝛾(𝑡). (4.15) Then we have 𝐾^′(𝑡)≥1−𝛾−𝜀𝑚+1𝑀𝑚+2¹−𝜀𝛾𝑀₂𝑄^−𝛾𝑄^′(𝑡)+𝜀𝜌+1_Ω𝑢₁^𝜌𝑢₁𝑢₀+𝜀𝑑𝑥𝑑𝑠𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌+2𝑑𝑥𝑑𝑠+𝜀𝑚+1𝑀𝑚+2¹+𝜀𝛾𝑀₂𝑄^−𝛾𝑄^′(0)𝑀−𝜀^1−𝑚−1𝑄𝑚+2^{𝛾(𝑚+1)𝑡}_0Ω|𝑢|^𝑚+2𝑑𝑥𝑑𝑠−𝜀𝑀₂⁻¹𝑄^𝛾𝛾𝑇𝐶₃−𝜀^𝑡₀‖Δ𝑢‖²₂𝑑𝑠+𝜀^𝑡₀‖𝑢‖^𝑝+1_𝑝+1𝑑𝑠+𝜀𝐶₄^𝑡₀‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠. (4.16) A simple computing implies 𝑄^𝛾=−^𝑡₀𝐸(𝑠)𝑑𝑠+(𝑜𝑡+𝜔)_Ω𝑢²₀𝑑𝑥^𝛾≤^𝑡₀‖𝑢‖^𝑝+1_𝑝+1𝑝+1+𝐶₂‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠+(𝑜𝑇+𝜔)𝑢₀²₂^𝛾≤2^𝛾−1𝐶₅(𝑜𝑇+𝜔)^𝛾𝑢₀₂^2𝛾+^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠^𝛾. (4.17) Using (4.17), (B2), embedding theorem, and Hӧlder inequality, we can get 𝑄^𝛾(𝑚+1)≤2^𝛾−1𝐶₅(𝑜𝑇+𝜔)^𝛾𝑢₀₂^2𝛾+^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠^𝛾(𝑚+1)≤2^{(𝛾−1)(𝑚+1)−1}𝐶₆^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠^𝛾(𝑚+1)+(𝑜𝑇+𝜔)^𝛾(𝑚+1)𝑢₀₂^{2𝛾(𝑚+1)},𝛼^𝑡_0Ω|𝑢|^𝑚+2𝑑𝑥𝑑𝑠≤𝛼^𝑡_0Ω|𝑢|^𝑝+1𝑑𝑥^{(𝑚+2)/(𝑝+1)}𝑑𝑠|Ω|^{(𝑝−𝑚−1)/(𝑝+1)}≤𝛼|Ω|^{(𝑝−𝑚−1)/(𝑝+1)}𝑇^{(𝑝−𝑚−1)/(𝑝+1)𝑡}₀‖𝑢‖^𝑝+1_𝑝+1𝑑𝑠^{(𝑚+2)/(𝑝+1)}=∶𝐶₈^𝑡₀‖𝑢‖^𝑝+1_𝑝+1𝑑𝑠^{(𝑚+2)/(𝑝+1)}, (4.18)(4.19) where 𝐶₈=𝛼|Ω|^{(𝑝−𝑚−1)/(𝑝+1)}𝑇^{(𝑝−𝑚−1)/(𝑝+1)}.
Consider (1−𝛼)^𝑡_0Ω|𝑢|^𝑚+2𝑑𝑥𝑑𝑠≤(1−𝛼)𝐵^𝑡₀‖∇𝑢‖₂^𝑚+2≤()𝐵𝑑𝑠1−𝛼^𝑡₀‖∇𝑢‖₂^𝑟+1𝑑𝑠^{(𝑚+2)/(𝑟+1)}𝑡^{(𝑟−𝑚−1)/(𝑟+1)}≤𝐶₇^𝑡₀‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠^{(𝑚+2)/(𝑟+1)}. (4.20) Using (4.18)–(4.20), (B1), we get 𝑄^{𝛾(𝑚+1)𝑡}_0Ω|𝑢|^𝑚+2𝑑𝑥𝑑𝑠≤2^{(𝛾−1)(𝑚+1)−1}𝐶₆^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠^𝛾(𝑚+1)+(𝑜𝑇+𝜔)^𝛾(𝑚+1)𝑢₀₂^{2𝛾(𝑚+1)}×𝐶₈^𝑡₀‖𝑢‖^𝑝+1_𝑝+1𝑑𝑠^{(𝑚+2)/(𝑝+1)}+𝐶₇^𝑡₀‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠^{(𝑚+2)/(𝑟+1)}≤2^{(𝛾−1)(𝑚+1)−1}𝐶₆𝐶₈^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠^{𝛾(𝑚+1)+((𝑚+2)/(𝑝+1))}+(𝑜𝑇+𝜔)^𝛾(𝑚+1)𝑢₀₂^{2𝛾(𝑚+1)}×𝐶₈^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠^{𝛾(𝑚+1)+((𝑚+2)/(𝑝+1))}+2^{(𝛾−1)(𝑚+1)−1}𝐶₆𝐶₇^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠^{𝛾(𝑚+1)+((𝑚+2)/(𝑟+1))}+(𝑜𝑇+𝜔)^𝛾(𝑚+1)𝑢₀₂^{2𝛾(𝑚+1)}×𝐶₇^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠^{𝛾(𝑚+1)+((𝑚+2)/(𝑟+1))}, (4.21) Using (4.9), (4.17), and Young inequality, we have 𝑄^{𝛾(𝑚+1)𝑡}_0Ω|𝑢|^𝑚+2𝑑𝑥𝑑𝑠≤𝐶₉1+^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠. (4.22) Similarly, we have 𝑇𝐶₃𝑄^𝛾≤𝐶₁₀1+^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠. (4.23) Additionally, we choose 𝜀 satisfying 𝜀≤1−𝛾((𝑚+1)/(𝑚+2))𝑀₁+𝛾𝑀₂. (4.24) Using (4.16), (4.21)–(4.24), we get 𝐾^′(𝑡)≥𝜀𝜌+1_Ω𝑢₁^𝜌𝑢₁𝑢₀𝜀𝑑𝑥𝑑𝑠+𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌+2𝑑𝑥𝑑𝑠−𝜀^𝑡₀‖Δ𝑢‖²₂𝑀𝑑𝑠−𝜀^{1(−𝑚−1)}𝐶𝑚+2⁹+𝐶₁₀𝛾𝑀₂×^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+‖∇𝑢‖^𝑟+1_𝑟+1𝑀−𝜀^{1(−𝑚−1)}𝐶𝑚+2⁹+𝐶₁₀𝛾𝑀₂+𝜀min1,𝐶₄^𝑡₀‖∇𝑢‖^𝑟+1_𝑟+1+‖𝑢‖^𝑝+1_𝑝+1+𝜉𝑄(𝑡)+𝜉^𝑁_𝑖=1^𝑡_0Ω^𝑢𝑥𝑖₀𝜎_𝑖(𝜏)𝑑𝜏𝑑𝑥𝑑𝑠−𝜉^𝑡₀‖𝑢‖^𝑝+1_𝑝+1𝜉𝑝+1𝑑𝑠+𝜌+1^𝑡₀𝑢_𝑠^𝜌+2_𝜌+2+𝜉𝑑𝑠2^𝑡₀‖Δ𝑢‖²₂𝑑𝑠−𝜉(𝑜𝑡+𝜔)𝑢₀²₂,_{𝑡0Ω𝑢𝑥𝑖0}𝜎_𝑖(𝜏)𝑑𝜏𝑑𝑥𝑑𝑠−^𝑡₀‖𝑢‖^𝑝+1_𝑝+1𝐶𝑝+1𝑑𝑠≥−²𝑟+1^𝑡₀‖∇𝑢‖^𝑟+1_𝑟+1𝑑𝑠−^𝑡₀‖𝑢‖^𝑝+1_𝑝+11𝑝+1𝑑𝑠≥−𝑝+1^𝑡₀‖∇𝑢‖^𝑟+1_𝑟+1+‖𝑢‖^𝑝+1_𝑝+1𝑑𝑠. (4.25) Using (4.25), we get 𝐾^′(𝑡)(𝑡)+𝜉≥𝜉𝑄+𝜀𝜌+2𝜌+1^𝑡₀𝑢_𝑠^𝜌+2_𝜌+2𝜉𝑑𝑠+𝜀2𝜀−1^𝑡₀‖Δ𝑢‖²₂𝑑𝑠+𝜀min1,𝐶₄−1𝜉𝑝+1𝜀−𝑀₁^−𝑚−1𝐶𝑚+2⁹−𝐶₁₀𝛾𝑀₂^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+‖∇𝑢‖^𝑟+1_𝑟+11+𝜀𝜌+1_Ω𝑢₁^𝜌𝑢₁𝑢₀𝜉𝑑𝑥−𝜀(𝑜𝑇+𝜔)^𝑡₀𝑢²₀𝑀𝑑𝑥−^1−𝑚−1𝐶𝑚+2⁹−𝐶₁₀𝛾𝑀₂, (4.26) choosing 2𝜀<𝜉<(𝑝+1)𝜀, 𝑢₀,𝑢₁ satisfying min1,𝐶₄−1𝜉𝑝+1𝜀1>0,𝜌+1_Ω𝑢₁^𝜌𝑢₁𝑢₀𝜉𝑑𝑥−𝜀(𝑜𝑇+𝜔)^𝑡₀𝑢²₀𝑑𝑥>0. (4.27) Then we choose 𝑀₁,𝑀₂ big enough, satisfying min1,𝐶₄−1𝜉𝑝+1𝜀−𝑀₁^−𝑚−1𝐶𝑚+2⁹−𝐶₁₀𝛾𝑀₂1>0𝜌+1_Ω𝑢₁^𝜌𝑢₁𝑢₀𝜉𝑑𝑥−𝜀(𝑜𝑇+𝜔)^𝑡₀𝑢²₀𝑀𝑑𝑥−^1−𝑚−1𝐶𝑚+2⁹−𝐶₁₀𝛾𝑀₂>0. (4.28) Using the two above inequalities, we have 𝐾^′(𝑡)≥𝐶𝜀𝑄(𝑡)+^𝑡₀𝑢_𝑠^𝜌+2_𝜌+2𝑑𝑠+^𝑡₀‖Δ𝑢‖²₂𝑑𝑠+^𝑡₀‖∇𝑢‖^𝑟+1_𝑟+1+‖𝑢‖^𝑝+1_𝑝+1𝑑𝑠+𝜈, (4.29) where 𝜈>0. So 𝐾(𝑡)>𝐾(0)≥𝜔_Ω𝑢²₀𝑑𝑥^1−𝛾>0. (4.30)
Using (B1)–(B3), (A4), and Hӧlder and Young inequalities, we have 1𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌𝑢_𝑠≤1𝑢𝑑𝑥𝑑𝑠𝜌+1^𝑡₀𝑢_𝑠^𝜌+1_𝜌+2‖∇𝑢‖_𝑟+1≤𝐶𝑑𝑠𝜌+1^𝑡₀𝑢_𝑠^(𝜌+1)𝜇_𝜌+2+‖∇𝑢‖^𝜃_𝑟+1𝑑𝑠, (4.31) where (1/𝜇)+(1/𝜃)=1.
So we have 1𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌𝑢_𝑠𝑢𝑑𝑥𝑑𝑠^1/(1−𝛾)≤𝐶^𝑡₀𝑢_𝑠^(𝜌+1)𝜇_𝜌+2𝑑𝑠^1/(1−𝛾)+^𝑡₀‖∇𝑢‖^𝜃_𝑟+1𝑑𝑠^1/(1−𝛾)≤𝐶^𝑡₀𝑢_𝑠^{(𝜌+1)𝜇⋅((𝜌+2)/((𝜌+1)𝜇))}_𝜌+2𝑑𝑠^{𝜇(𝜌+1)/(1−𝛾)(𝜌+2)}+𝐶^𝑡₀‖∇𝑢‖^{𝜃⋅((1+𝑟)/𝜃)}_𝑟+1𝑑𝑠^{𝜃/(1+𝑟)(1−𝛾)}=𝐶^𝑡₀𝑢_𝑠^𝜌+2_𝜌+2𝑑𝑠^{𝜇(𝜌+1)/(1−𝛾)(𝜌+2)}+𝐶^𝑡₀‖∇𝑢‖^1+𝑟_𝑟+1𝑑𝑠^{𝜃/(1+𝑟)(1−𝛾)}. (4.32) Using (4.9), (4.32) and choosing 𝜇 satisfy 𝜇(𝜌+1)/((𝜌+2)(1−𝛾))=1, then 𝜃/(1+𝑟)(1−𝛾)<1.
So 1𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌𝑢_𝑠𝑢𝑑𝑥𝑑𝑠^1/(1−𝛾)≤𝐶^𝑡₀𝑢_𝑠^𝜌+2_𝜌+2𝑑𝑠+^𝑡₀‖∇𝑢‖^1+𝑟_𝑟+1𝑑𝑠+𝛽≤𝐶^𝑡₀𝑢_𝑠^𝜌+2_𝜌+2𝑑𝑠+^𝑡₀‖∇𝑢‖^1+𝑟_𝑟+1+𝑑𝑠_𝑡0‖Δ𝑢‖²₂𝑑𝑠+^𝑡₀‖𝑢‖^𝑝+1_𝑝+1+𝛽, (4.33) where 𝛽>0 is a constant. Finally, we can easily get 𝐾^1/(1−𝛾)≤2^1/(1−𝛾)𝑄(𝑡)+𝜀^1/(1−𝛾)1𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌𝑢_𝑠𝑢𝑑𝑥𝑑𝑠^1/(1−𝛾). (4.34) Combining (4.29) and (4.33)-(4.34), we have 𝐾^′(𝑡)≥𝐶𝐾^1/(1−𝛾)(𝑡),𝑡≤𝑇, (4.35) for some constant 𝐶>0. Integrating the above inequality in (0,𝑡), we get 𝐾^1/(1−𝛾)≥1𝐾^{−𝛾/(1−𝛾)}(0)−𝑐𝑡^1/𝛾,∀𝑡≤𝑇. (4.36) The above inequality implies 𝐾(𝑡) blows-up on some time 𝑇^∗. Since 𝑢 exists globally, so we have 𝜀𝜌+1^𝑡_0Ω𝑢_𝑠^𝜌𝑢_𝑠𝑢𝑑𝑥𝑑𝑠<∞. (4.37) And we know that 𝐾(𝑡)→∞,as𝑡→𝑇^∗, so 𝑄^1−𝛾(𝑡)→∞; (4.38) this implies 𝑄(𝑡)→∞, that is to say −∫^𝑡₀𝐸(𝑠)𝑑𝑠→∞. Because −^𝑡₀𝐸(𝑠)𝑑𝑠≤−𝑡𝐸(𝑡), (4.39) we know 𝐸(𝑡)→−∞,as𝑡→𝑇^∗. (4.40) This contradicts with the assumption that 𝑢 is a global solution. So the solution of (1.1) blows-up on time 𝑇^∗.

Acknowledgment

This work was supported by the National Natural Science Foundation of China 10771032.