# Semilinear Parabolic Equations on the Heisenberg Group with a Singular Potential

Authors: Houda Mokrani; Fatimetou Mint Aghrabatt
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 1. Introduction

In this paper, we study a class of parabolic equations on the Heisenberg groupℍd. Let us recall that the Heisenberg group is the space ℝ2d+1 of the (noncommutative) law of product(1.1)(x,y,s)⋅(x′,y′,s′)=(x+x′,y+y′,s+s′+2((y∣x′)-(y′∣x))).The left invariant vector fields are(1.2)Xj=∂xj+2yj∂s,Yj=∂yj-2xj∂s,j=1,…,dS=∂s=14[Yj,Xj].In the sequel we will denote, we will denoteZj=Xj and Zj+d=Yj for j∈{1,…,d}. We fix here some notations:(1.3)z=(x,y)∈R2d,w=(z,s)∈Hd,ρ(z,s)=(|z|4+|s|2)1/4,whereρ is the Heisenberg distance. Moreover, the Laplacian-Kohn operator on ℍd and Heisenberg gradient is given by(1.4)ΔHd=∑j=1nXj2+Yj2;∇Hd=(Z1,…,Z2d).LetΩ be an open and bounded domain of ℍd, we define thus the associated Sobolev space as follows(1.5)H1(Ω,Hd)={f∈L2(Ω);∇Hdf∈L2(Ω)},andH01(Ω,ℍd) is the closure of C0∞(Ω) in H1(Ω,ℍd).We are concerned in the following semilinear parabolic problem(1.6)∂tu-ΔHdu-μ|z|2ρ4u=λu+|u|p-2u,w∈Ω,t>0,u(w,0)=u0(w),w∈Ω,u|∂Ω=0,t>0,whereλ 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]:(1.7)‖u‖L2*(Ω)≤CΩ‖u‖H1(Ω,Hd),∀u∈H01(Ω,Hd).D’Ambrosio in [3] has proved Hardy’s inequality: let u∈H01(Ω,ℍd), it holds that(1.8)μ¯∫Ω|z|2ρ(w)4|u(w)|2dw≤‖∇Hdu‖L2(Ω)2.And by the work of Dou et al. [4], we have the following Hardy inequality with remainder terms for all u∈C0∞(Ω∖{0}):(1.9)14∫Ω(ln⁡(Rρ(w)))-2|z|2ρ(w)4|u(w)|2dw+μ¯∫Ω|z|2ρ(w)4|u(w)|2dw≤‖∇Hdu‖L2(Ω)2,for anyR≥R0, where R0=supw∈Ωρ(w), and μ¯=d2. Moreover μ¯ is optimal and it is not attained in H01(Ω,ℍd).Stimulated by the recent paper in the Euclidean spaceℝd 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(1.10)-ΔHdu-μ|z|2ρ(w)4u=λu+|u|p-2uinΩ,u|∂Ω=0,with respect toλ. Let us recall some definitions on semiflows.Definition 1.1.
Let E be a complete metric space, a semiflow is a family of continuous maps𝒮(t):E→E, t≥0, satisfying the semigroup identities
(1.11)S(0)=I,S(t+t′)=S(t)S(t′).
For ℬ⊂E and t≥0,
(1.12)S(t)B:={u(t)=S(t)u0;u0∈B}.
The positive orbit of u through u0 is the set
(1.13)γ+(u0)={u(t)=S(t)u0,t≥0},
then the positive orbit of ℬ is the set γ+(ℬ)=∪t≥0𝒮(t)ℬ. The 𝒲-limit set of u0 is
(1.14)W(u0)={ϕ∈E:u(tj)=S(tj)u0⟶ϕ,tj⟶+∞}.
The α-limit set of u0 is
(1.15)α(u0)={ϕ∈E:u(tj)⟶ϕ,tj⟶-∞}.
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 functional𝒥: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.
Let𝒮(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 orbitu containing u0 lying in 𝒜(t), the limit sets α(u0) and 𝒲(u0) are connected subsets of ℰ on which 𝒥 is constant.
Ifℰ is totally disconnected (in particular countable), the limits
(1.16)ϕ-=lim⁡t→-∞u(t),ϕ+=lim⁡t→+∞u(t)
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 Section2, 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 ofH01(Ω,ℍd) with respect to the operator -Δℍd-μ(|z|2/ρ(w)4), where the singular potential V satisfies Hardy’s inequality (1.8), we have the following.Proposition 2.1.
Let0<μ≤μ¯. Then there exist 0<λ1<λ2≤λ3≤⋯≤λk≤⋯→+∞, such that for each k≥1, the following Dirichlet problem
(2.1)-ΔHdϕk-μ|z|2ρ(w)4ϕk=λkϕk,inΩϕk|∂Ω=0
admits a nontrivial solution in H01(Ω,ℍd). Moreover, {ϕk}k≥1 constitutes an orthonormal basis of Hilbert space H01(Ω,ℍd).For the proof of this proposition, we refer to [9].Remark that the first eigenvalueλ1,μ characterized by(2.2)λ1,μ=inf⁡u∈H01(Ω,Hd)∖{0}∫Ω(|∇Hdu(w)|2-μ(|z|2/ρ(w)4)|u(w)|2)dw‖u‖L2(Ω)2,
is simple with a positive associated eigenfunction ϕ1,μ.We discuss the behavior ofλ1,μ when 0<μ<μ¯ and μ↑μ¯.Proposition 2.2.
Let0<μ<μ¯ and μ↑μ¯. Then, (i)
(λ1,μ)μ is a decreasing sequence, and there exist λ*>0 such that λ1,μ→λ*.(ii)
The corresponding normalized eigenfunctionϕ1,μ converging weakly to 0, in H01(Ω,ℍd).Proof.
(i) Letμ1<μ2. The characterization (2.2) of λ1,μ implies that λ1,μ1>λ1,μ2.
The improved Hardy inequality (1.9) implies that λ1,μ is bounded from below by CΩ=1/4supΩ(ln(R/ρ(w)))-2(|z|2)/(ρ(w)4). So, there exist λ*>0 such that λ1,μ→λ*.
(ii) The eigenfunctionϕ1,μ satisfies, for any v∈C0∞(Ω):
(2.3)∫Ω∇Hdϕ1,μ∇Hdv¯dw-μ∫Ω|z|2ρ(w)4ϕ1,μv¯dw=λ1,μ∫Ωϕ1,μv¯dw.
We still denote by ϕ1,μ the sequence of normalized eigenfunction, forming a bounded sequence in H01(Ω,ℍd). Then there exists u∈H01(Ω,ℍd) such that
(2.4)ϕ1,μ⇀uinH01(Ω,Hd),ϕ1,μ→uinLq(Ω),forany2≤q<2*.
For some fixed small enough ɛ>0 and any v∈C0∞(Ω), we have
(2.5)∫Ω|z|2ρ(w)4(ϕ1,μ-u)v¯dw≤‖v‖L∞(Ω)(∫Ω|ϕ1,μ-u|Q-ɛ/Q-2-ɛdw)Q-2-ɛ/Q-ɛ×(∫Ω(|z|ρ(w)2)Q-ɛdw)2/Q-ɛ.
Thus,
(2.6)∫Ω|z|2ρ(w)4ϕ1,μv¯dw⟶∫Ω|z|2ρ(w)4uv¯dw,asμ↑μ¯.
We assume that u≠0, so passing to the limit in (2.3), we get that u is a nontrivial solution of the problem
(2.7)-ΔHdu-μ¯|z|2ρ(w)4u=λ*u,u∈H01(Ω,Hd).
However, μ¯ is not achieved in H01(Ω,ℍd), so u=0.Thanks to Hardy’s inequality (1.8) and Poincaré’s inequality,(2.8)‖u‖μ=(∫Ω[|∇Hdu(w)|2-μ|z|2ρ(w)4|u(w)|2]dw)1/2,
is equivalent to the norm on H01(Ω,ℍd) for all 0≤μ<μ¯, so that we will use ∥·∥μ as the norm of H01(Ω,ℍd).Theorem 2.3.
LetΩ∈ℍd a bounded domain and assume that 0<μ<μ¯. Then, there exists an unbounded component Cλ1,μ+⊂ℝ×H01(Ω,ℍd) of the set of positive solutions of (1.10) bifurcating from (λ1,μ,0).Proof.
We introduce the Banach spaceX=H01(Ω,ℍd), and the inner product in X is given by
(2.9)〈u,v〉X≡∫Ω[∇Hdu∇Hdv¯-μ|z|2ρ(w)4uv¯]dw-λ1,μ2∫Ωuv¯dw.
Let
(2.10)a(u,v)=∫Ωuvdw,∀u,v∈X.
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
(2.11)a(u,v)=〈Lu,v〉,∀u,v∈X.
The operator L is self-adjoint and compact and its largest eigenvalue is characterized by
(2.12)ν1=sup⁡u∈X〈Lu,u〉〈u,u〉X=sup⁡u∈X‖u‖L2(Ω)∫Ω[|∇Hdu|2-μ(|z|2/ρ(w)4)|u|2]dw=1λ1,μ.
We define the following energy functional onH01(Ω,ℍd):
(2.13)Iμ,λ(u)=12∫Ω[|∇Hdu|2-μ|z|2ρ(w)4|u|2]dw-1p∫Ω|u|pdw-λ2∫Ω|u|2dw.
Similar to the classical case,Iμ,λ(·) is well defined on H01(Ω,ℍd) and belongs to C1(H01(Ω,ℍd);ℝ), and we have
(2.14)〈Iμ,λ′(u),v〉=∫Ω[∇Hdu∇Hdv¯-μ|z|2ρ(w)4uv¯-|u|p-2uv¯-λuv¯]dw,
for any v∈H01(Ω,ℍd). Let N(λ,.):ℝ×X→X*, X* is the dual space of X, defined as by
(2.15)〈N(λ,u),v〉=∫Ω[∇Hdu∇Hdv¯-μ|z|2ρ(w)4uv¯-|u|p-2uv¯-λuv¯]dw,
for all v∈X. Since Iμ,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),
(2.16)〈H(u),v〉=∫Ω|u|p-2uv¯dw,∀v∈X.
So,(2.17)|〈H(u),v〉|≤‖u‖Lp(Ω)p-1‖v‖Lp(Ω).
By Sobolev embedding Sobolev theorem [10], we have
(2.18)1‖u‖X|〈H(u),v〉|≤‖u‖Xp-2‖v‖X.
Then,
(2.19)lim⁡∥u∥X→0‖H(u)‖X*‖u‖X=lim⁡∥u∥X→0sup⁡∥v∥X≤11‖u‖X|〈H(u),v〉|=0.
Consequently, hypotheses (HL) and (HR) of [8] are hold. If u∈H01(Ω,ℍd)∖{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:
(2.20)int⁡(P)={u∈H01(Ω,Hd):u>0inΩ,∂u∂n<0on⁡∂Ω}.
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 Cλ1,μ of the set of nontrivial solutions of (1.10), and the continuum Cλ1,μ consists of two subcontinua Cλ1,μ+ and Cλ1,μ-. However, this does not necessarily implies that the subcontinuum Cλ1,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 Cλ1,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 level(3.1)cμ,λ=inf⁡h∈Γmax⁡t∈[0,1]Iμ,λ(h(t)),whereΓ={h∈C([0,1];H01(Ω,Hd));h(0)=0andh(1)=e}.
In view of [9], since 2<p<2*, the functional Iμ,λ satisfies the Palais-Smale condition and admits at least a positive solution (called mountain pass solution).Proposition 3.1.
Letu0∈H01(Ω,ℍd), λ>0, and 0<μ<μ¯, the problem (1.6) has a unique weak solution u such that
(3.2)u∈C([0,T);H01(Ω,Hd))∩C1([0,T);H-1(Ω,Hd)),
and we have
(3.3)ddtIμ,λ(u(t))=-‖∂tu‖L2(Ω)2.Proof.
By means of the Hill-Yosida theorem,𝒯(t)={e-tLμ}t≥0 is the semigroup generated by the operator Lμ=-Δℍd-μ|z|2/ρ(z,s)4. Let f the function defined by f(t)=λt+|t|p-2t, for t∈ℝ. Since f:H01(Ω,ℍd)→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,Tmax), where 0<Tmax≤+∞ and
(3.4)u∈C([0,T);H01(Ω,Hd))∩C1([0,T);H-1(Ω)),
satisfying the variation of constants formula
(3.5)u(t)=T(t)u0+∫0tT(t-τ)f(u(τ))dτ.
Moreover, if Tmax<+∞, we say that Tmax is blow-up time, whereas if Tmax=+∞, we say that u is global solution.
We will show thatu satisfies (3.3): Let u∈D(Lμ), (D(Lμ) is the domain of definition of Lμ), and t∈[0,T),T<Tmax. Since Iμ,λ∈C1(H01(Ω,ℍd);ℝ), we have
(3.6)〈Iμ,λ′(u),ΔHdu+μ|z|2ρ(w)4u+f(u)〉=-∫Ω|ΔHdu+μ|z|2ρ(w)4u+f(u)|2dw=-∫Ω|∂tu|2dw.
Setg(t)=f(u(t)), and let gn∈C1([0,T];H01(Ω,ℍd)), u0n∈D(Lμ) such that
(3.7)gn⟶ginC1([0,T];H01(Ω,Hd)),u0n⟶u0inH01(Ω,Hd).
Define un(t)=𝒯(t)u0n+∫0t𝒯(t-τ)gn(τ)dτ, then, un∈C1([0,T];H01(Ω,ℍd)) and satisfies
(3.8)∂tun-ΔHdun-μVun=gnun⟶uinH01(Ω,Hd).
Thus, from (3.6),
(3.9)Iμ,λ(un(t))-Iμ,λ(u0n)=∫0t〈Iμ,λ′(un(τ)),ΔHdun+μ|z|2ρ(w)4un+gn(τ)〉dτ=-∫0t‖∂τun(τ)‖L2(Ω)2dτ+∫0t〈Iμ,λ′(un(τ)),gn(τ)-f(un(τ))〉dτ.
Passing to the limit, we deduce (3.3).Next, we introduce the following sets:(3.10)O+≡{u∈H01(Ω,Hd):Iμ,λ(u)<cμ,λ;〈Iμ,λ′(u),u〉>0},O-≡{u∈H01(Ω,Hd):Iμ,λ(u)<cμ,λ;〈Iμ,λ′(u),u〉<0},N≡{u∈H01(Ω,Hd):〈Iμ,λ′(u),u〉=0}.𝒩 is named the Nehari manifold relative to Iμ,λ. The mountain-pass level cμ,λ defined in (3.1) may also be characterized as(3.11)cμ,λ=inf⁡u∈NIμ,λ(u).Theorem 3.2.
If there existt0≥0 such that Iμ,λ(u(t0))≤0, then u(t) blows up in finite time.Proof.
Lett0≥0 such that Iμ,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
(3.12)Iμ,λ(u(t0))=Iμ,λ(u(t))+∫t0t‖∂τu(τ)‖L2(Ω)2dτ.
Set g(t)≡∫Ω|u(t)|2dw, then
(3.13)ddtg(t)=∫Ωu(t)∂tu(t)dw=-2∫Ω[|∇Hdu(t)|2-μ|z|2ρ(w)4|u(t)|2]dw+2λ∫Ω|u(t)|2dw+2∫Ω|u(t)|pdw=4∫t0t‖∂τu(τ)‖L2(Ω)2dτ-4Iμ,λ(u(t0))+2(1-2p)∫Ω|u(t)|pdw≥2(1-2p)∫Ω|u(t)|pdw>0.
Hence, we get for t≥t0, g(t)≥g(t0)=∫Ω|u(t0)|2dw.
Letϵ∈(1,p/2), so we deduce by (3.13), that for any t≥t0:
(3.14)-1ϵ-1ddtg1-ϵ(t)=g-ϵ(t)ddtg(t)≥2(1-2p)g-ϵ(t)∫Ω|u(t)|pdw≥Cg-ϵ(t)(∫Ω|u(t)|2dw)p/2≥C(∫Ω|u(t0)|2dw)(p/2)-ϵ.
Hence, for anyt≥t0 sufficiently large, we have
(3.15)0<(∫Ω|u(t)|2dw)1-ϵ=g1-ϵ(t)≤g1-ϵ(t0)+C(ϵ-1)g(p/2)-ϵ(t0)(t0-t).
Then
(3.16)-1<C(ϵ-1)g(p/2)-1(t0)(t0-t),
and so t<t0+[C(ϵ-1)gp/2-1(t0)]-1, which is a contradiction.Theorem 3.3.
Assume thatu0∈𝒪+ and λ<λ1,μ, then the problem (1.6) admits a global solution u(t). Moreover, there exists a positive number α such that
(3.17)‖u(t)‖=O(e-αt),ast→+∞.Proof.
Letu0∈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↦Iμ,λ(u(t)) is strictly decreasing, so
(3.18)Iμ,λ(u(t))≤Iμ,λ(u0)≤cμ,λ.
Suppose there exists t*∈(0,Tmax) such that u(t*)∉𝒪+. Then,
(3.19)〈Iμ,λ′(u(t*)),u(t*)〉≤0.
Moreover, since the application t↦〈Iμ,λ′(u(t)),u(t)〉 is continuous, there exists t0∈(0,t*] such that
(3.20)〈Iμ,λ′(u(t0)),u(t0)〉=0.
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,Tmax). Thus, u(t) is global by extending to 0 for all t≥Tmax, and so Iμ,λ(u(t))>0 for any t≥0 by Theorem 3.2. But Iμ,λ(u(t0))=0, which is a contradiction. So, we conclude that u(t)∈𝒪+ for all t∈[t0,Tmax).
On other hand, we can write(3.21)Iμ,λ(u(t))=1p〈Iμ,λ′(u(t)),u(t)〉+(12-1p)∫Ω[|∇Hdu(t)|2-μ|z|2ρ(w)4|u(t)|2]dw-(12-1p)λ∫Ω|u(t)|2dw>(12-1p)∫Ω[|∇Hdu(t)|2-μ|z|2ρ(w)4|u(t)|2]dw-(12-1p)λ∫Ω|u(t)|2dw≥(12-1p)(1-λλ1,μ)‖u(t)‖μ2>0.
Since u(t) satisfy (3.3), we have
(3.22)∫t0t‖∂τu(τ)‖L2(Ω)2dτ+(12-1p)(1-λλ1,μ)‖u(t)‖μ2≤Iμ,λ(u(t0))<cμ,λ.
Then we have
(3.23)∫t0t‖∂τu(τ)‖L2(Ω)2dτ<cμ,λ,‖u(t)‖μ2<[(12-1p)(1-λλ1,μ)]-1cμ,λ,
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 ∫t0t∥∂τu(τ)∥L2(Ω)2dτ is finitely determined. Therefore, there exists a sequence (tn)n⩾0 with tn→+∞ as n→+∞ such that
(3.24)∫Ω|∂τu(tn)|2dw⟶0,u(tn)⇀vinH01(Ω,Hd).
Letting tn→+∞, we obtain that v∈H01(Ω,ℍd) is a solution of problem (1.10). So
(3.25)〈I′μ,λ(v),v〉=0.
If v≠0, then v∈𝒩, and so
(3.26)Iμ,λ(v)≥cμ,λ.
Since u(tn) satisfies (3.3), it follows by Hölder inequality and from (3.24), that
(3.27)|〈Iμ,λ′(u(tn,.)),u(tn,.)〉|≤|∫Ωu(tn,w)∂tu(tn,w)dw|≤‖u(tn,.)∥L2(Ω)∥∂tu(tn,.)‖L2(Ω)≤λ1,μ‖u(tn,.)‖μ‖∂tu(tn,.)‖L2(Ω)≤C‖∂tu(tn,.)‖L2(Ω).
Therefore,
(3.28)lim⁡n→+∞〈Iμ,λ′(u(tn)),u(tn)〉=0.
We deduce by (3.22), (3.25), and (3.28) that
(3.29)Iμ,λ(v)=(12-1p)∫Ω[|∇Hdv(w)|2-μ|z|2ρ4(w)|v(w)|2]dw-(12-1p)λ∫Ω|v(w)|2dw≤lim⁡n→+∞(12-1p)∫Ω[|∇Hdu(tn,w)|2-μ|z|2ρ4(w)|v(tn,w)|2]dw-lim⁡n→+∞(12-1p)λ∫Ω|v(tn,w)|2dw+lim⁡n→+∞〈Iμ,λ′(u(tn)),u(tn)〉≤lim⁡n→+∞Iμ,λ(u(tn))≤Iμ,λ(u0)<cμ,λ,
which contradicts (3.26), and so v=0 in Ω. Hence, by (3.24), we have
(3.30)u(tn,.)⟶0inLq(Ω),2≤q<2*.
Since
(3.31)‖u(tn,.)‖μ2=〈Iμ,λ′(u(tn)),u(tn)〉+λ∫Ω|u(tn,w)|2dw+∫Ω|u(tn,w)|pdw→0,asn⟶+∞,
we have
(3.32)u(tn,.)⟶0inH01(Ω,Hd),asn→+∞.
For simplicity, let us denote by t the divergent sequence and by u(t)=u(tn,w). We have from (3.29) that
(3.33)Iμ,λ(u(t))=(12-1p)∫Ω[|∇Hdu(t,w)|2-μ|z|2ρ4(w)|u(t,w)|2]dw-(12-1p)λ∫Ω|u(t,w)|2dw=(12-1p)‖u(t)‖μ2-λ(12-1p)‖u(t)‖L2(Ω)2.
So, due to (3.3) we have
(3.34)‖u(t)‖μ2=2pp-2Iμ,λ(u(t))+λ‖u(t)‖L2(Ω)2≤2pp-2Iμ,λ(u0)+λ‖u(t)‖L2(Ω)2<2pp-2cμ,λ+o(1).
Therefore, there exists t0 such that for all t≥t0,
(3.35)‖u(t)‖μ2≤2pp-2cμ,λ.
On the other hand,
(3.36)∫Ω|u(t,w)|pdw≤CΩp(μ¯μ¯-μ)p/2‖u(t)‖μp≤CΩp(μ¯μ¯-μ)p/2[2pp-2cμ,λ]p-2/2‖u(t)‖μ2.
Let C1=CΩp(μ¯/μ¯-μ)p/2[(2p/p-2)cμ,λ](p-2)/2, we have
(3.37)(1-C1)‖u(t)‖μ2≤‖u(t)‖μ2-‖u(t)‖Lp(Ω)p≤〈Iμ,λ′(u(t)),u(t)〉+λ‖u(t)‖L2(Ω)2.
Let us recall that if we set g(t)≡∫Ω|u(t,w)|2dw, then
(3.38)ddtg(t)=2∫Ωu(t,w)∂tu(t,w)dw=-2∫Ω[|∇Hdu(t,w)|2-μ|z|2ρ4(w)|u(t,w)|2]dw+2λ∫Ω|u(t,w)|2dw+λ∫Ω|v(tn,w)|pdw=-2〈Iμ,λ′(u(t)),u(t)〉.
So we get from (3.22) that for any t≥t0, we have
(3.39)∫t+∞〈Iμ,λ′(u(τ)),u(τ)〉dτ=12‖u(t)‖L2(Ω)2≤12λμ,λ‖u(t)‖μ2≤p(p-2)λμ,λIμ,λ(u(t)).
So from (3.37) and (3.39), we have for any t≥t0 that
(3.40)∫t+∞Iμ,λ(u(τ))dτ≤12λμ,λ∫t+∞‖u(τ)‖μ2dτ≤(1-C1)-112λμ,λ[∫t+∞〈Iμ,λ′(u(τ)),u(τ)〉dτ+λ∫t+∞‖u(τ)‖L2(Ω)2dτ]≤(1-C1)-12λμ,λ2pp-2Iμ,λ(u(t))+(1-C1)-1λ2λμ,λ∫t+∞‖u(τ)‖L2(Ω)2dτ.
Since limt→+∞∥u(t)∥L2(Ω)2=0, there exists t1>t0 such that for any t≥t1, we have
(3.41)∫t+∞Iμ,λ(u(τ))dτ≤(1-C1)-12λμ,λ2pp-2Iμ,λ(u(t)).
Thus,
(3.42)∫t+∞Iμ,λ(u(τ))dτ≤C(t1)e-αt,
with α=((1-C1)-1/2λμ,λ2)(p/p-2). But we remark that
(3.43)Iμ,λ(u(t+1))≤∫tt+1Iμ,λ(u(τ))dτ<∫t+∞Iμ,λ(u(τ))dτ,
hence, we deduce that for any t≥t1, we have
(3.44)Iμ,λ(u(t+1))<C(t1)e-αt,
and we can conclude that for any t≥t1, we have
(3.45)‖u(t)‖μ=O(e-αt).Remark 3.4.
for smallu0, 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 inLp(Ω).Corollary 3.5.
Assume thatu0∈𝒪+ and λ<λ1,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 Lμ 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
(3.46)(12-1p)‖u(t)‖μ2=Iμ,λ(u(t))+λ(12-1p)‖u(t)‖L2(Ω)2≤Iμ,λ(u0)+λ(12-1p)‖u(t)‖L2(Ω)2≤cμ,λ+λλ1,μ(12-1p)‖u(t)‖μ2,
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⊂H01(Ω,ℍd). 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 thatu0∈𝒪-. Then the solution u(t) of the problem (1.6) blows up in finite time.Proof.
Letu0∈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↦Iμ,λ(u(t)) is strictly decreasing, so
(3.47)Iμ,λ(u(t))=Iμ,λ(u0)=cμ,λ.
Suppose there exists t̃∈(0,Tmax) such that u(t̃)∉𝒪-. Then
(3.48)〈Iμ,λ′(u(t)),u(t)〉≥0.
And since the application t↦〈Iμ,λ′(u(t)),u(t)〉 is continuous, there exists t̃0∈(0,t̃] such that
(3.49)〈Iμ,λ′(u(t0)),u(t0)〉=0.
Hence, u(t̃0)=0 in Ω or u(t̃0)∈𝒩. If u(t̃0)=0 in Ω, then by the uniquess of u(t), we conclude that u(t)=0 for any t∈[t̃0,Tmax). Thus, u(t) is global by extending to 0 for all t≥Tmax, and thanks to Theorem 3.2, Iμ,λ(u(t))>0 for any t≥0. But Iμ,λ(u(t̃0))=0, which is a contradiction, and so u(t̃0)∈𝒩. But by [21],
(3.50)cμ,λ=inf⁡u∈NIμ,λ(u),
then cμ,λ≤Iμ,λ(u(t̃0)), which contradicts (3.47). So, we conclude that u(t)∈𝒪- for all t∈[t̃0,Tmax). We suppose by contradiction that Tmax=+∞, that is, u(t)=u(t,·) exists for all t≥0. For u∈𝒪-, we have
(3.51)ddt‖u(t,⋅)‖L2(Ω)2=-2〈Iμ,λ′(u(t0)),u(t0)〉>0.
Then t↦∥u(t,·)∥L2(Ω) is strictly increasing and so
(3.52)lim⁡t→+∞‖u(t,⋅)‖L2(Ω)=c∈(0,+∞].
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→+∞,
(3.53)u(t,⋅)⟶0inH01(Ω,Hd).
Letting t→+∞ in the inequality
(3.54)λ1,μ‖u(t,⋅)‖L2(Ω)≤‖u(t,⋅)‖μ,
we get that 0<c≤0, which is a contradiction. So we conclude that
(3.55)lim⁡t→+∞‖u(t,⋅)‖L2(Ω)=+∞.
Set g(t)=∥u(t,·)∥L2(Ω)2, so
(3.56)-2p-2ddtg1-p/2(t)=g′(t)g-p/2(t)=-2‖u(t,⋅)‖L2(Ω)-p(‖u(t,⋅)‖μ2-λ‖u(t,⋅)‖L2(Ω)2‖u(t,⋅)‖Lp(Ω)2)≥-2‖u(t,⋅)‖L2(Ω)-p‖u(t,⋅)‖μ2+2‖u(t,⋅)‖L2(Ω)-p‖u(t,⋅)‖Lp(Ω)2.
By Hölder inequality, we have
(3.57)‖u(t,⋅)‖Lp(Ω)p≥|Ω|1-p/2‖u(t,⋅)‖L2(Ω)p,
and by (3.55), there exist t1>0 and a constant C1>0 such that for t≥t1, we have
(3.58)‖u(t,⋅)‖L2(Ω)≥C1.
Then, there exist t1>0 and a constant C2>0 such that for t≥t1, we have
(3.59)-2p-2ddtg1-p/2(t)≥-2λ1,μC12-p+2|Ω|1-p/2≥C2.
Hence, we have from (3.59), that for any t≥t1,
(3.60)0<g(t)≤g(t1)+p-22C2(t-t1),
which is a contradiction if t is sufficiently large. So we conclude that Tmax<+∞.



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*Source: 749683-2012-02-20.xml*