Semilinear Parabolic Equations on the Heisenberg Group with a Singular Potential

Abstract

We discuss the asymptotic behavior of solutions for semilinear parabolic equations on the Heisenberg group with a singular potential. The singularity is controlled by Hardy's inequality, and the nonlinearity is controlled by Sobolev's inequality. We also establish the existence of a global branch of the corresponding steady states via the classical Rabinowitz theorem.

1. Introduction

In this paper, we study a class of parabolic equations on the Heisenberg group ℍ^𝑑. Let us recall that the Heisenberg group is the space ℝ^2𝑑+1 of the (noncommutative) law of product𝑥,𝑦,𝑠⋅𝑥^′,𝑦^′,𝑠^′=𝑥+𝑥^′,𝑦+𝑦^′,𝑠+𝑠^′+2𝑦∣𝑥^′−𝑦^′∣𝑥. (1.1)

The left invariant vector fields are𝑋_𝑗=𝜕_𝑥𝑗+2𝑦_𝑗𝜕_𝑠,𝑌_𝑗=𝜕_𝑦𝑗−2𝑥_𝑗𝜕_𝑠,𝑗=1,…,𝑑𝑆=𝜕_𝑠=14𝑌_𝑗,𝑋_𝑗. (1.2)

In the sequel we will denote, we will denote 𝑍_𝑗=𝑋_𝑗 and 𝑍_𝑗+𝑑=𝑌_𝑗 for 𝑗∈{1,…,𝑑}. We fix here some notations:𝑧=𝑥,𝑦∈ℝ^2𝑑,𝑤=(𝑧,𝑠)∈ℍ^𝑑,𝜌(𝑧,𝑠)=|𝑧|⁴+|𝑠|^21/4, (1.3)

where 𝜌 is the Heisenberg distance. Moreover, the Laplacian-Kohn operator on ℍ^𝑑 and Heisenberg gradient is given byΔ_ℍ𝑑=^𝑛_𝑗=1𝑋²_𝑗+𝑌²_𝑗;∇_ℍ𝑑=𝑍₁,…,𝑍_2𝑑. (1.4)

Let Ω be an open and bounded domain of ℍ^𝑑, we define thus the associated Sobolev space as follows𝐻¹Ω,ℍ^𝑑=𝑓∈𝐿²(Ω);∇_ℍ𝑑𝑓∈𝐿²(Ω), (1.5)

and 𝐻¹₀(Ω,ℍ^𝑑) is the closure of 𝐶^∞₀(Ω) in 𝐻¹(Ω,ℍ^𝑑).

We are concerned in the following semilinear parabolic problem𝜕_𝑡𝑢−Δ_ℍ𝑑|𝑧|𝑢−𝜇²𝜌⁴𝑢=𝜆𝑢+|𝑢|^𝑝−2𝑢()𝑢,𝑤∈Ω,𝑡>0,𝑤,0=𝑢₀(𝑤)𝑢|,𝑤∈Ω,_𝜕Ω=0,𝑡>0, (1.6)

where 𝜆 is a real constant and 2<𝑝<2^∗; the index 2^∗=2+2/𝑑 is the critical index of Sobolev’s inequality on the Heisenberg group [1, 2]:‖𝑢‖_𝐿2^∗_(Ω)≤𝐶_Ω‖𝑢‖_{𝐻1(Ω,ℍ𝑑)},∀𝑢∈𝐻¹₀Ω,ℍ^𝑑. (1.7)

D’Ambrosio in [3] has proved Hardy’s inequality: let 𝑢∈𝐻¹₀(Ω,ℍ^𝑑), it holds that𝜇_Ω|𝑧|²𝜌(𝑤)⁴|𝑢(𝑤)|²𝑑𝑤≤∇_ℍ𝑑𝑢²_𝐿2(Ω). (1.8)

And by the work of Dou et al. [4], we have the following Hardy inequality with remainder terms for all u∈𝐶^∞₀(Ω⧵{0}):14_Ω𝑅ln𝜌(𝑤)⁻²|𝑧|²𝜌(𝑤)⁴|𝑢(𝑤)|²𝑑𝑤+𝜇_Ω|𝑧|²𝜌(𝑤)⁴|𝑢(𝑤)|²𝑑𝑤≤∇_ℍ𝑑𝑢²_𝐿2(Ω), (1.9)

for any 𝑅≥𝑅₀, where 𝑅₀=sup_𝑤∈Ω𝜌(𝑤), and 𝜇=𝑑². Moreover 𝜇 is optimal and it is not attained in 𝐻¹₀(Ω,ℍ^𝑑).

Stimulated by the recent paper in the Euclidean space ℝ^𝑑 of Karachalios and Zographopoulos [5] which studied the global bifurcation of nontrivial equilibrium solutions on the bounded domain case for a reaction term 𝑓(𝑠)=𝜆𝑠−|𝑠|²𝑠, where 𝜆 is a bifurcation parameter; our focus here is devoted to some results concerning the existence of a global attractor for the (1.6) and the existence of a global branch of the corresponding steady states−Δ_ℍ𝑑|𝑧|𝑢−𝜇²𝜌(𝑤)⁴𝑢=𝜆𝑢+|𝑢|^𝑝−2𝑢|𝑢inΩ,_𝜕Ω=0, (1.10)

with respect to 𝜆. Let us recall some definitions on semiflows.

Definition 1.1. Let E be a complete metric space, a semiflow is a family of continuous maps 𝒮(𝑡)∶𝐸→𝐸, 𝑡≥0, satisfying the semigroup identities 𝒮(0)=𝐼,𝒮𝑡+𝑡^′=𝒮(𝑡)𝒮𝑡^′. (1.11) For ℬ⊂𝐸 and 𝑡≥0, 𝒮(𝑡)ℬ∶=𝑢(𝑡)=𝒮(𝑡)𝑢₀;𝑢₀∈ℬ. (1.12) The positive orbit of 𝑢 through 𝑢₀ is the set 𝛾⁺𝑢₀=𝑢(𝑡)=𝒮(𝑡)𝑢₀,𝑡≥0, (1.13) then the positive orbit of ℬ is the set 𝛾⁺(ℬ)=∪_𝑡≥0𝒮(𝑡)ℬ. The 𝒲-limit set of 𝑢₀ is 𝒲𝑢₀=𝜙∈𝐸∶𝑢𝑡_𝑗=𝒮𝑡_𝑗𝑢₀⟶𝜙,𝑡_𝑗⟶+∞. (1.14) The 𝛼-limit set of 𝑢₀ is 𝛼𝑢₀=𝜙∈𝐸∶𝑢𝑡_𝑗⟶𝜙,𝑡_𝑗⟶−∞. (1.15) The subset 𝒜 attracts a set ℬ if dist(𝒮(𝑡)ℬ,𝒜)→0,𝑡→+∞. 𝒜 is invariant if 𝒮(𝑡)𝒜=𝒜 and for all 𝑡≥0.
The functional 𝒥∶𝐸→ℝ is a Lyapunov functional for the semiflow 𝒮(𝑡) if(i)𝒥 is continuous.(ii)𝒥(𝒮(𝑡)𝑢₀)≤𝒥(𝒮(𝑡′)𝑢₀) for 0≤𝑡′≤𝑡.(iii)𝒥(𝒮(𝑡)) is constant for some orbit 𝑢 and for all 𝑡∈ℝ.We have the following theorem from Ball [6, 7].

Theorem 1.2. Let 𝒮(𝑡) be an asymptotically compact semiflow, and suppose that there exists a Lyapunov functional 𝒥. Suppose further that the set ℰ is bounded, then 𝒮(𝑡) is dissipative, so there exists a global attractor 𝒜(𝑡).
For each complete orbit 𝑢 containing 𝑢₀ lying in 𝒜(𝑡), the limit sets 𝛼(𝑢₀) and 𝒲(𝑢₀) are connected subsets of ℰ on which 𝒥 is constant.
If ℰ is totally disconnected (in particular countable), the limits 𝜙₋=lim_{𝑡→−∞}𝑢(𝑡),𝜙₊=lim_𝑡→+∞𝑢(𝑡) (1.16) exist and are equilibrium points. furthermore, any solution 𝒮(𝑡)𝑢₀ tends to an equilibrium point as 𝑡→±∞.

The outline of the paper is as follows. In Section 2, we study the existence of an unbounded connected branch of positive solutions of (1.10) with respect to the parameter 𝜆 by using global bifurcation theorem introduced by López-Gómez and Molina-Meyer in [8]. In Section 3, we describe the asymptotic behavior of solutions of (1.6) when 𝑢₀ has low energy smaller than the mountain pass level.

2. Existence of a Global Branch of the Corresponding Steady States

From the study of spectral decomposition of 𝐻¹₀(Ω,ℍ^𝑑) with respect to the operator −Δ_ℍ𝑑−𝜇(|𝑧|²/𝜌(𝑤)⁴), where the singular potential 𝑉 satisfies Hardy’s inequality (1.8), we have the following.

Proposition 2.1. Let 0<𝜇≤𝜇. Then there exist 0<𝜆₁<𝜆₂≤𝜆₃≤⋯≤𝜆_𝑘≤⋯→+∞, such that for each 𝑘≥1, the following Dirichlet problem −Δ_ℍ𝑑𝜙_𝑘|𝑧|−𝜇²𝜌(𝑤)⁴𝜙_𝑘=𝜆_𝑘𝜙_𝑘𝜙,inΩ_𝑘𝜕Ω=0 (2.1) admits a nontrivial solution in 𝐻¹₀(Ω,ℍ^𝑑). Moreover, {𝜙_𝑘}_𝑘≥1 constitutes an orthonormal basis of Hilbert space 𝐻¹₀(Ω,ℍ^𝑑).

For the proof of this proposition, we refer to [9].

Remark that the first eigenvalue 𝜆_1,𝜇 characterized by𝜆_1,𝜇=inf_{𝑢∈𝐻10Ω,ℍ𝑑⧵{0}}∫_Ω∇_ℍ𝑑𝑢(𝑤)²−𝜇|𝑧|²/𝜌(𝑤)⁴|𝑢(𝑤)|²𝑑𝑤‖𝑢‖²_𝐿2(Ω), (2.2) is simple with a positive associated eigenfunction 𝜙_1,𝜇.

We discuss the behavior of 𝜆_1,𝜇 when 0<𝜇<𝜇 and 𝜇↑𝜇.

Proposition 2.2. Let 0<𝜇<𝜇 and 𝜇↑𝜇. Then, (i)(𝜆_1,𝜇)_𝜇 is a decreasing sequence, and there exist 𝜆_∗>0 such that 𝜆_1,𝜇→𝜆_∗.(ii)The corresponding normalized eigenfunction 𝜙_1,𝜇 converging weakly to 0, in 𝐻¹₀(Ω,ℍ^𝑑).

Proof. (i) Let 𝜇₁<𝜇₂. The characterization (2.2) of 𝜆_1,𝜇 implies that 𝜆_1,𝜇1>𝜆_1,𝜇2.
The improved Hardy inequality (1.9) implies that 𝜆_1,𝜇 is bounded from below by 𝐶_Ω=1/4sup_Ω(ln(𝑅/𝜌(𝑤)))⁻²(|𝑧|²)/(𝜌(𝑤)⁴). So, there exist 𝜆_∗>0 such that 𝜆_1,𝜇→𝜆_∗.
(ii) The eigenfunction 𝜙_1,𝜇 satisfies, for any 𝑣∈𝐶^∞₀(Ω): _Ω∇_ℍ𝑑𝜙_1,𝜇∇_ℍ𝑑𝑣𝑑𝑤−𝜇_Ω|𝑧|²𝜌(𝑤)⁴𝜙_1,𝜇𝑣𝑑𝑤=𝜆_1,𝜇Ω𝜙_1,𝜇𝑣𝑑𝑤. (2.3) We still denote by 𝜙_1,𝜇 the sequence of normalized eigenfunction, forming a bounded sequence in 𝐻¹₀(Ω,ℍ^𝑑). Then there exists 𝑢∈𝐻¹₀(Ω,ℍ^𝑑) such that 𝜙_1,𝜇⇀𝑢in𝐻¹₀Ω,ℍ^𝑑,𝜙_1,𝜇→𝑢in𝐿^𝑞(Ω),forany2≤𝑞<2^∗. (2.4) For some fixed small enough 𝜀>0 and any 𝑣∈𝐶^∞₀(Ω), we have _Ω|𝑧|²𝜌(𝑤)⁴𝜙_1,𝜇−𝑢𝑣𝑑𝑤≤‖𝑣‖_{𝐿∞(Ω)Ω}𝜙_1,𝜇−𝑢^{𝑄−𝜀/𝑄−2−𝜀}𝑑𝑤^{𝑄−2−𝜀/𝑄−𝜀}×_Ω|𝑧|𝜌(𝑤)^2𝑄−𝜀𝑑𝑤^{2/𝑄−𝜀}. (2.5) Thus, _Ω|𝑧|²𝜌(𝑤)⁴𝜙_1,𝜇𝑣𝑑𝑤⟶_Ω|𝑧|²𝜌(𝑤)⁴𝑢𝑣𝑑𝑤,as𝜇↑𝜇. (2.6) We assume that 𝑢≠0, so passing to the limit in (2.3), we get that 𝑢 is a nontrivial solution of the problem −Δ_ℍ𝑑𝑢−𝜇|𝑧|²𝜌(𝑤)⁴𝑢=𝜆_∗𝑢,𝑢∈𝐻¹₀Ω,ℍ^𝑑. (2.7) However, 𝜇 is not achieved in 𝐻¹₀(Ω,ℍ^𝑑), so 𝑢=0.

Thanks to Hardy’s inequality (1.8) and Poincaré’s inequality,‖𝑢‖_𝜇=_Ω∇_ℍ𝑑𝑢(𝑤)²|𝑧|−𝜇²𝜌(𝑤)⁴|𝑢(𝑤)|²𝑑𝑤^1/2, (2.8) is equivalent to the norm on 𝐻¹₀(Ω,ℍ^𝑑) for all 0≤𝜇<𝜇, so that we will use ‖⋅‖_𝜇 as the norm of 𝐻¹₀(Ω,ℍ^𝑑).

Theorem 2.3. Let Ω∈ℍ^𝑑 a bounded domain and assume that 0<𝜇<𝜇. Then, there exists an unbounded component 𝐶⁺_𝜆1,𝜇⊂ℝ×𝐻¹₀(Ω,ℍ^𝑑) of the set of positive solutions of (1.10) bifurcating from (𝜆_1,𝜇,0).

Proof. We introduce the Banach space 𝑋=𝐻¹₀(Ω,ℍ^𝑑), and the inner product in 𝑋 is given by ⟨𝑢,𝑣⟩_𝑋≡_Ω∇_ℍ𝑑𝑢∇_ℍ𝑑|𝑧|𝑣−𝜇²𝜌(𝑤)⁴𝑢𝑣𝜆𝑑𝑤−^1,𝜇2_Ω𝑢𝑣𝑑𝑤. (2.9) Let 𝑎(𝑢,𝑣)=_Ω𝑢𝑣𝑑𝑤,∀𝑢,𝑣∈𝑋. (2.10) The bilinear form 𝑎(𝑢,𝑣) is continuous in 𝑋, so the Riesz representation theorem implies that there exist a bounded linear operator 𝐿 such that 𝑎(𝑢,𝑣)=⟨𝐿𝑢,𝑣⟩,∀𝑢,𝑣∈𝑋. (2.11) The operator 𝐿 is self-adjoint and compact and its largest eigenvalue is characterized by 𝜈₁=sup_𝑢∈𝑋⟨𝐿𝑢,𝑢⟩⟨𝑢,𝑢⟩_𝑋=sup_𝑢∈𝑋‖𝑢‖_𝐿2(Ω)∫_Ω∇_ℍ𝑑𝑢²−𝜇|𝑧|²/𝜌(𝑤)⁴|𝑢|²=1𝑑𝑤𝜆_1,𝜇. (2.12)
We define the following energy functional on 𝐻¹₀(Ω,ℍ^𝑑): 𝐼_𝜇,𝜆(𝑢)=12_Ω∇_ℍ𝑑𝑢²|𝑧|−𝜇²𝜌(𝑤)⁴|𝑢|²1𝑑𝑤−𝑝_Ω|𝑢|^𝑝𝜆𝑑𝑤−2_Ω|𝑢|²𝑑𝑤. (2.13)
Similar to the classical case, 𝐼_𝜇,𝜆(⋅) is well defined on 𝐻¹₀(Ω,ℍ^𝑑) and belongs to 𝐶¹(𝐻¹₀(Ω,ℍ^𝑑);ℝ), and we have 𝐼^′_𝜇,𝜆(𝑢),𝑣=_Ω∇_ℍ𝑑𝑢∇_ℍ𝑑|𝑧|𝑣−𝜇²𝜌(𝑤)⁴𝑢𝑣−|𝑢|^𝑝−2𝑢𝑣−𝜆𝑢𝑣𝑑𝑤, (2.14) for any 𝑣∈𝐻¹₀(Ω,ℍ^𝑑). Let 𝑁(𝜆,.)∶ℝ×𝑋→𝑋^∗, 𝑋^∗ is the dual space of 𝑋, defined as by ⟨𝑁(𝜆,𝑢),𝑣⟩=_Ω∇_ℍ𝑑𝑢∇_ℍ𝑑|𝑧|𝑣−𝜇²𝜌(𝑤)⁴𝑢𝑣−|𝑢|^𝑝−2𝑢𝑣−𝜆𝑢𝑣𝑑𝑤, (2.15) for all 𝑣∈𝑋. Since 𝐼^′_𝜇,𝜆(𝑢) is a bounded linear functional, 𝑁(𝜆,⋅) is well defined, and 𝑁(𝜆,𝑢)=𝑢−𝐺(𝜆,𝑢), where 𝐺(𝜆,𝑢)=𝜆𝐿𝑢+𝐻(𝑢), ⟨𝐻(𝑢),𝑣⟩=_Ω|𝑢|^𝑝−2𝑢𝑣𝑑𝑤,∀𝑣∈𝑋. (2.16)
So, |⟨𝐻(𝑢),𝑣⟩|≤‖𝑢‖_𝐿^𝑝−1_𝑝(Ω)‖𝑣‖_𝐿𝑝(Ω). (2.17) By Sobolev embedding Sobolev theorem [10], we have 1‖𝑢‖_𝑋|⟨𝐻(𝑢),𝑣⟩|≤‖𝑢‖_𝑋^𝑝−2‖𝑣‖_𝑋. (2.18) Then, lim_{‖𝑢‖𝑋→0}‖𝐻(𝑢)‖_𝑋∗‖𝑢‖_𝑋=lim_{‖𝑢‖𝑋→0}sup_{‖𝑣‖𝑋≤1}1‖𝑢‖_𝑋|⟨𝐻(𝑢),𝑣⟩|=0. (2.19)
Consequently, hypotheses (HL) and (HR) of [8] are hold. If 𝑢∈𝐻¹₀(Ω,ℍ^𝑑)⧵{0} is a nonnegative solution of (1.10), then it follows from the strong maximum principle of J.-M. Bony [11] and the generalization of the Hopf boundary point lemma on the Heisenberg group [12], that 𝑢 lies in the interior of the cone: int(𝒫)=𝑢∈𝐻¹₀Ω,ℍ^𝑑∶𝑢>0inΩ,𝜕𝑢𝜕𝑛<0on𝜕Ω. (2.20) Hence, the assumption (HP) of [8] is fulfilled.

Remark 2.4. According to the theory of Rabinowitz [13], we can see that there is a continuum C_𝜆1,𝜇 of the set of nontrivial solutions of (1.10), and the continuum C_𝜆1,𝜇 consists of two subcontinua C⁺_𝜆1,𝜇 and C⁻_𝜆1,𝜇. However, this does not necessarily implies that the subcontinuum C⁺_𝜆1,𝜇 satisfies the global alternative of Rabinowitz [13] by the reasons already explained by Dancer [14], López-Gómez and Molina-Meyer [8, 15]. Instead, the existence of a global subcontinuum C⁺_𝜆1,𝜇 of the set of positive solutions follows by slightly adapting [8, Theorem 1.1].

3. Asymptotic Behavior of Solutions for Problem (1.6)

Similar to [16, 17], we are interested here in the description of the behavior of solutions of (1.6) when 𝑢₀ has low energy smaller than the mountain pass level𝑐_𝜇,𝜆=inf_ℎ∈Γmax_𝑡∈[0,1]𝐼_𝜇,𝜆(ℎ(𝑡[])),whereΓ=ℎ∈𝐶0,1;𝐻¹₀Ω,ℍ^𝑑;ℎ(0)=0andℎ(1)=𝑒. (3.1) In view of [9], since 2<𝑝<2^∗, the functional 𝐼_𝜇,𝜆 satisfies the Palais-Smale condition and admits at least a positive solution (called mountain pass solution).

Proposition 3.1. Let 𝑢₀∈𝐻¹₀(Ω,ℍ^𝑑), 𝜆>0, and 0<𝜇<𝜇, the problem (1.6) has a unique weak solution 𝑢 such that 𝑢∈𝐶[0,𝑇);𝐻¹₀Ω,ℍ^𝑑∩𝐶¹[0,𝑇);𝐻⁻¹Ω,ℍ^𝑑, (3.2) and we have 𝑑𝐼𝑑𝑡^𝜇,𝜆(𝑢(𝑡))=−𝜕_𝑡𝑢²_𝐿2(Ω). (3.3)

Proof. By means of the Hill-Yosida theorem, 𝒯(𝑡)={𝑒^{−𝑡𝐿𝜇}}_𝑡≥0 is the semigroup generated by the operator 𝐿_𝜇=−Δ_ℍ𝑑−𝜇|𝑧|²/𝜌(𝑧,𝑠)⁴. Let 𝑓 the function defined by 𝑓(𝑡)=𝜆𝑡+|𝑡|^𝑝−2𝑡, for 𝑡∈ℝ. Since 𝑓∶𝐻¹₀(Ω,ℍ^𝑑)→𝐻⁻¹(Ω) is locally Lipschitz, so by Pazy [18, Theorem 1.4] or Cazenave and Haraux [19, Theorem 6.2.2], there exists a unique solution of (1.6) defined on a maximal interval [0,𝑇_max), where 0<𝑇_max≤+∞ and 𝑢∈𝐶[0,𝑇);𝐻¹₀Ω,ℍ^𝑑∩𝐶¹[0,𝑇);𝐻⁻¹(Ω), (3.4) satisfying the variation of constants formula 𝑢(𝑡)=𝒯(𝑡)𝑢₀+^𝑡₀𝒯(𝑡−𝜏)𝑓(𝑢(𝜏))𝑑𝜏. (3.5) Moreover, if 𝑇_max<+∞, we say that 𝑇_max is blow-up time, whereas if 𝑇_max=+∞, we say that 𝑢 is global solution.
We will show that 𝑢 satisfies (3.3): Let 𝑢∈𝐷(𝐿_𝜇), (𝐷(𝐿_𝜇) is the domain of definition of 𝐿_𝜇), and 𝑡∈[0,𝑇),𝑇<𝑇_max. Since 𝐼_𝜇,𝜆∈𝐶¹(𝐻¹₀(Ω,ℍ^𝑑);ℝ), we have 𝐼^′_𝜇,𝜆(𝑢),Δ_ℍ𝑑|𝑧|𝑢+𝜇²𝜌(𝑤)⁴𝑢+𝑓(𝑢)=−_ΩΔ_ℍ𝑑|𝑧|𝑢+𝜇²𝜌(𝑤)⁴𝑢+𝑓(𝑢)²𝑑𝑤=−_Ω𝜕_𝑡𝑢²𝑑𝑤. (3.6)
Set 𝑔(𝑡)=𝑓(𝑢(𝑡)), and let 𝑔_𝑛∈𝐶¹([0,𝑇];𝐻¹₀(Ω,ℍ^𝑑)), 𝑢_0𝑛∈𝐷(𝐿_𝜇) such that 𝑔_𝑛⟶𝑔in𝐶¹[0,𝑇];𝐻¹₀Ω,ℍ^𝑑,𝑢_0𝑛⟶𝑢₀in𝐻¹₀Ω,ℍ^𝑑. (3.7) Define 𝑢_𝑛(𝑡)=𝒯(𝑡)𝑢_0𝑛+∫^𝑡₀𝒯(𝑡−𝜏)𝑔_𝑛(𝜏)𝑑𝜏, then, 𝑢_𝑛∈𝐶¹([0,𝑇];𝐻¹₀(Ω,ℍ^𝑑)) and satisfies 𝜕_𝑡𝑢_𝑛−Δ_ℍ𝑑𝑢_𝑛−𝜇𝑉𝑢_𝑛=𝑔_𝑛𝑢_𝑛⟶𝑢in𝐻¹₀Ω,ℍ^𝑑. (3.8) Thus, from (3.6), 𝐼_𝜇,𝜆𝑢_𝑛(𝑡)−𝐼_𝜇,𝜆𝑢_0𝑛=^𝑡₀𝐼^′_𝜇,𝜆𝑢_𝑛(𝜏),Δ_ℍ𝑑𝑢_𝑛|𝑧|+𝜇²𝜌(𝑤)⁴𝑢_𝑛+𝑔_𝑛(𝜏)𝑑𝜏=−^𝑡₀𝜕_𝜏𝑢_𝑛(𝜏)²_𝐿2(Ω)𝑑𝜏+^𝑡₀𝐼^′_𝜇,𝜆𝑢_𝑛(𝜏),𝑔_𝑛(𝜏)−𝑓𝑢_𝑛(𝜏)𝑑𝜏. (3.9) Passing to the limit, we deduce (3.3).

Next, we introduce the following sets:𝒪⁺≡𝑢∈𝐻¹₀Ω,ℍ^𝑑∶𝐼_𝜇,𝜆(𝑢)<𝑐_𝜇,𝜆;𝐼^′_𝜇,𝜆(𝑢),𝒪,𝑢>0₋≡𝑢∈𝐻¹₀Ω,ℍ^𝑑∶𝐼_𝜇,𝜆(𝑢)<𝑐_𝜇,𝜆;𝐼^′_𝜇,𝜆(𝑢),𝑢<0,𝒩≡𝑢∈𝐻¹₀Ω,ℍ^𝑑∶𝐼^′_𝜇,𝜆(𝑢),𝑢=0. (3.10)

𝒩 is named the Nehari manifold relative to 𝐼_𝜇,𝜆. The mountain-pass level 𝑐_𝜇,𝜆 defined in (3.1) may also be characterized as𝑐_𝜇,𝜆=inf_𝑢∈𝒩𝐼_𝜇,𝜆(𝑢). (3.11)

Theorem 3.2. If there exist 𝑡₀≥0 such that 𝐼_𝜇,𝜆(𝑢(𝑡₀))≤0, then 𝑢(𝑡) blows up in finite time.

Proof. Let 𝑡₀≥0 such that 𝐼_𝜇,𝜆(𝑢(𝑡₀))≤0, and we suppose that 𝑢(𝑡) is a global solution for the problem (1.6). Since 𝑢(𝑡) satisfy (3.3), we have 𝐼_𝜇,𝜆𝑢𝑡₀=𝐼_𝜇,𝜆(𝑢(𝑡))+^𝑡_𝑡0𝜕_𝜏𝑢(𝜏)²_𝐿2(Ω)𝑑𝜏. (3.12) Set 𝑔(𝑡)≡∫_Ω|𝑢(𝑡)|²𝑑𝑤, then 𝑑𝑔(𝑡)=𝑑𝑡_Ω𝑢(𝑡)𝜕_𝑡𝑢(𝑡)𝑑𝑤=−2_Ω∇_ℍ𝑑𝑢(𝑡)²|𝑧|−𝜇²𝜌(𝑤)⁴|𝑢(𝑡)|²𝑑𝑤+2𝜆_Ω|𝑢(𝑡)|²𝑑𝑤+2_Ω|𝑢(𝑡)|^𝑝𝑑𝑤=4^𝑡_𝑡0𝜕_𝜏𝑢(𝜏)²_𝐿2(Ω)𝑑𝜏−4𝐼_𝜇,𝜆𝑢𝑡₀2+21−𝑝_Ω|𝑢(𝑡)|^𝑝2𝑑𝑤≥21−𝑝_Ω|𝑢(𝑡)|^𝑝𝑑𝑤>0. (3.13) Hence, we get for 𝑡≥𝑡₀, 𝑔(𝑡)≥𝑔(𝑡₀)=∫_Ω|𝑢(𝑡₀)|²𝑑𝑤.
Let 𝜖∈(1,𝑝/2), so we deduce by (3.13), that for any 𝑡≥𝑡₀: −1𝑑𝜖−1𝑔𝑑𝑡^1−𝜖(𝑡)=𝑔^−𝜖(𝑡)𝑑𝑔(𝑡)2𝑑𝑡≥21−𝑝𝑔^−𝜖(𝑡)_Ω|𝑢(𝑡)|^𝑝𝑑𝑤≥𝐶𝑔^−𝜖(𝑡)_Ω|𝑢(𝑡)|²𝑑𝑤^𝑝/2≥𝐶_Ω𝑢𝑡₀²𝑑𝑤^{(𝑝/2)−𝜖}. (3.14)
Hence, for any 𝑡≥𝑡₀ sufficiently large, we have 0<_Ω|𝑢(𝑡)|²𝑑𝑤^1−𝜖=𝑔^1−𝜖(𝑡)≤𝑔^1−𝜖𝑡₀+𝐶(𝜖−1)𝑔^{(𝑝/2)−𝜖}𝑡₀𝑡₀−𝑡. (3.15) Then −1<𝐶(𝜖−1)𝑔^(𝑝/2)−1𝑡₀𝑡₀−𝑡, (3.16) and so 𝑡<𝑡₀+[𝐶(𝜖−1)𝑔^𝑝/2−1(𝑡₀)]⁻¹, which is a contradiction.

Theorem 3.3. Assume that 𝑢₀∈𝒪⁺ and 𝜆<𝜆_1,𝜇, then the problem (1.6) admits a global solution 𝑢(𝑡). Moreover, there exists a positive number 𝛼 such that ‖𝑢(𝑡)‖=𝑂𝑒^−𝛼𝑡,as𝑡→+∞. (3.17)

Proof. Let 𝑢₀∈𝒪⁺, and let 𝑢(𝑡)=𝑢(𝑤,𝑡,𝑢₀) be the unique solution established in Proposition 3.1. From inequality (3.3), we have that 𝑡↦𝐼_𝜇,𝜆(𝑢(𝑡)) is strictly decreasing, so 𝐼_𝜇,𝜆(𝑢(𝑡))≤𝐼_𝜇,𝜆𝑢₀≤𝑐_𝜇,𝜆. (3.18) Suppose there exists 𝑡^∗∈(0,𝑇_max) such that 𝑢(𝑡^∗)∉𝒪⁺. Then, 𝐼^′_𝜇,𝜆𝑢𝑡^∗,𝑢𝑡^∗≤0. (3.19) Moreover, since the application 𝑡↦⟨𝐼^′_𝜇,𝜆(𝑢(𝑡)),𝑢(𝑡)⟩ is continuous, there exists 𝑡₀∈(0,𝑡^∗] such that 𝐼^′_𝜇,𝜆𝑢𝑡₀,𝑢𝑡₀=0. (3.20) Hence, 𝑢(𝑡₀)=0 in Ω or 𝑢(𝑡₀)∈𝒩. If 𝑢(𝑡₀)=0 in Ω, then by the uniquess of 𝑢(𝑡), we conclude that 𝑢(𝑡)=0 for any 𝑡∈[𝑡₀,𝑇_max). Thus, 𝑢(𝑡) is global by extending to 0 for all 𝑡≥𝑇_max, and so 𝐼_𝜇,𝜆(𝑢(𝑡))>0 for any 𝑡≥0 by Theorem 3.2. But 𝐼_𝜇,𝜆(𝑢(𝑡₀))=0, which is a contradiction. So, we conclude that 𝑢(𝑡)∈𝒪⁺ for all 𝑡∈[𝑡₀,𝑇_max).
On other hand, we can write 𝐼_𝜇,𝜆(𝑢(𝑡=1))𝑝𝐼^′_𝜇,𝜆(𝑢(𝑡(𝑡+1)),𝑢)2−1𝑝_Ω∇_ℍ𝑑𝑢(𝑡)²|𝑧|−𝜇²𝜌(𝑤)⁴|𝑢(𝑡)|²1𝑑𝑤−2−1𝑝𝜆_Ω|𝑢(𝑡)|²>1𝑑𝑤2−1𝑝_Ω∇_ℍ𝑑𝑢(𝑡)²|𝑧|−𝜇²𝜌(𝑤)⁴|𝑢(𝑡)|²1𝑑𝑤−2−1𝑝𝜆_Ω|𝑢(𝑡)|²≥1𝑑𝑤2−1𝑝𝜆1−𝜆_1,𝜇‖𝑢(𝑡)‖²_𝜇>0. (3.21) Since 𝑢(𝑡) satisfy (3.3), we have ^𝑡_𝑡0𝜕_𝜏𝑢(𝜏)²_𝐿2(Ω)1𝑑𝜏+2−1𝑝𝜆1−𝜆_1,𝜇‖𝑢(𝑡)‖²_𝜇≤𝐼_𝜇,𝜆𝑢𝑡₀<𝑐_𝜇,𝜆. (3.22) Then we have ^𝑡_𝑡0𝜕_𝜏𝑢(𝜏)²_𝐿2(Ω)𝑑𝜏<𝑐_𝜇,𝜆,‖𝑢(𝑡)‖²_𝜇<12−1𝑝𝜆1−𝜆_1,𝜇⁻¹𝑐_𝜇,𝜆, (3.23) which implies that 𝑢(𝑡) is a global solution of the problem (1.6), and 𝒪⁺ is invariant set. Letting 𝑡→+∞ in (3.23), the integral ∫^𝑡_𝑡0‖𝜕_𝜏𝑢(𝜏)‖²_𝐿2(Ω)𝑑𝜏 is finitely determined. Therefore, there exists a sequence (𝑡_𝑛)𝑛⩾0 with 𝑡_𝑛→+∞ as 𝑛→+∞ such that _Ω𝜕_𝜏𝑢𝑡_𝑛²𝑑𝑤⟶0,𝑢𝑡_𝑛⇀𝑣in𝐻¹₀Ω,ℍ^𝑑. (3.24) Letting 𝑡_𝑛→+∞, we obtain that 𝑣∈𝐻¹₀(Ω,ℍ^𝑑) is a solution of problem (1.10). So 𝐼^′_𝜇,𝜆(𝑣),𝑣=0. (3.25) If 𝑣≠0, then 𝑣∈𝒩, and so 𝐼_𝜇,𝜆(𝑣)≥𝑐_𝜇,𝜆. (3.26) Since 𝑢(𝑡_𝑛) satisfies (3.3), it follows by Hölder inequality and from (3.24), that 𝐼^′_𝜇,𝜆𝑢𝑡_𝑛,.,𝑢𝑡_𝑛,.≤_Ω𝑢𝑡_𝑛,𝑤𝜕_𝑡𝑢()≤𝑢𝑡𝑡𝑛,𝑤𝑑𝑤_𝑛,.‖_𝐿2(Ω)‖𝜕_𝑡𝑢𝑡_𝑛,._𝐿2(Ω)≤𝜆_1,𝜇𝑢𝑡_𝑛,._𝜇𝜕_𝑡𝑢𝑡_𝑛,._L2(Ω)≤𝐶𝜕_𝑡𝑢𝑡_𝑛,._𝐿2(Ω). (3.27) Therefore, lim_𝑛→+∞𝐼^′_𝜇,𝜆𝑢𝑡_𝑛,𝑢𝑡_𝑛=0. (3.28) We deduce by (3.22), (3.25), and (3.28) that 𝐼_𝜇,𝜆(𝑣)=12−1𝑝_Ω∇_ℍ𝑑𝑣(𝑤)²|𝑧|−𝜇²𝜌⁴(𝑤)|𝑣(𝑤)|²1𝑑𝑤−2−1𝑝𝜆_Ω|𝑣(𝑤)|²𝑑𝑤≤lim_𝑛→+∞12−1𝑝_Ω∇_ℍ𝑑𝑢𝑡_𝑛,𝑤²|𝑧|−𝜇²𝜌⁴(𝑤)𝑣𝑡_𝑛,𝑤²𝑑𝑤−lim_𝑛→+∞12−1𝑝𝜆_Ω𝑣𝑡_𝑛,𝑤²𝑑𝑤+lim_𝑛→+∞𝐼^′_𝜇,𝜆𝑢𝑡_𝑛,𝑢𝑡_𝑛≤lim_𝑛→+∞𝐼_𝜇,𝜆𝑢𝑡_𝑛≤𝐼_𝜇,𝜆𝑢₀<𝑐_𝜇,𝜆, (3.29) which contradicts (3.26), and so 𝑣=0 in Ω. Hence, by (3.24), we have 𝑢𝑡_𝑛,.⟶0in𝐿^𝑞(Ω),2≤𝑞<2^∗. (3.30) Since 𝑢𝑡_𝑛,.²_𝜇=𝐼^′_𝜇,𝜆𝑢𝑡_𝑛,𝑢𝑡_𝑛+𝜆_Ω𝑢𝑡_𝑛,𝑤²𝑑𝑤+_Ω𝑢𝑡_𝑛,𝑤^𝑝𝑑𝑤→0,as𝑛⟶+∞, (3.31) we have 𝑢𝑡_𝑛,.⟶0in𝐻¹₀Ω,ℍ^𝑑,as𝑛→+∞. (3.32) For simplicity, let us denote by 𝑡 the divergent sequence and by 𝑢(𝑡)=𝑢(𝑡_𝑛,𝑤). We have from (3.29) that 𝐼_𝜇,𝜆(𝑢(𝑡=1))2−1𝑝_Ω∇_ℍ𝑑𝑢(𝑡,𝑤)²|𝑧|−𝜇²𝜌⁴(𝑤)|𝑢(𝑡,𝑤)|²−1𝑑𝑤2−1𝑝𝜆_Ω|𝑢(𝑡,𝑤)|²=1𝑑𝑤2−1𝑝‖𝑢(𝑡)‖²_𝜇1−𝜆2−1𝑝‖𝑢(𝑡)‖²_𝐿2(Ω). (3.33) So, due to (3.3) we have ‖𝑢(𝑡)‖²_𝜇=2𝑝𝐼𝑝−2^𝜇,𝜆(𝑢(𝑡))+𝜆‖𝑢(𝑡)‖²_𝐿2(Ω)≤2𝑝𝐼𝑝−2^𝜇,𝜆𝑢₀+𝜆‖𝑢(𝑡)‖²_𝐿2(Ω)<2𝑝𝑐𝑝−2^𝜇,𝜆+𝑜(1). (3.34) Therefore, there exists 𝑡₀ such that for all 𝑡≥𝑡₀, ‖𝑢(𝑡)‖²_𝜇≤2𝑝𝑐𝑝−2^𝜇,𝜆. (3.35) On the other hand, _Ω|𝑢(𝑡,𝑤)|^𝑝𝑑𝑤≤𝐶^𝑝_Ω𝜇𝜇−𝜇^𝑝/2‖𝑢(𝑡)‖^𝑝_𝜇≤𝐶^𝑝_Ω𝜇𝜇−𝜇^𝑝/22𝑝𝑐𝑝−2^{𝜇,𝜆𝑝−2/2}‖𝑢(𝑡)‖²_𝜇. (3.36) Let 𝐶₁=𝐶^𝑝_Ω(𝜇/𝜇−𝜇)^𝑝/2[(2𝑝/𝑝−2)𝑐_𝜇,𝜆]^(𝑝−2)/2, we have 1−𝐶₁‖𝑢(𝑡)‖²_𝜇≤‖𝑢(𝑡)‖²_𝜇−‖𝑢(𝑡)‖^𝑝_𝐿𝑝(Ω)≤𝐼^′_𝜇,𝜆(𝑢(𝑡)),𝑢(𝑡)+𝜆‖𝑢(𝑡)‖²_𝐿2(Ω). (3.37) Let us recall that if we set 𝑔(𝑡)≡∫_Ω|𝑢(𝑡,𝑤)|²𝑑𝑤, then 𝑑𝑔(𝑡)𝑑𝑡=2_Ω𝑢(𝑡,𝑤)𝜕_𝑡𝑢(𝑡,𝑤)𝑑𝑤=−2_Ω∇_ℍ𝑑𝑢(𝑡,𝑤)²|𝑧|−𝜇²𝜌⁴(𝑤)|𝑢(𝑡,𝑤)|²𝑑𝑤+2𝜆_Ω|𝑢(𝑡,𝑤)|²𝑑𝑤+𝜆_Ω𝑣𝑡_𝑛,𝑤^𝑝𝐼𝑑𝑤=−2^′_𝜇,𝜆(𝑢(𝑡)),𝑢(𝑡). (3.38) So we get from (3.22) that for any 𝑡≥𝑡₀, we have _𝑡^+∞𝐼^′_𝜇,𝜆(𝑢(𝜏(𝜏1)),𝑢)d𝜏=2‖𝑢(𝑡)‖²_𝐿2(Ω)≤12𝜆_𝜇,𝜆‖𝑢(𝑡)‖²_𝜇≤𝑝𝑝−2𝜆_𝜇,𝜆𝐼_𝜇,𝜆(𝑢(𝑡)). (3.39) So from (3.37) and (3.39), we have for any 𝑡≥𝑡₀ that _𝑡^+∞𝐼_𝜇,𝜆(𝑢(𝜏1))𝑑𝜏≤2𝜆_{𝜇,𝜆𝑡}^+∞‖𝑢(𝜏)‖²_𝜇≤𝑑𝜏1−𝐶₁⁻¹12𝜆_{𝜇,𝜆𝑡}^+∞𝐼^′_𝜇,𝜆(𝑢(𝜏)),𝑢(𝜏)𝑑𝜏+𝜆_𝑡^+∞‖𝑢(𝜏)‖²_𝐿2(Ω)≤𝑑𝜏1−𝐶₁⁻¹2𝜆²_𝜇,𝜆𝑝𝐼𝑝−2^𝜇,𝜆(𝑢(𝑡))+1−𝐶₁⁻¹𝜆2𝜆_{𝜇,𝜆𝑡}^+∞‖𝑢(𝜏)‖²_𝐿2(Ω)𝑑𝜏. (3.40) Since lim_𝑡→+∞‖𝑢(𝑡)‖²_𝐿2(Ω)=0, there exists 𝑡₁>𝑡₀ such that for any 𝑡≥𝑡₁, we have _𝑡^+∞𝐼_𝜇,𝜆(𝑢(𝜏))𝑑𝜏≤1−𝐶₁⁻¹2𝜆²_𝜇,𝜆𝑝𝐼𝑝−2^𝜇,𝜆(𝑢(𝑡)). (3.41) Thus, _𝑡^+∞𝐼_𝜇,𝜆(𝑢(𝜏))𝑑𝜏≤𝐶𝑡₁𝑒^−𝛼𝑡, (3.42) with 𝛼=((1−𝐶₁)⁻¹/2𝜆²_𝜇,𝜆)(𝑝/𝑝−2). But we remark that 𝐼_𝜇,𝜆(𝑢(𝑡+1))≤_𝑡^𝑡+1𝐼_𝜇,𝜆(𝑢(𝜏))𝑑𝜏<_𝑡^+∞𝐼_𝜇,𝜆(𝑢(𝜏))𝑑𝜏, (3.43) hence, we deduce that for any 𝑡≥𝑡₁, we have 𝐼_𝜇,𝜆(𝑢(𝑡+1))<𝐶𝑡₁𝑒^−𝛼𝑡, (3.44) and we can conclude that for any 𝑡≥𝑡₁, we have ‖𝑢(𝑡)‖_𝜇=𝑂𝑒^−𝛼𝑡. (3.45)

Remark 3.4. for small 𝑢₀, Theorem 3.3 is an immediate consequence from the fact that, according to the linearized stability principle, the trivial solution is linearly asymptotically stable. In other words, from the fact that the principle eigenvalue of the linearization at 𝑢=0 is positive.
Questions of stability for nonlinear systems are frequently resolved via linearized stability or Lyapunov-type methods. Here, we proved the asymptotic stability under Lyapunov function to obtain estimates in 𝐿^𝑝(Ω).

Corollary 3.5. Assume that 𝑢₀∈𝒪⁺ and 𝜆<𝜆_1,𝜇. Then any solution 𝑢(𝑡) of (1.6) tends to the trivial equilibrium point, as 𝑡→+∞.

Proof. It follows from (3.45) that the semiflow 𝒯(𝑡) is eventually bounded, see [7]. Since the resolvent of the operator 𝐿_𝜇 is compact, 𝒯(𝑡) is compact for 𝑡>0 (see [20, Theorem 3.3], thus by [18, Corollary 3.2.2], 𝒯(𝑡) is asymptotically smooth and so by [7, Proposition 2.3] is asymptotically compact. It remains to shows that ℰ, the set of equilibrium points of 𝒯(𝑡), is bounded: 𝑢(𝑡)∈ℰ, so 𝑢(𝑡)∈𝒩. Then from (3.3) and Poincaré’s inequality, we have 12−1𝑝‖𝑢(𝑡)‖²_𝜇=𝐼_𝜇,𝜆(𝑢(𝑡1))+𝜆2−1𝑝‖𝑢(𝑡)‖²_𝐿2(Ω)≤𝐼_𝜇,𝜆𝑢₀1+𝜆2−1𝑝‖𝑢(𝑡)‖²_𝐿2(Ω)≤𝑐_𝜇,𝜆+𝜆𝜆_1,𝜇12−1𝑝‖𝑢(𝑡)‖²_𝜇, (3.46) which implies that the set ℰ is bounded. Then, by Theorem 1.2, 𝒯(𝑡) is dissipative and by (3.45), we have dist(𝒯(𝑡)ℬ,0)→0 as 𝑡→+∞, for every bounded set ℬ⊂𝐻¹₀(Ω,ℍ^𝑑). So, we conclude that the global attractor 𝒜=0, and that any solution 𝑢(𝑡)=𝒮(𝑡)𝑢₀ tends to the trivial equilibrium point as 𝑡→+∞, when 𝑢₀∈𝒪⁺.

Theorem 3.6. Assume that 𝑢₀∈𝒪⁻. Then the solution 𝑢(𝑡) of the problem (1.6) blows up in finite time.

Proof. Let 𝑢₀∈𝒪⁻, and let 𝑢(𝑡)=𝑢(𝑤,𝑡,𝑢₀) be the unique solution, the existence of which has been proved in Proposition 3.1. From the inequality (3.3), we have that 𝑡↦𝐼_𝜇,𝜆(𝑢(𝑡)) is strictly decreasing, so 𝐼_𝜇,𝜆(𝑢(𝑡))=𝐼_𝜇,𝜆𝑢₀=𝑐_𝜇,𝜆. (3.47) Suppose there exists 𝑡̃∈(0,𝑇_max) such that 𝑢(𝑡)̃∉𝒪⁻. Then 𝐼^′_𝜇,𝜆(𝑢(𝑡)),𝑢(𝑡)≥0. (3.48) And since the application 𝑡↦⟨𝐼^′_𝜇,𝜆(𝑢(𝑡)),𝑢(𝑡)⟩ is continuous, there exists 𝑡₀̃∈(0,𝑡]̃ such that 𝐼^′_𝜇,𝜆𝑢𝑡₀,𝑢𝑡₀=0. (3.49) Hence, 𝑢(𝑡₀̃)=0 in Ω or 𝑢(𝑡₀̃)∈𝒩. If 𝑢(𝑡₀̃)=0 in Ω, then by the uniquess of 𝑢(𝑡), we conclude that 𝑢(𝑡)=0 for any 𝑡∈[𝑡₀̃,𝑇_max). Thus, 𝑢(𝑡) is global by extending to 0 for all 𝑡≥𝑇_max, and thanks to Theorem 3.2, 𝐼_𝜇,𝜆(𝑢(𝑡))>0 for any 𝑡≥0. But 𝐼_𝜇,𝜆(𝑢(𝑡₀̃))=0, which is a contradiction, and so 𝑢(𝑡₀̃)∈𝒩. But by [21], 𝑐_𝜇,𝜆=inf_𝑢∈𝒩𝐼_𝜇,𝜆(𝑢), (3.50) then 𝑐_𝜇,𝜆≤𝐼_𝜇,𝜆(𝑢(𝑡₀̃)), which contradicts (3.47). So, we conclude that 𝑢(𝑡)∈𝒪⁻ for all 𝑡∈[𝑡₀̃,𝑇_max). We suppose by contradiction that 𝑇_max=+∞, that is, 𝑢(𝑡)=𝑢(𝑡,⋅) exists for all 𝑡≥0. For 𝑢∈𝒪⁻, we have 𝑑‖𝑢(𝑑𝑡𝑡,⋅)‖²_𝐿2(Ω)=−2𝐼^′_𝜇,𝜆𝑢𝑡₀,𝑢𝑡₀>0. (3.51) Then 𝑡↦‖𝑢(𝑡,⋅)‖_𝐿2(Ω) is strictly increasing and so lim_t→+∞‖𝑢(𝑡,⋅)‖_𝐿2(Ω)=𝑐∈(0,+∞]. (3.52) We suppose that 𝑐<+∞. Following the same reasoning as in the proof of Theorem 3.3, we deduce that we can select a divergent subsequence, still denoted by 𝑡, such that when 𝑡→+∞, 𝑢(𝑡,⋅)⟶0in𝐻¹₀Ω,ℍ^𝑑. (3.53) Letting 𝑡→+∞ in the inequality 𝜆_1,𝜇‖𝑢(𝑡,⋅)‖_𝐿2(Ω)≤‖𝑢(𝑡,⋅)‖_𝜇, (3.54) we get that 0<𝑐≤0, which is a contradiction. So we conclude that lim_𝑡→+∞‖𝑢(𝑡,⋅)‖_𝐿2(Ω)=+∞. (3.55) Set 𝑔(𝑡)=‖𝑢(𝑡,⋅)‖²_𝐿2(Ω), so −2𝑑𝑝−2𝑔𝑑𝑡^1−𝑝/2(𝑡)=𝑔^′(𝑡)𝑔^−𝑝/2(𝑡)=−2‖𝑢(𝑡,⋅)‖_𝐿^−𝑝_2(Ω)‖𝑢(𝑡,⋅)‖²_𝜇−𝜆‖𝑢(𝑡,⋅)‖²_𝐿2(Ω)‖𝑢(𝑡,⋅)‖²_𝐿𝑝(Ω)≥−2‖𝑢(𝑡,⋅)‖_𝐿^−𝑝_2(Ω)‖𝑢(𝑡,⋅)‖²_𝜇+2‖𝑢(𝑡,⋅)‖_𝐿^−𝑝_2(Ω)‖𝑢(𝑡,⋅)‖²_𝐿𝑝(Ω). (3.56) By Hölder inequality, we have ‖𝑢(𝑡,⋅)‖^𝑝_𝐿𝑝(Ω)≥|Ω|^1−𝑝/2‖𝑢(𝑡,⋅)‖^𝑝_𝐿2(Ω), (3.57) and by (3.55), there exist 𝑡₁>0 and a constant 𝐶₁>0 such that for 𝑡≥𝑡₁, we have ‖𝑢(𝑡,⋅)‖_𝐿2(Ω)≥𝐶₁. (3.58) Then, there exist 𝑡₁>0 and a constant 𝐶₂>0 such that for 𝑡≥𝑡₁, we have −2𝑑𝑝−2𝑔𝑑𝑡^1−𝑝/2(𝑡)≥−2𝜆_1,𝜇𝐶₁^2−𝑝+2|Ω|^1−𝑝/2≥𝐶₂. (3.59) Hence, we have from (3.59), that for any 𝑡≥𝑡₁, 0<𝑔(𝑡)≤𝑔𝑡₁+𝑝−22𝐶₂𝑡−𝑡₁, (3.60) which is a contradiction if 𝑡 is sufficiently large. So we conclude that 𝑇_max<+∞.

Acknowledgment

The author is glad to thank the referee for a careful and very constructive reading of the paper and making many good suggestions.