On Asymptotic Behavior and Blow-Up of Solutions for a Nonlinear Viscoelastic Petrovsky Equation with Positive Initial Energy

Abstract

This paper deals with the initial boundary value problem for the nonlinear viscoelastic Petrovsky equation 𝑢_𝑡𝑡+Δ²𝑢−∫^𝑡₀𝑔(𝑡−𝜏)Δ²𝑢(𝑥,𝜏)d𝜏−Δ𝑢_𝑡−Δ𝑢_𝑡𝑡+|𝑢_𝑡|^𝑚−1𝑢_𝑡=|𝑢|^𝑝−1𝑢. Under certain conditions on 𝑔 and the assumption that 𝑚<𝑝, we establish some asymptotic behavior and blow-up results for solutions with positive initial energy.

1. Introduction

In this work, we investigate the following nonlinear viscoelastic Petrovsky problem: 𝑢_𝑡𝑡+Δ²𝑢−^𝑡₀𝑔(𝑡−𝜏)Δ²𝑢(𝑥,𝜏)d𝜏−Δ𝑢_𝑡−Δ𝑢_𝑡𝑡+𝑢_𝑡^𝑚−1𝑢_𝑡=|𝑢|^𝑝−1𝑢()=()𝑢,𝑥∈Ω,𝑡≥0,𝑥,𝑡𝜕𝑢𝑥,𝑡𝑢()𝜕𝜈=0,𝑥∈𝜕Ω,𝑡≥0,𝑥,0=𝑢₀(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥),𝑥∈Ω, (1) where Ω is a bounded domain in ℝ^𝑛 (𝑛≥1) with a smooth boundary 𝜕Ω, 𝑚,𝑝>1; 𝜈 is the unit outer normal on 𝜕Ω; and 𝑔 is a nonnegative memory term.

In the absence of viscoelastic term (i.e., 𝑔=0), Guesmia [1] considered the following equation: 𝑢_𝑡𝑡+Δ²𝑢+𝑞(𝑥)𝑢(𝑥,𝑡)+𝑔𝑢_𝑡=0,𝑥∈Ω,𝑡>0, (2) where 𝑞∶Ω→ℝ⁺ is a bounded function and 𝑔 is continuous and increasing function satisfying 𝑔(0)=0. Under suitable growth conditions on 𝑔, the author established global existence, uniqueness, and decay results by using the semigroup method. Messaoudi [2] investigated a nonlinearly damped semilinear Petrovsky equation 𝑢_𝑡𝑡+Δ²𝑢+𝑎𝑢_𝑡^𝑚−2𝑢_𝑡=𝑏|𝑢|^𝑝−2𝑢,𝑥∈Ω,𝑡>0, (3) where 𝑎,𝑏>0 and 𝑝,𝑚>2 and proved that the solution is global when 𝑚≥𝑝 while the solution blows up in finite time with negative initial energy when 𝑚<𝑝. Later, this blow-up result was improved by Chen and Zhou [3] with positive initial energy. For a related study, we may see the work of Wu and Tsai [4]. In [5], Amroun and Benaissa studied (3) by generalizing the damping term into the form of 𝑔(𝑢_𝑡) and obtained the global existence of the solutions by means of the stable set method combined with the Faedo-Galerkin procedure. Very recently, in the presence of the strong damping, Li et al. [6] considered the following Petrovsky equation: 𝑢_𝑡𝑡+Δ²𝑢−Δ𝑢_𝑡+𝑢_𝑡^𝑚−1𝑢_𝑡=|𝑢|^𝑝−1𝑢,𝑥∈Ω,𝑡≥0. (4) Without any interaction between 𝑚 and 𝑝, the authors obtained the global existence and uniform decay of solutions when the initial data are in some stable set. And a blow-up result was established when 𝑚<𝑝 and the initial energy is less than the potential well depth.

In the presence of the viscoelastic term (i.e., 𝑔≠0), Muñoz Rivera et al. [7] studied the following equation: 𝑢_𝑡𝑡+Δ²𝑢−^𝑡₀𝑔(𝑡−𝜏)Δ²𝑢(𝑥,𝜏)d𝜏−𝛾Δ𝑢_𝑡𝑡=0,𝑥∈Ω,𝑡≥0. (5) They proved that the memory effect produces strong dissipation capable of making uniform rate of decay for the energy. Later, in the presence of strong damping term, M. M. Cavalcanti et al. [8] considered 𝑢_𝑡^𝜌𝑢_𝑡𝑡−Δ𝑢+^𝑡₀𝑔(𝑡−𝜏)Δ𝑢(𝑥,𝜏)d𝜏−𝛾Δ𝑢_𝑡−Δ𝑢_𝑡𝑡=0,𝑥∈Ω,𝑡≥0, (6) and obtained a global existence for 𝛾≥0 and uniform exponential decay for 𝛾>0. This work was extended by Messaoudi and Tatar [9] to a situation where a nonlinear source term is competing with the damping induced by −𝛾Δ𝑢_𝑡 and the integral term. Then in the case of 𝛾=0, the same authors [10] showed that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set by introducing a new functional and using the potential well method. Recently, Wu [11] improved [10] by considering the nonlinear equation: 𝑢_𝑡^𝜌𝑢_𝑡𝑡−Δ𝑢−Δ𝑢_𝑡𝑡+^𝑡₀𝑔(𝑡−𝜏)Δ𝑢(𝑥,𝜏)d𝜏+𝑢_𝑡^𝑚𝑢_𝑡=|𝑢|^𝑝𝑢, (7) and a general decay result was obtained. In the presence of strong damping term Δ𝑢_𝑡 and dispersive term Δ𝑢_𝑡𝑡, Xu et al. [12] considered the initial boundary value problem for the following viscoelastic wave equation: 𝑢_𝑡𝑡−Δ𝑢+^𝑡₀𝑔(𝑡−𝜏)Δ𝑢(𝑥,𝜏)d𝜏−Δ𝑢_𝑡−Δ𝑢_𝑡𝑡+𝑢_𝑡=|𝑢|^𝑝−1𝑢. (8) By introducing a family of potential wells, the authors not only obtained the invariant sets, but also proved the existence and nonexistence of global weak solution under some conditions with low initial energy. Furthermore, they established a blow-up result for certain solutions with arbitrary positive initial energy (high energy case). Very recently, Tahamtani and Peyravi [13] considered problem (1) and obtained the exponential decay of the energy under some assumptions on 𝑔 without any interaction between source term and damping term. Under an appropriate restriction on 𝑔, they also proved that the 𝐿^𝑝+1 norm of any solution grows as an exponential function if 𝑚<𝑝 and the initial energy is negative. For other related works, we refer the readers to [14–24] and the references therein.

Motivated by the above works, in this paper, we intend to consider problem (1) and establish some asymptotic behavior and blow-up results for solutions with positive initial energy. For our purpose, we use the functional 𝐿(𝑡)=‖𝑢‖²₂+‖∇𝑢‖²₂ and give a modified manner to estimate the term |∫_Ω𝑢|𝑢_𝑡|^𝑚−1𝑢_𝑡d𝑥| so that the appearance of the form like 𝛾=𝑅𝐻^−𝜎(𝑡) (for constants 𝛾, 𝑅, and 𝜎) which has been used in many earlier works (e.g., in [3, 19, 25]) can be avoided.

The paper is organized as follows. In Section 2 we present some assumptions and known results and state the main results. Section 3 is devoted to proof the the blow-up result—Theorem 4.

2. Preliminaries and Main Results

In this section, we first present some assumptions and known results which will be used throughout this work.

For the relaxation function 𝑔, we give the following assumptions:(G1)𝑔(𝑡)∶ℝ⁺→ℝ⁺ is a 𝐶¹ function such that 𝑔(0)>0,1−^∞₀𝑔(𝑠)d𝑠=𝑙>0, (9)(G2)𝑔^′(𝑡)≤0, for all 𝑡≥0.

Lemma 1 (Sobolev-Poincaré inequality). Let 𝑞 be a number with 2≤𝑞<∞(𝑛=1,2,3,4) or 2≤𝑞≤2𝑛/(𝑛−4)(𝑛≥5); then for 𝑢∈𝐻²₀(Ω) there exists a positive 𝐶_∗=𝐶(Ω,𝑞) such that ‖𝑢‖_𝑞≤𝐶_∗‖Δ𝑢‖₂. (10)

We assume that 𝑚,𝑝 satisfy ()𝑛1<𝑝<∞𝑛=1,2,3,4or1<𝑝≤(),()𝑛−4𝑛≥51<𝑚<∞𝑛=1,2,3,4or1<𝑚≤𝑛+4().𝑛−4𝑛≥5 (11) We state a local existence theorem that can be established by adopting the arguments of [2, 5, 22].

Theorem 2. Assume that (11), (G1), and (G2) hold. Let (𝑢₀,𝑢₁)∈𝐻²₀(Ω)×𝐻¹₀(Ω) be given. Then there exists a unique weak solution 𝑢(𝑡) such that 𝑢∈𝐶[0,𝑇];𝐻²₀(Ω)∩𝐶¹[0,𝑇];𝐿²(Ω,𝑢_𝑡)∈𝐿²[0,𝑇];𝐻¹₀(Ω)∩𝐿^𝑚+1(Ω×(0,𝑇)), (12) for 𝑇>0 small enough.

Next, we define the following functionals: 𝐼(𝑡)(𝑢(𝑡=∶=𝐼))1−^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢‖²₂+𝑔∘Δ𝑢(𝑡)−‖𝑢‖^𝑝+1_𝑝+1,𝐽(𝑡)(𝑢(𝑡=1∶=𝐽))21−^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢‖²₂+12(𝑡)−1𝑔∘Δ𝑢‖𝑢‖𝑝+1^𝑝+1𝑝+1,𝐸(𝑡)(𝑢(𝑡=1∶=𝐸))2𝑢_𝑡²₂+12∇𝑢_𝑡²₂+𝐽(𝑡), (13) where 𝑔∘𝑣(𝑡)=^𝑡_0Ω𝑔(𝑡−𝜏)|𝑣(𝑡)−𝑣(𝜏)|²d𝑥d𝜏. (14)

Remark 3. A multiplication of (1) by 𝑢_𝑡 and integration over Ω easily yield 𝐸^′(𝑡)1=−2𝑔(𝑡)‖Δ𝑢‖²₂−∇𝑢_𝑡²₂+12𝑔^′∘Δ𝑢(𝑡)−𝑢_𝑡^𝑚+1_𝑚+1≤−∇𝑢_𝑡²₂−𝑢_𝑡^𝑚+1_𝑚+1≤0, (15) since 𝑔^′(𝑡)≤0.
Our main results read as follows.

Theorem 4. Suppose that (11), (G1), and (G2) hold and 𝑚<𝑝. Let 𝑢 be the unique solution to problem (1) and denote 𝑑₁=((𝑝−1)/2(𝑝+1))(𝑙/𝐶²_∗)^{(𝑝+1)/(𝑝−1)}. Then, for any fixed 0<𝜃<1, 𝐸(0)=𝜃𝑑₁<𝑑₁, 𝐼(0)<0, and 𝑔 satisfies ^∞₀𝑔(𝜏)()d𝜏<𝑝−11−𝜃𝑝−1+1/𝑝+1−𝜃𝑝−1, (16) either there exists some 𝑇^∗ such that the solution of problem (1) blows up in 𝑇^∗ in the sense of lim_𝑡→𝑇^∗−𝐸(𝑡)=−∞, or one has lim_𝑡→+∞𝐸(𝑡)=−∞.

3. Proof of the Main Results

We denote 𝑑(𝑡)=inf_{𝑢∈𝐻20(Ω)⧵{0}}sup_𝜆≥0𝐽(𝜆𝑢),𝒩=𝑢∈𝐻²₀(Ω)⧵{0}∶𝐼(𝑢(𝑡))=0, (17) then we can prove the following lemma.

Lemma 5. For 𝑡≥0, we have 0<𝑑₁≤𝑑(𝑡)≤𝑑₂(𝑢)=sup_𝜆≥0𝐽(),𝑑(𝑡)𝜆𝑢=inf_𝑢∈𝒩𝐽(𝑢). (18)(19)

Proof. Obviously, 𝑑(𝑡)≤𝑑₂(𝑢)=sup_𝜆≥0𝐽(𝜆𝑢). (20) Straightforward computations yield 𝐽()=𝜆𝜆𝑢²21−^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+(𝑡−𝜆𝑔∘Δ𝑢)_𝑝+1‖𝑢(𝑡𝑝+1)‖^𝑝+1_𝑝+1, (21) then d𝐽()d𝜆𝜆𝑢=𝜆1−^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+𝑔∘Δ𝑢(𝑡)−𝜆^𝑝‖𝑢(𝑡)‖^𝑝+1_𝑝+1. (22) Let d𝐽()d𝜆𝜆𝑢=0, (23) which leads to 𝜆₁=0,𝜆₂=1−∫^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+𝑔∘Δ𝑢(𝑡)^1/(𝑝−1)‖𝑢(𝑡)‖^𝑝+1_𝑝+1^1/(𝑝−1). (24) An elementary calculation shows d²d𝜆²𝐽𝜆₁𝑢d>0,²d𝜆²𝐽𝜆₂𝑢<0. (25) Using (G1) and Lemma 1 we arrive at sup_𝜆≥0𝐽(𝜆𝑢)=𝐽𝜆₂𝑢=𝑝−12×𝑝+11−∫^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+𝑔∘Δ𝑢(𝑡)‖𝑢(𝑡)‖^𝑝+1_𝑝+1^{2/(𝑝+1)(𝑝+1)/(𝑝−1)}≥𝑝−12𝑙‖(𝑡𝑝+1Δ𝑢)‖²₂‖𝑢(𝑡)‖^𝑝+1_𝑝+1^{2/(𝑝+1)(𝑝+1)/(𝑝−1)}≥𝑝−12𝑙𝑝+1𝐶²_∗^{(𝑝+1)/(𝑝−1)}=𝑑₁>0, (26) which implies that 𝑑(𝑡)≥𝑑₁.
To get (19), straightforward computations lead to 1−^𝑡₀𝑔(𝜏)d𝜏Δ𝜆₂𝑢²₂+𝑔∘Δ𝜆₂𝑢(𝑡)−𝜆₂𝑢^𝑝+1_𝑝+1=1−^𝑡₀𝑔(𝜏)×d𝜏1−∫^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+𝑔∘Δ𝑢(𝑡)^2/(𝑝−1)‖𝑢(𝑡)‖^𝑝+1_𝑝+1^2/(𝑝−1)×‖Δ𝑢(𝑡)‖²₂+1−∫^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+𝑔∘Δ𝑢(𝑡)^2/(𝑝−1)‖𝑢(𝑡)‖^𝑝+1_𝑝+1^2/(𝑝−1)×(𝑡)−𝑔∘Δ𝑢1−∫^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+𝑔∘Δ𝑢(𝑡)^{(𝑝+1)/(𝑝−1)}‖𝑢(𝑡)‖^𝑝+1_𝑝+1^{(𝑝+1)/(𝑝−1)}×‖𝑢(𝑡)‖^𝑝+1_𝑝+1=1−∫^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+𝑔∘Δ𝑢(𝑡)^{(𝑝+1)/(𝑝−1)}‖𝑢(𝑡)‖^{2(𝑝+1)/(𝑝−1)}_𝑝+1×1−∫^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂1−∫^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+(𝑡)+(𝑡)𝑔∘Δ𝑢𝑔∘Δ𝑢1−∫^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+(𝑡)−‖𝑢(𝑡𝑔∘Δ𝑢)‖^𝑝+1_𝑝+1‖𝑢(𝑡)‖^𝑝+1_𝑝+1=0, (27) which implies that 𝜆₂𝑢∈𝒩. Also, for any 𝑢∈𝒩, we note that 𝜆₂(𝑢)=1−∫^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+𝑔∘Δ𝑢(𝑡)^1/(𝑝−1)‖𝑢(𝑡)‖^𝑝+1_𝑝+1^1/(𝑝−1)=1. (28) Therefore we have 𝜆₂(𝑢)𝑢=𝑢 for all 𝑢∈𝒩. Hence, we complete the proof.

Lemma 6. Suppose that (11), (G1), and (G2) hold; 𝐸(0)<𝑑₁; and 𝐼(0)<0; then we have 𝐼(𝑡)<0, for all 𝑡∈[0,𝑇) and 𝑑₁<𝑝−12𝑝+11−^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+𝑡<𝑔∘Δ𝑢()𝑝−12‖𝑢‖𝑝+1^𝑝+1𝑝+1. (29)

Proof. Using (15), we have 𝐸(𝑡)<𝑑₁, for all 𝑡∈[0,𝑇). To show that 𝐼(𝑡)<0 on [0,𝑇), we proceed by contradiction. Assume there exists 𝑡₀∈[0,𝑇) such that 𝐼(𝑡₀)≥0. Since 𝐼(0)<0, it follows that there exists some 𝑡̃∈(0,𝑡₀] such that 𝐼(𝑢(𝑡)̃)=0. Now, we define 𝑡^∗=inf𝑡̃∈0,𝑡₀∶𝑡̃1−₀𝑔(𝜏)×𝑡̃d𝜏Δ𝑢²₂+𝑔∘Δ𝑢𝑡̃=𝑢𝑡̃^𝑝+1_𝑝+1. (30) Then, we have 𝐼(𝑢(𝑡^∗))=0 and 1−^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢‖²₂+𝑡)<‖𝑢‖𝑔∘Δ𝑢(_𝑝+1𝑝+1,∀0≤𝑡<𝑡^∗. (31)
Suppose that ‖Δ𝑢(𝑡^∗)‖²₂=0, by the regularity of 𝑢(𝑡), we have lim_{𝑡→𝑡∗−}‖Δ𝑢(𝑡)‖²₂=0. (32) On the other hand, applying Lemma 1 to (31), we obtain 1−^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢‖²₂+𝑡)<‖𝑢‖𝑔∘Δ𝑢(_𝑝+1𝑝+1≤𝐶_∗^𝑝+1‖Δ𝑢‖₂^𝑝+1,0≤𝑡<𝑡^∗. (33) From the above inequality, we can easily get ‖Δ𝑢(𝑡)‖²₂≠0, for all 𝑡∈[0,𝑡^∗), and 𝑙𝐶_∗^𝑝+1<‖Δ𝑢‖₂^𝑝−1, (34) which implies lim_{𝑡→𝑡∗−}‖Δ𝑢(𝑡)‖₂>𝑙𝐶_∗^{𝑝+11/(𝑝−1)} (35) and this contradicts to (32).
Suppose that ‖Δ𝑢(𝑡^∗)‖²₂≠0. Applying Lemma 5, we see that 𝑑(𝑡) is the infimum of 𝐽(𝑢(𝑡)) over all functions 𝑢 in 𝒩 and 𝐽(𝑢(𝑡^∗))≥𝑑(𝑡)≥𝑑₁, which contradicts to 𝐽(𝑢(𝑡^∗))≤𝐸(𝑡^∗)<𝑑₁. Thus, we conclude that 𝐼(𝑡)<0 for all 𝑡∈[0,𝑇).
To get (29), using Lemma 5, (31) and the conclusion that 𝐼(𝑡)<0, for all 𝑡∈[0,𝑇), we obtain 𝑑₁≤𝑝−12×𝑝+11−^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+(𝑡×𝑔∘Δ𝑢)1−^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+𝑔∘Δ𝑢(𝑡)^2/(𝑝−1)×‖𝑢‖^𝑝+1_𝑝+1^{2/(𝑝−1)−1}<𝑝−12𝑝+11−^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+(𝑡<𝑔∘Δ𝑢)𝑝−12‖𝑢‖𝑝+1^𝑝+1𝑝+1,0≤𝑡<𝑇, (36) which completes the proof.

Proof of Theorem 4. Suppose on the contrary that there exists a positive constant 𝐶₀ such that 𝐸(𝑡)≥−𝐶₀, for all 𝑡≥0. An integration of (15) over [0,𝑡] yields 𝑑+𝐶₀>𝐸(0)−𝐸(𝑡)≥^𝑡₀∇𝑢_𝑡(𝜏)²₂+𝑢_𝑡(𝜏)^𝑚+1_𝑚+1d𝜏,∀𝑡≥0. (37) Set 𝐿(𝑡)=‖𝑢‖²₂+‖∇𝑢‖²₂; then we have 𝐿^′(𝑡)=2_Ω𝑢𝑢_𝑡d𝑥+2_Ω∇𝑢⋅∇𝑢_𝑡𝐿d𝑥,_′′(𝑡)=2_Ω𝑢²_𝑡d𝑥+2_Ω𝑢𝑢_𝑡𝑡d𝑥+2_Ω∇𝑢²_𝑡d𝑥+2_Ω∇𝑢⋅∇𝑢_𝑡𝑡𝑢d𝑥=2_𝑡²₂+∇𝑢_𝑡²₂−‖Δ𝑢‖²₂+2_ΩΔ𝑢^𝑡₀𝑔(𝑡−𝜏)Δ𝑢(𝑥,𝜏)d𝜏d𝑥−2_Ω∇𝑢⋅∇𝑢_𝑡d𝑥−2_Ω𝑢𝑢_𝑡^𝑚−1𝑢_𝑡d𝑥+2‖𝑢‖^𝑝+1_𝑝+1. (38)(39) We estimate (39) as follows. By using Cauchy-Schwarz inequality and Young's inequality, we get for any 𝜂>0_Ω^𝑡₀𝑔()(𝑡)(𝜏)=𝑡−𝜏Δ𝑢Δ𝑢d𝜏d𝑥_Ω𝑡0𝑔()(𝑡(𝜏)(𝑡+𝑡−𝜏Δ𝑢)(Δ𝑢−Δ𝑢))d𝜏d𝑥_𝑡0𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂≥−^𝑡₀𝑔(𝑡−𝜏)‖Δ𝑢(𝑡)‖₂‖Δ𝑢(𝜏)−Δ𝑢(𝑡)‖₂+d𝜏_𝑡0𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂(𝑡)+1≥−𝜂𝑔∘Δ𝑢1−4𝜂^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂. (40) Using Young's inequality, we get for any 𝜀₁>02_Ω∇𝑢⋅∇𝑢_𝑡d𝑥≤𝜀₁‖∇𝑢‖²₂+𝐶𝜀₁∇𝑢_𝑡²₂. (41) By Hölder and Young inequalities, we arrive at 2_Ω𝑢𝑢_𝑡𝑢_𝑡^𝑚−1‖𝑢‖d𝑥≤2_𝑚+1𝑢_𝑡^𝑚_𝑚+1≤2|Ω|^{(𝑝−𝑚)/((𝑝+1)(𝑚+1))}‖𝑢‖_𝑝+1𝑢_𝑡^𝑚_𝑚+1≤2|Ω|^{(𝑝−𝑚)/((𝑝+1)(𝑚+1))}‖𝑢‖^{1−(𝑝+1)/(𝑚+1)}_𝑝+1×‖𝑢‖^{(𝑝+1)/(𝑚+1)}_𝑝+1𝑢_𝑡^𝑚_𝑚+1<2|Ω|^{(𝑝−𝑚)/((𝑝+1)(𝑚+1))}2𝑝+1𝑑₁𝑝−1^{(1/(𝑝+1))−(1/(𝑚+1))}×‖𝑢‖^{(𝑝+1)/(𝑚+1)}_𝑝+1𝑢_𝑡^𝑚_𝑚+1≤𝐶_𝑑𝜀₂‖𝑢‖^𝑝+1_𝑝+1+𝐶𝜀₂𝑢_𝑡^𝑚+1_𝑚+1, (42) where 𝐶_𝑑=2|Ω|^{(𝑝−𝑚)/((𝑝+1)(𝑚+1))}(2(𝑝+1)𝑑₁/(𝑝−1))^{(1/(𝑝+1))−(1/(𝑚+1))}. Combining (40)–(42), we obtain 𝐿^′′(𝑡)+𝐶𝜀₁∇𝑢_𝑡²₂+𝐶_𝑑𝐶𝜀₂𝑢_𝑡^𝑚+1_𝑚+1>2𝑢_𝑡²₂+∇𝑢_𝑡²₂1−21−1−4𝜂^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂−2𝜂𝑔∘Δ𝑢(𝑡)+2−𝐶_𝑑𝜀₂‖𝑢‖^𝑝+1_𝑝+1−𝜀₁‖∇𝑢‖²₂≥2𝑢_𝑡²₂+∇𝑢_𝑡²₂𝜀−21+¹𝐵2−11−4𝜂^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂−2𝜂𝑔∘Δ𝑢(𝑡)+2−𝐶_𝑑𝜀₂‖𝑢‖^𝑝+1_𝑝+1, (43) where 𝐵 is the Poincaré constant. Since 𝜀−21+¹𝐵2−11−4𝜂^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂−2𝜂𝑔∘Δ𝑢(𝑡)+2−𝐶_𝑑𝜀₂‖𝑢‖^𝑝+1_𝑝+1𝜀≥−21+¹𝐵2−11−4𝜂^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂−2𝜂𝑔∘Δ𝑢(𝑡)+2−𝐶_𝑑𝜀₂‖𝑢‖^𝑝+1_𝑝+1(𝐸(𝑡)(0≥𝜎+𝜎−𝐸))2−2−𝜀₁𝜎𝐵−21−21−4𝜂^𝑡₀𝑔(𝜏)×‖(𝑡d𝜏Δ𝑢)‖²₂+𝜎2𝑢_𝑡²₂+∇𝑢_𝑡²₂+𝜎2(𝑡)+−2𝜂𝑔∘Δ𝑢2−𝐶_𝑑𝜀₂−𝜎‖𝑢‖𝑝+1^𝑝+1𝑝+1−𝜎𝐸(0),∀𝑡≥0, (44) where 𝜎>0 will be chosen later. Then, we have 𝐿^′′(𝑡)+𝐶𝜀₁∇𝑢_𝑡²₂+𝐶_𝑑𝐶𝜀₂𝑢_𝑡^𝑚+1_𝑚+1>𝜎2+2𝑢_𝑡²₂+∇𝑢_𝑡²₂+𝜎2−2−𝜀₁𝜎𝐵−21−21−4𝜂^𝑡₀𝑔(𝜏)×‖(𝑡d𝜏Δ𝑢)‖²₂+𝜎2(𝑡)+−2𝜂𝑔∘Δ𝑢2−𝐶_𝑑𝜀₂−𝜎‖𝑢‖𝑝+1^𝑝+1𝑝+1−𝜎𝐸(0),∀𝑡≥0. (45) Choosing 𝜂<(𝜎/4)−(𝜃(𝑝−1)/2) and 𝜀₁ small enough and using (16), we can get a certain constant 𝜎∈(2𝜃(𝑝−1)+4,2(𝑝+1)), such that 𝜎2−2−𝜀₁𝜎𝐵−21−21−4𝜂^𝑡₀𝑔(𝜏),𝜎d𝜏>𝜃𝑝−12−2𝜂>𝜃𝑝−1. (46) Therefore, it follows from (G1) and Lemma 6 that 𝜎2−2−𝜀₁𝜎𝐵−21−21−4𝜂^𝑡₀𝑔(𝜏)d𝜏‖Δ𝑢(𝑡)‖²₂+𝜎2(𝑡)(0)‖(𝑡−2𝜂𝑔∘Δ𝑢−𝜎𝐸>𝜃𝑝−1Δ𝑢)‖²₂+(𝑡(0)𝑑𝑔∘Δ𝑢)−𝜎𝐸>2𝜃𝑝+1₁(0)𝐸(0)(0)−𝜎𝐸≥2𝑝+1−𝜎𝐸>0, (47) since the choice of 𝜎. Finally, we take 𝜀₂ small enough so that 2−𝐶_𝑑𝜀₂−(𝜎/(𝑝+1))∶=𝐶₁>0 and use Lemma 6 to get 𝐿^′′(𝑡)+𝐶𝜀₁∇𝑢_𝑡²₂+𝐶_𝑑𝐶𝜀₂𝑢_𝑡^𝑚+1_𝑚+1>𝐶₁‖𝑢‖^𝑝+1_𝑝+1>2𝐶₁𝑝+1𝑑𝑝−1¹. (48) Integrate twice the above inequality over [0,𝑡] and use (37); we have 𝐿(𝑡)>𝐶₁𝑝+1𝑑₁𝑡𝑝−1²+𝐿^′(0)−𝐶𝜀₁+𝐶𝜀₂𝐶₀+𝑑₁𝑡+𝐿(0), (49) which implies that 𝐿(𝑡) grows more quickly than the linear growth for 𝑡≥0.
On the other hand, by (37) and Hölder's inequality, we have ‖𝑢(𝑡)‖₂≤𝑢₀₂+^𝑡₀𝑢_𝑡(𝜏)₂≤𝑢d𝜏₀₂+𝐶𝐶₀+𝑑₁^1/(𝑚+1)𝑡^{(𝑚−1)/(𝑚+1)},‖∇𝑢‖₂≤∇𝑢₀₂+^𝑡₀∇𝑢_𝑡(𝜏)₂d𝜏≤∇𝑢₀₂+𝐶₀+𝑑₁^1/2. (50) Thus, 𝐿(𝑡)=‖𝑢‖²₂+‖∇𝑢‖²₂≤𝐶₂𝑡^{2(𝑚−1)/(𝑚+1)}+𝐶₃𝑡+𝐶₄,∀𝑡≥0, (51) where 𝐶₂, 𝐶₃, and 𝐶₄ are positive constants. Obviously, (51) contradicts (49).

Acknowledgments

This work was sponsored by Qing Lan Project of Jiangsu Province and was partly supported by the Tianyuan Fund of Mathematics (Grant no. 11026211), the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant no. 09KJB110005), and the JSPS Innovation Program (Grant no. CXLX12_0490).