# Dynamic Analysis of a Nonlinear Timoshenko Equation

Authors: Jorge Alfredo Esquivel-Avila
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We characterize the global and nonglobal solutions of the Timoshenko equation in a bounded
domain. We consider nonlinear dissipation and a nonlinear source term. We prove blowup of
solutions as well as convergence to the zero and nonzero equilibria, and we give rates of decay to the
zero equilibrium. In particular, we prove instability of the ground state. We show existence of global
solutions without a uniform bound in time for the equation with nonlinear damping. We define and
use a potential well and positive invariant sets.


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## Body

## 1. Introduction

We consider(1.1)utt+kΔ2u-M(‖∇u‖22)Δu+g(ut)=f(u)inΩ,
with initial conditions
(1.2)u(x,0)=u0,ut(x,0)=v0,x∈Ω,
and with one set of the following boundary conditions:
(1.3)u=0,Δu=0on∂Ω,
or
(1.4)u=0,∂u∂ν=0on∂Ω,
where Ω⊂ℝn is a bounded domain with sufficiently smooth boundary, ∥·∥2 is the norm in L2(Ω),
(1.5)M(s2)=α+βs2γ,α≥0,β≥0,α+β>0,γ≥1,k=1,(1.6)g(ut)=δut|ut|λ-2,δ>0,λ≥2,(1.7)f(u)=μu|u|r-2,μ>0,r>0.
When the source term f≡0, there is a considerable set of works studying several properties of equation (1.1), see for instance, the early papers by Ball [1, 2], Haraux and Zuazua [3], and the books by Hale [4], Haraux [5], and references therein. For a destabilizing source term, sf(s)>0,s∈ℝ∖{0}, in the works of Payne and Sattinger [6], Georgiev and Todorova [7], and Ikehata [8], qualitative properties of (1.1) are studied, when k=0=β. To understand the dynamics of second-order equations in time, similar to (1.1), active research is reported in Alves and Cavalcanti [9], Barbu et al. [10], Cavalcanti et al. [11–16], Rammaha [17] Rammaha and Sakuntasathien [18], and Todorova and Vitillaro [19], Vitillaro [20]. For the Timoshenko equation, with g≡0, Bainov and Minchev [21] gave sufficient conditions for the nonexistence of smooth solutions of (1.1), with negative initial energy, and gave an upper bound of the maximal time of existence. For positive and sufficiently small initial energy, blowup and globality properties are characterized in Esquivel-Avila [22]. For the Kirchhoff equation, that is, (1.1) with k=0, the nonexistence of global solutions is studied in [23]. In [24, 25], we characterized properties such as blowup and asymptotic behavior of solutions, for (1.1) with k=0 and β=0. To the knowledge of the author, such problems are still open for the Timoshenko equation (1.1). Here, we want to give some results about the dynamics of problem (1.1). To do that we will generalize the concept of the depth of the potential well in such manner that our results of the dynamics be as sharp as the ones in [24, 25]. Furthermore, for particular cases, our definition of depth of the potential well will coincide with the one introduced in [6].


## 2. Preliminaries and Framework

We begin this section with an existence, uniqueness, and continuation theorem for (1.1). The proof is similar to the ones in [7, 8], where semilinear wave equations are studied.Theorem 2.1.
Assume thatr>2 and r≤2(n-2)/(n-4) if n≥5. For every initial data (u0,v0)∈H≡B×L2(Ω), where B is defined either by B≡H2(Ω)∩H01(Ω), or B≡H02(Ω), there exists a unique (local) weak solution (u(t),v(t))≡S(t)(u0,v0) of problem (1.1), that is,
(2.1)ddt(v(t),w)2+(Δu(t),Δw)2+M(‖∇u(t)‖22)(∇u(t),∇w)2+(g(v(t)),w)2=(f(u(t)),w)2,
a.e. in (0,T) and for every w∈B∩Lλ(Ω), such that
(2.2)u∈C([0,T);B)∩C1([0,T);L2(Ω)),v≡ut∈Lλ((0,T)×Ω).
Here,S(t) denotes the corresponding semigroup on H, generated by problem (1.1), and (·,·)2 is the inner product in L2(Ω).
The following energy equation holds:(2.3)E0=E(t)+∫0tδ‖v(τ)‖λλdτ,
where
(2.4)E(t)≡E(u(t),v(t))≡12‖v(t)‖22+J(u(t)),(2.5)J(u)≡12a(u)+12(γ+1)c(u)-1rb(u),
with
(2.6)a(u)≡‖u‖B2,b(u)≡μ‖u‖rr,c(u)≡β‖∇u‖22(γ+1).
Here,E0≡E(u0,v0) is the initial energy, and ∥·∥q denotes the norm in the Lq(Ω) space.
If the maximal time of existenceTM<∞, then S(t)(u0,v0)→∞ as t↗TM, in the norm of H:
(2.7)‖(u,v)‖H2≡‖u‖B2+‖v‖22≡‖Δu‖22+α‖∇u‖22+‖v‖22.
In that case, from (2.3)–(2.6), ∥u(t)∥r→∞ as t↗TM.Now, we define, respectively, the stable (potential well) and unstable sets:(2.8)W≡([I(u)>0]∪{0})∩[J(u)<d],V≡[I(u)<0]∩[J(u)<d],
where
(2.9)I(u)≡a(u)+c(u)-b(u).Here,[I(u)<0] denotes the set of u∈B with that property, and the depth of the potential well is defined as follows:
(2.10)d≡r-22rSr/(r-2),
where
(2.11)S≡inf⁡0≠u∈Bb̂(u)>0(â(u)1/2b̂(u)1/r),(2.12)â(u)≡a(u)+κ1c(u),b̂(u)≡b(u)+κ2c(u).
with
(2.13)κ1≡r-2(γ+1)(r-2)(γ+1),κ2≡κ1-1=-rγ(r-2)(γ+1).We assume thatr≥2(γ+1), and since γ≥1, then κ1∈[0,1/2), and κ2∈[-1,-1/2). Also note that if r=2(γ+1), then κ1=0, κ2=-1, and we have the following characterization of the depth of the potential well (2.10)-(2.11):
(2.14)d=inf⁡0≠u∈Bsup⁡λ≥0J(λu),
which is the definition given in [6], where a nondissipative nonlinear wave equation is studied.Consider anyu∈B, r>2, and r≤2n/(n-4) if n≥5, then
(2.15)â(u)≥C(Ω)b(u)2/r+κ1c(u)≥C(Ω)b(u)2/r,
where C(Ω)>0, is any constant in the Sobolev-Poincaré’s inequality
(2.16)(‖Δu‖22+α‖∇u‖22)1/2≥C(Ω)μ1/r‖u‖r.Moreover, ifb̂(u)>0, from (2.15) and since b(u)≥b̂(u),
(2.17)â(u)≥C(Ω)b̂(u)2/r.Hence,S≥C(Ω), and d≥D≡((r-2)/2r)C(Ω)r/(r-2)>0.Ifue denotes any nonzero equilibria of equation (1.1),
(2.18)E≡[0≠ue∈B:Δ2ue-M(‖u‖22)Δue=f(ue)],
then, by (2.1) in Theorem 2.1 with u(t)=ue=w, we get that ue belongs to the Nehari manifold, 𝒩, that is,
(2.19)E⊂N≡[0≠u∈B:Î(u)=I(u)=0],
where Î(u)≡â(u)-b̂(u).Consequently,b̂(ue)=â(ue)>0. Furthermore, from (2.17) which is an equality when C(Ω)=S, we conclude that the Nehari manifold can be represented by the line: y=x, in the plane with axes x=b̂(u) and y=â(u), beginning at the point: y=x=Sr/(r-2)=(2r/(r-2))d. We also note that
(2.20)J(u)=12â(u)-1rb̂(u).From these facts it follows that the depth of the potential well (2.10) is characterized by
(2.21)d=inf⁡u∈NJ(u)=r-22rϱ,
where
(2.22)0<ϱ≡inf⁡u∈Nâ(u)=inf⁡u∈Nb̂(u).Hence, any equilibrium is such thatue∈[J(u)≥d]. Moreover, like in [6], the set of extremals of (2.21) is characterized by set of equilibria with least energy, that is the ground state
(2.23)N*≡[ue∈E:J(ue)=d]=[ue∈E:â(ue)=b̂(ue)=ϱ].
Observe that J(u)=d is a tangent line to the curve defined by the equality in (2.17) with C(Ω)=S, at the point 𝒩*, which holds if b̂(u)>0. On the other hand, we notice that
(2.24)κ1b̂(u)-κ2â(u)=κ1b(u)-κ2a(u)>0,
and is equal to zero if and only if a(u)=0=b(u). Hence, if b̂(u)<0, then
(2.25)â(u)>r-2(γ+1)-rγb̂(u).Therefore, next results about the stable and unstable sets follow.Lemma 2.2.
The following properties ofV and  W hold: (i)
W is a neighborhood of 0∈B.(ii)
0∉[I(u)<0]¯ (closure in B), in particular 0∉V¯.(iii)
W=W+∪W-∪{0}, where
(2.26)W+≡W∩[b̂(u)>0]=[ϱ(r-2)/rb̂(u)2/r≤â(u)<2rb̂(u)+r-2rϱ,0<b̂(u)<ϱ],W-≡W∩[b̂(u)<0]⊂[r-2(γ+1)-rγb̂(u)<â(u)<2rb̂(u)+r-2rϱ,-γϱ<b̂(u)<0].(iv)
V=[ϱ(r-2)/rb̂(u)2/r≤â(u)<2rb̂(u)+r-2rϱ,b̂(u)>ϱ].(v)
𝒩*=W¯∩V¯=W+¯∩V¯=[ue∈𝒩,â(ue)=b̂(ue)=ϱ].(vi)
W=[I(u)<0]c∩[J(u)<d], V=([I(u)>0]∪{0})c∩[J(u)<d].The following result follows easily like in [23].Lemma 2.3.
One has that(2.27)J(u)>r-22râ(u)>r-22rb̂(u),(2.28)J(u)>r-22ra(u)+r-2(γ+1)2r(γ+1)c(u)>γ2(γ+1)a(u)+r-2(γ+1)2r(γ+1)b(u),
for any u∈B such that I(u)>0, in particular if 0≠u∈W, and
(2.29)d<r-22râ(u)<r-22rb̂(u),(2.30)d<r-22ra(u)+r-2(γ+1)2r(γ+1)c(u)<γ2(γ+1)a(u)+r-2(γ+1)2r(γ+1)b(u),
for any u∈B, such that I(u)<0, in particular if u∈V.A set𝒱⊂H is positive invariant, with respect to problem (1.1), if the corresponding generated semigroup S(t) on H is such that(2.31)S(t)V⊂V.Lemma 2.4.
Let(u,v) denote any solution of (1.1), given by Theorem 2.1. Then, the sets
(2.32)S≡[E(u,v)<d]∩[(u,v)∈H:u∈W],(2.33)U≡[E(u,v)<d]∩[(u,v)∈H:u∈V],
are positive invariant.Proof.
First, we show that𝒮 is positive invariant. In order to do that, we take (u0,v0)∈𝒮. Then, by (2.4), J(u(t))≤E(u(t),v(t))≤E0<d, for any t≥0. Now, if 𝒮 is not positive invariant, there exists some t̂>0, such that I(u(t̂))=0, with u(t̂)≠0. Then, by (2.21), d≤J(u(t̂)). But this is impossible because J(u(t̂))<d. The proof of the positive invariance of 𝒰 is quite similar. Indeed, if this is not true there exists some t̂>0, such that I(u(t̂))=0. From (ii) of Lemma 2.2  u(t̂)≠0, and this implies the same contradiction as before.Next result gives an interpretation of sets𝒮 and 𝒰 and follows from Lemma 2.2.Lemma 2.5.
The sets𝒮 and  𝒰 have the properties
(2.34)S⊂[E(u,v)<d]∩[(u,v)∈H:ϱ(r-2)/rb̂(u)2/r≤â(u)<2rb̂(u)+r-2rϱ,0<b̂(u)<ϱ],∪[E(u,v)<d]∩[(u,v)∈H:r-2(γ+1)-rγb̂(u)<â(u)<2rb̂(u)+r-2rϱ,-γϱ<b̂(u)<0]U=[E(u,v)<d]∩[(u,v)∈H:ϱ(r-2)/rb̂(u)2/r≤â(u)<2rb̂(u)+r-2rϱ,b̂(u)>ϱ],,
(2.35)S¯∩U¯=[(ue,0)∈H:ue∈N*]=[(ue,0)∈H:ue∈N,â(ue)=b̂(ue)=ϱ].The following result is a direct consequence of (vi) in Lemma2.2 and Lemma 2.4.Lemma 2.6.
For every solution of (1.1), only one of the following holds: (i)
there exists somet0≥0 such that (u(t0),v(t0))∈𝒮, and remains there for every t>t0,(ii)
there exists somet0≥0 such that (u(t0),v(t0))∈𝒰, and remains there for every t>t0,(iii)
(u(t),v(t))∈[E(u,v)≥d] for every t≥0.Hence, we notice that the sets𝒮 and 𝒰 play an important role in the dynamics of (1.1). Moreover, we will prove that any solution eventually contained in 𝒮 converges to the zero equilibrium. If enters in 𝒰, either blowups in a finite time or it is global but without a uniform bound in H for every t≥0, in the case that λ>2, in (1.6). Also, we will prove that any solution with (u(t),v(t))∈[E(u,v)≥d], for every t≥0, is bounded and converges to the set of nonzero equilibria ℰ.We will need the following inequalities to show blowup and convergence to the zero equilibrium, respectively, in the dissipative case.Lemma 2.7.
Letℱ∈Wloc1,1(ℝ+) be a nonnegative function such that
(2.36)Ḟ(t)≥CFa(t)a.e.fort≥0,
with a>1 and C>0.
Then, there exists someT*>0 such that limt↗T*ℱ(t)=∞.Proof.
Define𝒢(t)≡ℱ1-a(t), then
(2.37)Ġ(t)≤(1-a)C<0a.e.fort≥0.
Hence, 0<G(0)+(1-a)Ct, which is only possible if t<T*≡(1/C(a-1))ℱ1-a(0).Lemma 2.8.
Letℱ∈Wloc1,1(ℝ+) be a nonnegative function such that
(2.38)Ḟ(t)≤-CFa(t)a.e.fort≥0,
with a≥1 and C>0.
Then, fort≥0, if a>1
(2.39)F(t)≤F0{1+tC(a-1)F0a-1}-1/(a-1),
and, if a=1
(2.40)F(t)≤F0e-Ct.Proof.
Considera>1, and notice that (ℱ(1-a))̇(t)≥(a-1)C. Then, we integrate and obtain the first inequality. Now, let a→1, and the second one follows.


## 3. Timoshenko Equation

Due to our assumptions onr and γ, we restrict our analysis to dimensions n≤5. Indeed, since γ≥1, 2(γ+1)<r and r≤2(n-2)/(n-4), if n≥5, then our analysis considers, n=5 whenever γ<2. We also notice that in any case we do not consider the interval 2<r≤4. Moreover, r≤6 whenever n=5. We begin with a characterization of blowup when δ>0 and λ≥2.Theorem 3.1.
Let(u(t),v(t))=S(t)(u0,v0) be a solution of problem (1.1), and suppose that r>2(γ+1). A necessary and sufficient condition for nonglobality, blowup by Theorem 2.1, is that λ<r and there exists t0≥0 such that (u(t0),v(t0))∈𝒰.Proof.
Sufficiency
By Lemma2.4, (u(t),v(t))∈𝒰 for all t>t0.
Now, we consider the function defined, along the solution, by(3.1)V(t)≡d-E(t),
and notice that because of energy equation (2.3),
(3.2)V(t)≥d-E0≡V0>0,
where, now E0≡E(u(t0),v(t0)).
Notice that from (2.29) in Lemma 2.3,
(3.3)V(t)≤d-J(u(t))≤d-rr-2d+1rb̂(u(t))=-2r-2d+1rb(u(t))-γ(r-2)(γ+1)c(u(t)).
We will need some estimates. First, we notice that from energy equation in terms of𝒱(t) and (3.3),
(3.4)|δ(u(t),v(t)|v(t)|λ-2)2|≤δ‖u(t)‖λ‖v(t)‖λλ-1≤C(Ω)δ‖u(t)‖r‖v(t)‖λλ-1≤C(Ω)δ‖u(t)‖r1-k‖u(t)‖rk‖v(t)‖λλ-1≤C(Ω)δ‖u(t)‖r1-k[ν‖u(t)‖rkλ+1C(ν)‖v(t)‖λλ]<CV(1-k)/r(t)[νδ‖u(t)‖rkλ+1C(ν)V̇(t)],
where k∈(1,r/λ), C≡C(Ω)(r/μ)(1-k)/r, C(Ω)>0 is the constant in the continuous embedding Lr(Ω)⊂Lλ(Ω), C(ν)>0, and ν>0 will be chosen later.
Consider a positive numberq to be chosen later, from (3.2)-(3.3), we obtain
(3.5)-I(u(t))=-qJ(u(t))+q-22a(u(t))+q-2(γ+1)2(γ+1)c(u(t))+r-qrb(u(t))≥q(V0-d)+q-22a(u(t))+q-2(γ+1)2(γ+1)c(u(t))+r-qrb(u(t)).
If𝒱0≥d, we choose q≡2(γ+1), and from (3.5) we get
(3.6)-I(u(t))≥r-2(γ+1)rb(u(t)).
If𝒱0<d, then we notice that from (3.2)-(3.3),
(3.7)V0-d≥(r-2)(V0-d)(r-2)V0+2d(1rb(u(t))-γ(r-2)(γ+1)c(u(t))).
Hence and from (3.5), we have the estimate
(3.8)-I(u(t))≥q-22a(u(t))+q2(γ+1)(q-2(γ+1)q+2γ(d-V0)(r-2)V0+2d)c(u(t))+qr(r-qq-(r-2)(d-V0)(r-2)V0+2d)b(u(t)).
In this case, we choose the numberq so that the coefficient of c(u(t)) in (3.8) be equal to zero, then
(3.9)q≡2(γ+1)((r-2)V0+2d)(r-2(γ+1))V0+2(γ+1)d.
We note that2<q<2(γ+1), and we get
(3.10)-I(u(t))≥γra(u(t))+(r-2(γ+1))b(u(t))(r-2(γ+1))V0+2(γ+1)dV0.
Therefore, from (3.6) and (3.10),
(3.11)-I(u(t))≥Ĉb(u(t)),
where
(3.12)Ĉ≡r-2(γ+1)rmin⁡(1,rV0(r-2(γ+1))V0+2(γ+1)d)>0.
Now, we define the function, along the solution, by(3.13)F(t)≡V1/a(t)+ϵ(u(t),v(t))2,
where a≡(1+(1-k)/r)-1∈(1,2) and ϵ>0 will be choosen later.
We intend to apply Lemma2.7 to functional (3.13). First, we calculate the derivative, along solutions, with respect to t. Let us start with the second term of (3.13). From (3.2)–(3.4) and (3.11), one has
(3.14)ddt(u(t),v(t))2=‖v(t)‖22-I(u(t))-δ(u(t),v(t)|v(t)|λ-2)2≥‖v(t)‖22+Ĉμ‖u(t)‖rr-CV(1-k)/r(t)[νδ‖u(t)‖rkλ+1C(ν)V̇(t)]≥‖v(t)‖22+[Ĉμ-νCδ(rμ)(kλ-r)/rV0b]‖u(t)‖rr-CaC(ν)V̇1/a(t)≥‖v(t)‖22+Ĉμ2‖u(t)‖rr-CaC(ν)V̇1/a(t),
where b≡(k(λ-1)-(r-1))/r<0, and ν>0 is sufficiently small.
Consequently, ifϵ>0 is sufficiently small,
(3.15)Ḟ(t)≥C̃[‖v(t)‖22+‖u(t)‖rr]>0,
where C̃≡ϵmin(1,Ĉμ/2)>0.
From (3.15) and choosing ϵ>0 small enough, we get
(3.16)F(t)≥F0≡V0+ϵ(u(t0),v(t0))2>0.
Utilizing two times (3.3), we get
(3.17)Fa(t)≤2a-1[V(t)+ϵa|(u(t),v(t))2|a]≤2a-1[μr‖u(t)‖rr+ϵaC(Ω)a‖u(t)‖ra‖v(t)‖2a]≤2a-1[μr‖u(t)‖rr+ϵaC(Ω)a(‖u(t)‖r2a/(2-a)+‖v(t)‖22)]≤2a-1[μr∥u(t)∥rr+ϵaC(Ω)a((μ(r-2)2rd)c‖u(t)‖rr+‖v(t)‖22)]≤C̅[‖u(t)‖rr+‖v(t)‖22],
where C(Ω)>0 is the imbedding constant of Lr(Ω)⊂L2(Ω), c≡(1-2a)/r(2-a)>0, and C̅>0.
Hence and from (3.15), we obtain the inequality in order to apply Lemma 2.7. Therefore, the maximal time of existence is finite: T<∞.
Necessity
Suppose thatλ≥r. Define the function, along the solution, by
(3.18)W(t)≡E(t)+2μr‖u(t)‖rr.
Then,(3.19)Ẇ(t)=Ė(t)+2μ(u(t)|u(t)|r-2,v(t))2≤-δ‖v(t)‖λλ+δ2‖v(t)‖λr+C‖u(t)‖rr≤Ĉ(W(t)+1),
where Ĉ≡max(δ/2,Cr/μ), C≡2μC(Ω)(δ/2)r, and C(Ω)>0 is the imbedding constant of Lλ(Ω)⊂Lr(Ω).
Hence, by Gronwall inequality, it follows that(u,v) is bounded in H for any finite time. A contradiction.
Proceeding again by contradiction suppose that, for allt≥0, (u(t),v(t))∉𝒰. Then, by Lemma 2.6, we have either u(t)∈𝒮 for all t≥0, or E(t)≥d for all t≥0. In the first case, from (2.28) in Lemma 2.3(3.20)E0≥12‖v(t)‖22+r-22ra(u(t)),
that is, (u(t),v(t)) is bounded in H. This is not possible. In the second case,
(3.21)δ∫0t‖v(τ)‖λλdτ≤E0-d,
where E0≡E(u0,v0). Hence, by the Hölder inequality,
(3.22)δt-(λ-1)‖∫0tv(τ)dτ‖λλ≤E0-d,
and consequently
(3.23)‖u(t)‖λ≤C(T),
for t∈[0,T], where C(T)≡∥u0∥λ+((E0-d)/δ)1/λT(λ-1)/λ.
From Theorem2.1,
(3.24)lim⁡t↗TMAX‖u(t)‖r=∞;
hence, by Sobolev-Poincaré's inequality (2.16), for every M>E0, there exists some t̂>0, such that
(3.25)M<r-22ra(u(t)),
for every t≥t̂. This implies the first inequality of (2.29) in Lemma 2.3, replacing d by M. Now, we consider the function (3.13)
(3.26)F(t)≡V(t)1/a+ϵ(u(t),v(t))2,
defined for t≥t̂, where a∈(1,2), ϵ>0 is sufficiently small, and here
(3.27)V(t)≡M-E(t)>0,
and repeat the sufficiency part of the proof. Then, by Lemma 2.7, ℱ(t) blowups as t↗T*, T*>t̂. Moreover, for t̂≤t<T*,
(3.28)F(t)≥F(t̂)(1-(t-t̂)/(T*-t̂))1/(a-1),
hence and from (3.26), (3.27), and since E(t)≥d,
(3.29)‖u(t)‖22≥‖u(t̂)‖22+2ϵ{∫t̂t(F(t̂)(1-(τ-t̂)/(T*-t̂))1/(a-1)-V1/a(τ))dτ}≥‖u(t̂)‖22-2(t-t̂)ϵ(M-d)1/a+2(a-1)ϵ(2-a)F(t̂)(T*-t̂){(1-t-t̂T*-t̂)-((2-a)/(a-1))-1}.
Consequently,(3.30)lim⁡t↗T*‖u(t)‖22=∞.
But this contradicts (3.23), since Lλ(Ω)⊂L2(Ω). The proof is complete.Remark 3.2.
From the last result, ifλ=2 and r>2(γ+1), any solution of problem (1.1), (u(t),v(t)), is global if and only if either (i) there exists t0≥0 such that (u(t0),v(t0))∈𝒮 or (ii) (u(t),v(t))∈[E(u,v)≥d], for every t≥0. On the other hand, if λ>2 and r>2(γ+1), then any solution is global if and only if one of the following holds: (i), (ii), or (iii) λ≥r and there exists t0≥0 such that (u(t0),v(t0))∈𝒰.We next prove a characterization of convergence to the zero equilibrium, and we give rates of decay.Theorem 3.3.
Let(u(t),v(t))=S(t)(u0,v0) be a solution of problem (1.1) with  λ≥2. Suppose that  r>2(γ+1) and that  λ≤10, if  n=5. A necessary and sufficient condition for  (u(t),v(t))→(0,0), strongly in  H as  t→∞, is that there exists  t0≥0 such that  (u(t0),v(t0))∈𝒮.
In this case, ifℱ(t) denotes either the energy
(3.31)E(t)≡E(u(t),v(t)),
or the norm of the solution in H
(3.32)ω(t)≡‖(u(t),v(t))‖H2≡a(u(t))+‖v(t)‖22,
One has the rates of decay, for t≥T,
(3.33)F(t)≤K0{1+t(λ-22)K1K0(λ-2)/2}-2/(λ-2),
and, for linear dissipation, λ=2,
(3.34)F(t)≤K0e-K1t,
where T>0 is sufficiently large, and K0>0, K1>0 are constants depending only on initial conditions.Proof.
Necessity
By (ii) in Lemma2.2, (0,0)∉𝒰¯, and since the equilibrium (0,0)∉[E(u(t),v(t))≥d]¯, strong closures in H, then, by Lemma 2.6, the solution must eventually enter in 𝒮.Sufficiency
By energy equation and (2.27) in Lemma 2.3, the solution must be global and uniformly bounded in the norm of H, that is ω(t)<2rd/(r-2), for any t≥0. Hence, there exists a sequence of times, {tn}, such that if n→∞ then tn→∞, (u(tn),v(tn))→(û,v̂) weakly in H and, since the embedding B⊂Lr(Ω) is compact, b(u(tn))→b(û). Also, notice that the energy is such that
(3.35)0≤E∞≡lim⁡t→∞E(t)=inf⁡t≥0E(t)<∞.
Consequently, from the energy equation and the continuous embeddingLλ(Ω)⊂L2(Ω),
(3.36)lim⁡t→∞∫tt+1‖v(τ)‖2λdτ=0,
in particular, for any sequence of times {sn} such that sn→∞ as n→∞,
(3.37)lim⁡n→∞∫01hn(τ)dτ=0,
where hn(τ)≡∥v(sn+τ)∥2λ, for τ∈[0,1]. By Fatou Lemma,
(3.38)liminf⁡n→∞‖v(sn+τ)‖2λ=liminf⁡n→∞hn(τ)=0,
for a.e. τ∈[0,1], and by the weak convergence to v̂,
(3.39)‖v̂‖2≤liminf⁡n→∞‖v(tn)‖2=0,
where we choose {sn} such that tn=sn+τ0, for some τ0∈[0,1].
It can be shown that the semigroup generated by problem (1.1) is continuous in H with the weak topology, and then that the weak limit set is positive invariant, see Ball [26]. Consequently (û,v̂)=(ue,0) must be an equilibrium of (1.1). Furthermore, by the lower-semicontinuity of the norm in H, one has
(3.40)b̂(ue)=â(ue)≤liminf⁡n→∞{‖v(tn)‖22+â(u(tn))}=lim⁡n→∞{2E(tn)+2rb̂(u(tn))}=2E∞+2rb̂(ue).
Hence,(3.41)r-22rb̂(ue)≤E∞<d.
Then, by (2.19) and (2.21), ue=0, and
(3.42)lim⁡t→∞(u(t),v(t))=(0,0)weaklyinH.
Strong convergence follows if we get the rates of decay in our statement. Here, we will adapt the technique used in Haraux and Zuazua [3], to (1.1). That technique is based on the construction of suitable Liapunov functions defined along solutions and the application of Lemma 2.8. One of them is the energy, and we will need one more, defined by
(3.43)W(t)≡E(t)+κE(t)λ/2-1(u(t),v(t))2,
where κ>0 is a constant to be chosen later. We next prove that W(t) is equivalent to both, the energy E(t) and the norm ω(t) of the solution, in the sense of (2.30) and (2.28) below. First we note that from (2.27) in Lemma 2.3,
(3.44)r-22rω(t)≤E(t)≤K2ω(t),
where K-1≡(β/α(γ+1))(2rd/α(r-2))γ.
Also, notice that from (3.43),
(3.45)|W(t)-E(t)|≤κE0λ/2-1‖u(t)‖2‖v(t)‖2≤κE0λ/2-1C1(Ω)ω(t),
where C1(Ω)>0 is a constant that depends on the continuous embedding B⊂L2(Ω). Hence and from (3.44), if κ is sufficiently small, then
(3.46)12E(t)≤W(t)≤32E(t).
We will need the following estimate:(3.47)δ(u(t),v(t)|v(t)|λ-1)2≤δ‖u(t)‖λ‖v(t)‖λλ-1≤δC2(Ω)a(u(t))(λ-2)/2λa(u(t))1/λ‖v(t)‖λλ-1≤δC2(Ω)(2rr-2E0)(λ-2)/2λa(u(t))1/λ(-Ė(t))(λ-1)/λ≤1λa(u(t))-ĈĖ(t),
where we applied (3.44) in the third step and Young inequality in last step, and the constants C2(Ω)>0, Ĉ>0 depend on the continuous embedding B⊂Lλ(Ω), and Ĉ also depends on E0.
It follows that, by (3.42) and since B⊂Lr(Ω) is compact, for any ϵ>0, there exists some T>0 such that for any t>T
(3.48)b(u(t))≤C3(Ω)b(u(t))(r-2)/ra(u(t))≤ϵE(t),
where C3(Ω)>0 is the corresponding embedding constant and we used (3.44) in the last step.
Since we will apply Lemma2.8, we need to calculate the time derivative of (3.43) and we begin with
(3.49)ddt(u(t),v(t))2=‖v(t)‖22-a(u(t))-c(u(t))+b(u(t))-δ(u(t),v(t)|v(t)|λ-1)2≤‖v(t)‖22-12a(u(t))-c(u(t))-ĈĖ(t)+ϵE(t)≤32‖v(t)‖22-(1-ϵ)E(t)-ĈĖ(t),
which holds for any t>T, and where we used (3.47), (3.48) and definition of E(t).
We notice that for any smallη>0, and by Young inequality and energy equation
(3.50)Eλ/2-1(t)32‖v(t)‖22≤Eλ/2-1(t)C4(Ω)‖v(t)‖λ2≤ηEλ/2(t)-C(η)Ė(t),
where C4(Ω)>0, C(η)>0 depend on the continuous embedding Lλ(Ω)⊂L2(Ω), and C(η) depends on η.
Then, forϵ and η sufficiently small, (3.49) and (3.50), imply
(3.51)Eλ/2-1(t)ddt(u(t),v(t))2≤-12Eλ/2(t)-C̃Ė(t),
for any t>T, where C̃≡C(η)+ĈE0λ/2-1.
Consequently, forκ sufficiently small and any t>T
(3.52)Ẇ(t)≤Ė(t)-κλ-2r-2rC1(Ω)E0λ/2-1Ė(t)-κ(12Eλ/2(t)+C̃Ė(t))≤-κ2Eλ/2(t)≤-κ0Wλ/2(t),
where κ0≡(κ/2)(2/3)λ/2 and C1(Ω)>0 is the constant in (3.45); also we used (3.44), the fact that the energy is decreasing and (3.46). Then, from (3.52) and Lemma 2.8, we obtain the desired rates of decay for W(t). The result now follows by (3.46) and (3.44), and the proof is complete.Remark 3.4.
By (2.35), the ground state is: [(u,0)∈H:u∈𝒩*]=𝒮¯∩𝒰¯. Then, in any H-neighborhood of that subset of nonzero equilibria, one can choose initial conditions either in 𝒰 or in 𝒮. Hence, by Theorem 3.1 and (3.3), the ground state is unstable in the sense of Liapunov when the dissipation term g(ut) is either linear or nonlinear.Next we will study the behavior of solutions such that(u(t),v(t))∈[E(u,v)≥d] for all t≥0. First, we prove that those solutions are uniformly bounded in time. To that end we will study the cases: λ=2 and λ>2 separately, First, we consider the case λ=2.Theorem 3.5.
Let(u(t),v(t))=S(t)(u0,v0) be a solution of problem (1.1). Assume that  r>2(γ+1), and  λ=2. Also, assume that  r≤2(6/n+1) if  n≥2. If  (u(t),v(t))∈[E(u,v)≥d] for all  t≥0, then the solution is global and uniformly bounded in  H, for all  t≥0.Proof.
Suppose that(u(t),v(t)) is not global, then by Theorem 2.1 blowups and by Theorem 3.1, (u(t0),v(t0))∈𝒰 for some t0≥0. Hence, (u(t),v(t))∈[E(u,v)<d] for all t≥t0. A contradiction.
Next, we will prove that∥u(t)∥2 is uniformly bounded for all t≥0.
Letℱ(t)≡(1/2)∥u(t)∥22-C, where C>0 is the constant given below. Then, we obtain
(3.53)F̈(t)+δḞ(t)=‖v(t)‖22-â(u(t))+b̂(u(t))=r+22‖v(t)‖22+r-22â(u(t))-rE(t)≥C(Ω)(r-2)F(t),
where C(Ω)>0 is the imbedding constant of B⊂L2(Ω), and C≡rE0/(r-2)C(Ω).
We define𝒲(t)≡ℱ+(t)≡sup{ℱ(t),0}, the positive part of ℱ(t). We claim that, along solutions of (1.1), the time derivative satisfies 𝒲̇(t)≤0. Indeed, if this is no the case, there exists some t0>0 such that
(3.54)F(t0)>0,Ḟ(t0)>0.
By a standard comparison result for ordinary differential equations, (3.53) and (3.54) imply that ℱ(t)→∞ as t→∞. Consequently, for any constant C>E0, there exists some t0>0, such that for t≥t0
(3.55)C<r-22râ(u(t)).
This is (2.29) in Lemma 2.3, replacing d by C. If we now define, for t≥t0, the function
(3.56)V(t)≡C-E(t),
we can repeat the sufficiency part of the proof of Theorem 3.1 and show that the solution blowups in a finite time, consequently is nonglobal. A contradiction. Then, ∥u(t)∥2≤C<∞, for all t∈ℝ+, and some constant C>0.
Next, we will prove that uniform boundedness of∥u(t)∥2 implies uniform boundedness of (u(t),v(t)) in H, for all t∈ℝ+. To that end, we consider the functions ℋ(t)≡𝒢(t)-kE0≡ℱ̇(t)+δℱ(t)-kE0, where now ℱ(t)≡(1/2)∥u(t)∥22 and k>0 is defined below. From the second line in (3.53),
(3.57)Ḣ(t)≥KH(t),
where K≡min{r+2,αC(Ω)(r-2)/(1+δ)}>0 and k≡r/K.
Hence, for0≤s≤τ,
(3.58)H(τ)≥H(s)e(K(τ-s)),
and consequently, from definition of ℋ(t), for 0≤s≤τ≤t,
(3.59)F(t)=F(s)e-δ(t-s)+∫st(H(τ)+kE0)eδ(τ-t)dτ≥F(s)e-δ(t-s)+∫st(H(s)eK(τ-s)+kE0)eδ(τ-t)dτ≥H(s)δ+K(eK(t-s)-eδ(s-t))+kE0δ(1-eδ(s-t)).
Notice that ifℋ(s)>0, for some s≥0, we obtain from (3.59) that limt→∞ℱ(t)=∞. A contradiction. Then, for all t≥0,
(3.60)G(t)≤kE0.
Now, we defineℒ(t)≡𝒢(t)+k̃E0, and like in (3.57)
(3.61)L̇(t)≥-K̃L(t),
where K̃≡min{r+2,C(Ω)(r-2)}>0 and k̃≡r/K̃. Hence,
(3.62)L(t)≥L(0)e-K̃t≥min⁡{L(0),0},
and consequently,
(3.63)G(t)≥min⁡{G(0),-k̃E0}.
Hence and from (3.60), 𝒢(t) is uniformly bounded in time.
We integrate the second line of (3.53) in terms of 𝒢(t) and, by the energy equation, we obtain
(3.64)G(t+1)-G(t)≥r-2(γ+1)2∫tt+1(‖v(τ)‖22+a(u(τ))+1γ+1c(u(τ)))dτ-rE0.
Hence and since𝒢(t) is uniformly bounded in time,
(3.65)∫tt+1ω(τ)dτ≤C,
where C>0 is a constant, and
(3.66)ω(t)≡12‖v(t)‖22+12a(u(t))+12(γ+1)c(u(t))=E(t)+1rb(u(t)).
Next, we will show that there exists a constantκ>0, such that
(3.67)ω(t)≤κ(ω(s)+1),
for any 0≤s≤t≤s+1.
To this end, we calculate(3.68)ω̇(t)=(v(t),f(u(t)))2-δ‖v(t)‖22≤μ‖v(t)‖2‖u(t)‖2(r-1)r-1.
If n=1, we integrate and obtain, for 0≤s≤t≤s+1, that
(3.69)ω(t)≤ω(s)+μ∫st‖v(τ)‖2‖u(τ)‖2(r-1)r-1dτ≤ω(s)+μ2∫st(‖v(τ)‖22+‖u(τ)‖2(r-1)2(r-1))dτ≤ω(s)+μ2∫ss+1‖u(τ)‖2(r-1)2(r-1)dτ+μ∫stω(τ)dτ.
By Gronwall inequality and (3.65),
(3.70)ω(t)≤(ω(s)+μ2‖u‖L2(r-1)(Ω×(s,s+1))2(r-1))eμ≤C̃{ω(s)+[∫ss+1(‖u(τ)‖22+α‖∇u(τ)‖22+‖v(τ)‖22)dτ]r-1}≤Ĉ{ω(s)+[∫ss+1ω(τ)dτ]r-1}≤Ĉ{ω(s)+Cr-1},
where C̃>0, Ĉ>0 depend on the continuous embeddings B⊂L2(Ω), and H1(Ω×(s,s+1))⊂L2(r-1)(Ω×(s,s+1)).
If2≤n≤5, we use Galiardo-Niremberg's inequality,
(3.71)‖u‖2(r-1)r-1≤C(Ω)‖u‖B(r-1)a‖u‖2(r-1)(1-a),
where C(Ω)>0, and a=n(r-2)/4(r-1). Notice that a<1 if 2≤n≤4, and a≤1 if n=5, because r≤2(n-2)/(n-4)=6. Then, from
(3.72)ω̇(t)≤μ‖v(t)‖2‖u(t)‖2(r-1)r-1,
we integrate and apply Gronwall inequality, for 0≤s≤t≤s+1,
(3.73)ω(t)≤ω(s)+μ∫st‖v(τ)‖2‖u(τ)‖2(r-1)r-1dτ≤ω(s)+μC(Ω)∫st‖v(τ)‖2‖u(τ)‖B(r-1)a‖u(τ)‖2(r-1)(1-a)dτ≤ω(s)+μC(Ω)∫stω(τ)‖u(τ)‖B(r-1)a-1‖u(τ)‖2(r-1)(1-a)dτ≤ω(s)exp⁡{μC(Ω)∫st‖u(τ)‖B(r-1)a-1‖u(τ)‖2(r-1)(1-a)dτ}≤ω(s)exp⁡{Ĉ∫st‖u(τ)‖B(r-1)a-1dτ},
where Ĉ≡μC(Ω)supt≥0∥u(t)∥2(r-1)(1-a).
Notice that(r-1)a-1≤2 because by hypothesis r≤2(6/n+1), then we use the Hölder inequality, and from (3.65) we get
(3.74)ω(t)≤ω(s)exp⁡{Ĉ(∫st‖u(τ)‖B2dτ){(r-1)a-1}/2}≤ω(s)exp⁡{C̃(∫ss+1ω(τ)dτ){(r-1)a-1}/2}≤ω(s)exp⁡{C̃C{(r-1)a-1}/2}.
Then (3.67) holds for any n≥1, under our assumptions on r.
Consequently, (3.65) and (3.67) imply that
(3.75)‖v(t)‖22+‖u(t)‖B2≤∫t-1t2ω(t)ds≤2κ∫t-1t(ω(s)+1)ds≤2κ(C+1),
and the proof is complete.Next, we consider the caseλ>2. Due to our assumptions on r, we restrict our analysis to nγ<4. Since γ≥1, our analysis considers, at most, dimensions n≤3, whenever γ<4/3.Theorem 3.6.
Let(u(t),v(t))=S(t)(u0,v0) be a solution of problem (1.1). Assume that  r>2(γ+1), and  λ>2 with  λ≤2(n+1)/(n-1) if  n≥2. Also assume that  r<2(4/n+1) for  n≥1. If  (u(t),v(t))∈[E(u,v)≥d] for all  t≥0, then the solution is global and uniformly bounded in  H, for all  t≥0.Proof.
Globality follows like in last Theorem. Suppose that(3.76)‖u(t)‖22≤K,
for some constant K>0, and every t≥0.
We recall Galiardo-Nirenberg's inequality(3.77)‖u‖rr≤C(Ω)a(u)ar/2‖u‖2r(1-a),
where C(Ω)>0 is a constant, a=(n/2)((r-2)/2r), and a∈(0,1].
Hence and from (3.76) in the energy equation, we obtain, for any time t≥0,
(3.78)E0≥12‖v(t)‖22+12a(u(t))-Ĉa(u(t))ar/2,
where Ĉ≡Kr(1-a)/2C(Ω)μ/r, and therefore (u(t),v(t)), t≥0, is uniformly bounded in H, since ar<2 if and only if r<2(4/n+1). This implies, since r>2(γ+1), that nγ<4.
Now, we prove (3.76). First, we notice that from energy equation,
(3.79)δ∫0t‖v(τ)‖λλdτ≤E0-d.
Hence, by Hölder inequality,(3.80)δt-(λ-1)‖∫0tv(τ)dτ‖λλ≤E0-d,
and then
(3.81)‖u(t)‖λ≤C(t),
for every t≥0, where C(t)≡∥u0∥λ+((E0-d)/δ)1/λt(λ-1)/λ.
Next, we define(3.82)2ω(t)≡‖(u(t),v(t))‖H2,
and obtain the following estimate for t∈(0,T), and T>0 finite and arbitrary:
(3.83)∫0tδ(u(τ),|v(τ)|λ-2v(τ))2dτ≤∫0tδ‖u(τ)‖λλ‖v(τ)‖λλ-1dτ≤(∫0t‖u(τ)‖λλdτ)1/λδ(∫0t‖v(τ)‖λλdτ)(λ-1)/λ≤C0‖u‖Lλ(Ω×(0,T))≤Ĉ(∫0t(‖u(τ)‖22+α‖∇u(τ)‖22+‖v(τ)‖22)dτ)1/2≤C̃(∫0tω(τ)dτ)1/2.
Here, we used the Hölder inequality, the energy equation and the fact thatE(t)≥d, also C0≡(E0-d)(λ-1)/λδ1/λ>0, and Ĉ>0, C̃>0 depend on the continuous embedding H1(Ω×(0,T))⊂Lλ(Ω×(0,T)), valid for λ>2 and λ≤2(n+1)/(n-1) if n≥2. C̃ also depends on the embedding B⊂L2(Ω).
Now, we define the function(3.84)F(t)≡12‖u(t)‖22-κ,
where κ>0 is the constant given below. Then, the second derivative is
(3.85)F̈(t)=‖v(t)‖22-a(u(t))-c(u(t))-δ(u(t),|v(t)|λ-2v(t))2+b(u(t))=r+22‖v(t)‖22+r-22a(u(t))+r-2(γ+1)2(γ+1)c(u(t))-δ(u(t),|v(t)|λ-2v(t))2-rE(t)≥(r-2)ω(t)-δ(u(t),|v(t)|λ-2v(t))2-rE0.
If we integrate (3.85), and we use (3.83) and that ω(t)≥E(t)≥d, we obtain
(3.86)Ḟ(t)≥Ḟ(0)+(r-2)∫0tω(τ)dτ-C̃(∫0tω(τ)dτ)1/2-rE0t≥Ḟ(0)+(r-2)∫0tω(τ)dτ{1-C̃(r-2)Tmd}-rE0t=Ḟ(0)+(r-2)2∫0tω(τ)dτ-rE0t≥Ḟ(0)+(r-2)C(Ω)2∫0tF(τ)dτ,
for t∈(Tm,T), where Tmd≡2C̃/(r-2)>0, C(Ω)>0 is the embedding constant of B⊂L2(Ω), and κ≡2rE0/(r-2)C(Ω)>0.
Now, we define for everyt≥0,
(3.87)G(t)≡∫0tF(τ)dτ+Ḟ(0)C,
where C≡(r-2)C(Ω)/2. Hence, (3.86) has the form
(3.88)G̈(t)-CG(t)≥0,
for every t∈(Tm,T).
We defineℋ(t)≡ℱ+(t)≡sup{ℱ(t),0}, the positive part of ℱ(t). We claim that, for every t>Tm, the time derivative satisfies ℋ̇(t)≤0. Otherwise, there exists some t0∈(Tm,T) such that
(3.89)F(t0)>0,Ḟ(t0)>0,
that is,
(3.90)Ġ(t0)>0,G̈(t0)>0.
By a standard comparison result for ordinary differential equations, (3.88) and (3.90) imply that
(3.91)F(t)=Ġ(t)≥M0eCt,
for t∈(t1,T) and some t1∈(t0,T), where M0>0 depends on ℱ(t0) and ℱ̇(t0). Furthermore, for any constant M>0, there exists some t2∈(t1,T), such that M0eCt>M(1+t2(λ-1)/λ), for t∈(t2,T). Then, (3.91) contradicts (3.81), since Lλ(Ω)⊂L2(Ω). Consequently, (3.76) holds with K≡max{max0≤t≤Tm∥u(t)∥22,2κ}, and the proof is complete.From Remark3.2 and Theorems 3.1, 3.3, and 3.5, any global solution of (1.1), with λ=2, is necessarily uniformly bounded in H, for all t≥0. However, this is false when λ>2, as we show next.Theorem 3.7.
Let(u(t),v(t))=S(t)(u0,v0) be a solution of problem (1.1). Assume the conditions, on  r and  λ, made in Theorem 3.6. Then,  (u(t),v(t)) is global and not uniformly bounded in  H, for  t≥0, if and only if  λ≥r and there exists  t0≥ 0 such that  (u(t0),v(t0))∈𝒰.Proof.
Sufficiency
By Lemma2.4, (u(t),v(t))∈𝒰, for every t≥t0. This solution must be global. Otherwise, by Theorem 2.1, blowups in a finite time and, by Theorem 3.1, necessarily λ<r. A contradiction.
Now, suppose that, fort≥t0, the solution is uniformly bounded in H. Hence, there exists a sequence of times {tn}, such that if n→∞, then tn→∞, (u(tn),v(tn))→(û,v̂) weakly in H and, since the imbedding B⊂H1(Ω)∩Lr(Ω) is compact, b̂(u(tn))→b̂(û). Moreover, since the energy is nonincreasing and bounded, E∞≡limt→∞E(t)∈ℝ. Consequently, from the energy equation and since Lλ(Ω)⊂L2(Ω),
(3.92)lim⁡t→∞∫tt+1‖v(τ)‖2λdτ=0,
in particular, for any sequence of times {sn} such that sn→∞ as n→∞,
(3.93)lim⁡n→∞∫01hn(τ)dτ=0,
where hn(τ)≡∥v(sn+τ)∥2λ, for τ∈[0,1]. By Fatou Lemma,
(3.94)liminf⁡n→∞‖v(sn+τ)‖2λ=liminf⁡n→∞hn(τ)=0,
for a.e. τ∈[0,1], and by the weak convergence v(tn)→v̂, in L2(Ω),
(3.95)‖v̂‖2≤liminf⁡n→∞‖v(tn)‖2=0,
where we choose {sn} such that tn=sn+τ0, for some τ0∈[0,1].
It can be shown that the semigroup generated by problem (1.1) is continuous in H with the weak topology, and then that the weak limit set is positive invariant; see Ball [26]. Consequently (û,v̂)=(ue,0) must be an equilibrium of (1.1). Since (u(tn),v(tn))∈𝒰 then, by definition (2.33) and (2.29),
(3.96)b̂(ue)=lim⁡n→∞b̂(u(tn))≥2rdr-2.
Since the norm inH is weak lower-semicontinuous, from (2.4), (2.19), and (2.20), we get that
(3.97)b̂(ue)=â(ue)≤liminf⁡n→∞{‖v(tn)‖22+â(u(tn))}=lim⁡n→∞{2E(tn)+2rb̂(u(tn))}=2E∞+2rb̂(ue).
Hence,(3.98)r-22rb̂(ue)≤E∞<d.
This contradicts (3.96). Then, the solution cannot be uniformly bounded in H, for all t≥0.
Necessity
This follows from Lemma2.6 and Theorems 3.1, 3.3, and 3.6. The proof is complete.Next, we characterize the convergence to the set of nonzero equilibria of equation (1.1). Due to our assumptions on r and γ, our result considers, at most, dimensions n≤3 for λ>2, and n≤5 for λ=2.Theorem 3.8.
Let(u(t),v(t))=S(t)(u0,v0) be a solution of problem (1.1). If  λ>2, assume the conditions, on  r and  λ, made in Theorem 3.6. If  λ=2, and assume the conditions, on  r made in Theorem 3.5. Then,  (u(t),v(t))→ℰ∞, strongly in  H as  t→∞, if and only if  (u(t),v(t))∈[E(u,v)≥d] for all  t≥0, where  ℰ∞≡[(ue,0)∈ℰ:J(ue)=E∞≥d] and  E∞≡limt→∞E(u(t),v(t)).Proof.
Sufficiency
By Theorems3.5 and 3.6, the solution is global and uniformly bounded in H, that is, ω(t)≡∥(u(t),v(t))∥H2≤K, for all t≥0, and some constant K>0. Then, like in the sufficiency part of the proof of Theorem 3.7, there exists a sequence of times such that tn→∞ and (u(tn),v(tn))→(ue,0), weakly in H, as n→∞, where (ue,0) is an equilibrium. If
(3.99){(u(t),v(t))}t≥0isprecompactinH,
then strong convergence to (ue,0) follows. In this situation, (u(tn),v(tn)) converges to the set ℰ∞, because (0,0)∉[E(u,v)≥d]¯, strong closure in H.
Haraux [5], developed a technique to prove precompactness of bounded orbits of some kind of semilinear wave equations. We will follow that method to prove (3.99). To this end we define, for every ϵ>0, and t≥0,
(3.100)uϵ(t)≡u(t+ϵ)-u(t),vϵ(t)≡v(t+ϵ)-v(t),2ωϵ(t)≡‖(uϵ(t),vϵ(t))‖H2,
and note that from (2.1), we get the energy equation for (uϵ(t),vϵ(t))
(3.101)ωϵ(0)=ωϵ(t)+∫0t(gϵ(τ)-fϵ(τ)-m̂ϵ(τ),vϵ(τ))2dτ,
where,
(3.102)gϵ(t)≡g(v(t+ϵ))-g(v(t)),fϵ(t)≡f(u(t+ϵ))-f(u(t)),m̂ϵ(t)≡m(‖u(t+ϵ)‖22)Δu(t+ϵ)-m(‖u(t)‖22)Δu(t),m(‖∇u(t)‖22)≡β‖∇u(t)‖22γ.
However, in order to handle the nonlinearitym̂ϵ(t), we need to introduce the function
(3.103)Wϵ(t)≡ωϵ(t)+12m(‖∇u(t+ϵ)‖22)‖∇uϵ(t)‖22.
Hence, the corresponding energy equation forWϵ(t) is
(3.104)Wϵ(0)=Wϵ(t)+∫0t(gϵ(τ)-fϵ(τ)-mϵ(τ),vϵ(τ))2dτ-∫0t(nϵ(τ),uϵ(τ))2dτ,
where,
(3.105)mϵ(t)≡(m(‖∇u(t+ϵ)‖22)-m(‖∇u(t)‖22))Δu(t),nϵ(t)≡m′(‖∇u(t+ϵ)‖22)(Δu(t+ϵ),v(t+ϵ))2Δuϵ(t).
Notice that since the solution is uniformly bounded byK, there exists a constant K̂>0, depending on K, such that
(3.106)ωϵ(t)≤Wϵ(t)≤K̂ωϵ(t).
We will prove that for anyη>0, there exists ϵ(η)>0, such that
(3.107)ωϵ(t)≤η,
for every t≥0, and ϵ∈(0,ϵ(η)), that is, t↦(u(t),v(t))∈H, is uniformly continuous.
For everyt≥0, we have one of the following two cases:
(3.108)Wϵ(t+1)≤Wϵ(t),(3.109)Wϵ(t+1)>Wϵ(t).
From (3.109) and (3.104),
(3.110)0>Wϵ(t)-Wϵ(t+1)=∫tt+1(gϵ(τ)-fϵ(τ)-mϵ(τ),vϵ(τ))2dτ-∫tt+1(nϵ(τ),uϵ(τ))2dτ.
Notice that, by a well-known inequality,(3.111)(gϵ(t),vϵ(t))2≥22-λδ‖vϵ(t)‖λλ.
Also, we recall the inequality(3.112)|fϵ(t)|≤σ(r)μ(|u(t+ϵ)|r-2+|u(t)|r-2)|uϵ(t)|,
where σ(r)=1, if r∈[2,3] and σ(r)=(r-1)/2, if r>3.
From the Hölder inequality, (3.112) yields
(3.113)(∫tt+1‖fϵ(τ)‖22dτ)1/2≤2σ(r)μC(Ω)sup⁡t≥0a(u(t))(r-2)/2×(∫tt+1‖uϵ(τ)‖2(r-1)2(r-1)dτ)1/2(r-1),
where C(Ω)>0, is an embedding constant in B⊂L2(r-1)(Ω).
We claim thatt↦u(t)∈L2(r-1)(Ω) must be uniformly continuous. Otherwise, there exists some η0>0, and sequences {ϵn}n≥1, {tn}n≥1, such that ϵn→0, and tn→∞, as n→∞, and
(3.114)‖uϵn(tn)‖2(r-1)>η0,
for every n≥1. By assumption, B⊂L2(r-1)(Ω) is compact, then {u(tn+ϵn)}n≥1, {u(tn)}n≥1 are precompact in L2(r-1)(Ω), and we can extract subsequences {u(tn′+ϵn′)}n≥1, {u(tn′)}n≥1, such that for some fixed n0, sufficiently big, and every n≥n0,
(3.115)‖u(tn′+ϵn′)-u(tn′)‖2(r-1)≤‖u(tn′+ϵn′)-u(tn0+ϵn0)‖2(r-1)+‖u(tn0+ϵn0)-u(tn0)‖2(r-1)+‖u(tn0)-u(tn′)‖2(r-1)≤η03+η03+η03=η0.
This contradicts (3.114). Hence, for any η>0 there exists some ϵ̂(η)>0, such that for every t≥0, and every ϵ∈(0,ϵ̂(η)),
(3.116)(∫tt+1‖uϵ(τ)‖2(r-1)2(r-1)dτ)1/2(r-1)≤η4(λ-1).
Consequently, from (3.113), (3.116) and the Hölder inequality
(3.117)|∫tt+1(fϵ(τ),vϵ(τ))2dτ|≤Cη4(λ-1)(∫tt+1‖vϵ(τ)‖λλdτ)1/λ,
where C>0 depends on K, μ, r, C(Ω) and the inclusion Lλ(Ω)⊂L2(Ω).
Now notice that,(3.118)‖mϵ(t)‖2≤sup⁡t≥0{m′(‖∇u(t)‖22)}|‖∇u(t+ϵ)‖22-‖∇u(t)‖22|‖Δu(t)‖2=sup⁡t≥0{m′(‖∇u(t)‖22)}|(∇u(t+ϵ)+∇u(t),∇u(t+ϵ)-∇u(t))2|‖Δu(t)‖2≤sup⁡t≥0{m′(‖∇u(t)‖22)‖Δu(t+ϵ)+Δu(t)‖2‖Δu(t)‖2}‖uϵ(t)‖2≤C(K)‖uϵ(t)‖2,
and that
(3.119)|(nϵ(t),uϵ(t))2|≤m′(‖∇u(t+ϵ)‖22)‖Δu(t+ϵ)‖2‖v(t+ϵ)‖2‖Δuϵ(t)‖2‖uϵ(t)‖2≤C(K)‖uϵ(t)‖2,
where C(K)>0 is a constant.
SinceB⊂L2(Ω) is compact, we show like in (3.116), that
(3.120)(∫tt+1‖uϵ(τ)‖22dτ)1/2≤η4(λ-1).
Consequently, from (3.118)–(3.120) and the Hölder inequality, we obtain
(3.121)|∫tt+1{(mϵ(τ),vϵ(τ))2+(nϵ(τ),uϵ(τ))2}dτ|≤Ĉη4(λ-1){(∫tt+1‖vϵ(τ)‖λλdτ)1/λ+1},
where Ĉ>0 depends on K and the inclusion Lλ(Ω)⊂L2(Ω).
From (3.117), (3.121), and (3.111) in (3.110), we have
(3.122)∫tt+1‖vϵ(τ)‖λλdτ≤C̃η4(λ-1){(∫tt+1‖vϵ(τ)‖λλdτ)1/λ+1},
where C̃>0 is a constant. Consequently, for η sufficiently small, we obtain
(3.123)∫tt+1‖vϵ(τ)‖λλdτ≤η3λ,(3.124)∫tt+1‖vϵ(τ)‖22dτ≤η25.
We apply inequality (3.112) to gϵ, and by the Hölder inequality we get
(3.125)∫tt+1(gϵ(τ),uϵ(τ))2dτ≤σ(λ)δ(∫tt+1‖vϵ(τ)‖λλdτ)1/λ(∫tt+1‖uϵ(τ)‖λλdτ)1/λ×[(∫tt+1‖v(τ+ϵ)‖λλdτ)(λ-2)/λ+(∫tt+1‖v(τ)‖λλdτ)(λ-2)/λ].
Notice that by (2.3) and since E(t)≥d,
(3.126)∫0∞‖v(t)‖λλdt≤E0-dδ.
By assumptionB⊂Lλ(Ω), then
(3.127)(∫tt+1‖uϵ(τ)‖λλdτ)1/λ≤2C(Ω)sup⁡t≥0‖u(t)‖B≤C,
where C>0 depends on the embedding constant C(Ω) and K.
Therefore, from (3.123), (3.126), and (3.127) in (3.125), we get, for η sufficiently small
(3.128)|∫tt+1(gϵ(τ),uϵ(τ))2dτ|≤2σ(λ)δC(E0-dδ)(λ-2)/λη3≤η25.
By (3.113), (3.116), (3.120) and the Hölder inequality, for small η, we obtain
(3.129)∫tt+1(fϵ(τ),uϵ(τ))2dτ≤Cη8(λ-1)≤η25.
one has that(3.130)(m̂ϵ(t),uϵ(t))2=-m(‖∇u(t+ϵ)‖22)‖∇uϵ(t)‖22+{m(‖∇u(t+ϵ)‖22)-m(‖∇u(t)‖22)}(Δu(t),uϵ(t))2≤C(K)‖uϵ(t)‖2,
where C(K) depends on K. Hence, by (3.120) and the Hölder inequality, and again for small η,
(3.131)∫tt+1(m̂ϵ(τ),uϵ(τ))2dτ≤η25.
From (2.1), one has the identity
(3.132)ddt(vϵ(t),u(t)ϵ)2-‖vϵ(t)‖22+‖Δuϵ(t)‖22+α‖∇uϵ(t)‖22+(gϵ(t),uϵ(t))2=(m̂ϵ(t),uϵ(t))2+(fϵ(t),uϵ(t))2.
Hence, (3.124), (3.128), (3.129), and (3.131) in (3.132) yield
(3.133)∫tt+1‖uϵ(τ)‖B2dτ≤2sup⁡t≥0{‖vϵ(t)‖2‖uϵ(t)‖2}+4η25.
For everyt≥0, one has
(3.134)‖uϵ(t)‖2≤ϵsup⁡s∈[t,t+1]‖v(s)‖2≤ϵ2K,‖vϵ(t)‖2≤2sup⁡t≥0‖v(t)‖2≤22K
then, for ϵ∈(0,ϵ̂(η)), with ϵ̂(η) sufficiently small, (3.133) is
(3.135)∫tt+1‖uϵ(τ)‖B2dτ≤η2.
Hence, in case (3.109), from (3.124) and (3.135), we conclude that
(3.136)∫tt+1ωϵ(τ)dτ≤3η25.
And, from (3.106) with η small,
(3.137)∫tt+1Wϵ(τ)dτ≤3η5.
From (3.104), (3.111), (3.117), (3.121) and (3.124), we have, for any s∈[t,t+1], t≥0, and η sufficiently small, that
(3.138)Wϵ(t+1)=Wϵ(s)+∫st+1(mϵ(τ)+fϵ(τ)-gϵ(τ),vϵ(τ))2dτ+∫0t(nϵ(τ),uϵ(τ))2dτ≤Wϵ(s)+|∫tt+1(fϵ(τ),vϵ(τ))2dτ|+|∫tt+1(mϵ(τ),vϵ(τ))2dτ|+|∫tt+1(nϵ(τ),uϵ(τ))2dτ|≤Wϵ(s)+2η5.
Therefore, by (3.137), we obtain
(3.139)Wϵ(t+1)≤∫tt+1Wϵ(s)ds+2η5≤η.
Consequently, in both cases, (3.108) and (3.109),
(3.140)Wϵ(t+1)≤max⁡{η,Wϵ(t)},
and then
(3.141)Wϵ(t)≤max⁡{η,max⁡s∈[0,1]Wϵ(s)},
for any t≥0.
Since the solution(u,v):[0,1]→H, is uniformly continuous, for any η>0 there exists some ϵ̃(η)>0, such that
(3.142)max⁡s∈[0,1]Wϵ(s)≤η,
for any ϵ∈(0,ϵ̃(η)). Then, (3.141) and (3.106) imply (3.107) for any t≥0, η>0, and 0<ϵ<ϵ(η)≡min{ϵ̂(η),ϵ̃(η)}.
Next, we will prove that the orbit{(u(t),v(t))}t≥0, is a precompact subset of H. We start with {v(t)}t≥0⊂L2(Ω).
Notice that because of (3.107)
(3.143)‖v(t)-1ϵ∫tt+ϵv(τ)dτ‖2≤1ϵ∫tt+ϵ‖v(t)-v(τ)‖2dτ≤sup⁡τ∈[t,t+ϵ]‖v(t)-v(τ)‖2≤2η.
Since{u(t)}t≥0 is bounded in B, then
(3.144)‖1ϵ(∫tt+ϵv(τ)dτ)‖B≤1ϵ‖u(t+ϵ)-u(t)‖B≤2ϵsup⁡t≥0‖u(t)‖B≤2ϵ2K.
Consequently,{(1/ϵ)∫tt+ϵv(τ)dτ}t≥0 is precompact or, equivalently, totally bounded in L2(Ω), because B⊂L2(Ω) is compact. Hence, by (3.143), {v(t)}t≥0 is precompact in L2(Ω).
Like in (3.143), from (3.107), we obtain that
(3.145)‖u(t)-1ϵ∫tt+ϵu(τ)dτ‖B≤sup⁡τ∈[t,t+ϵ]‖u(t)-u(τ)‖B≤2η.
If{Δ2∫tt+ϵu(τ)dτ}t≥0 is precompact in B′≡ dual space of B, then precompactness of {u(t)}t≥0 in B follows from (3.145), since
(3.146)L-1≡(Δ2)-1:B′⟶B,
is a linear and continuous operator.
According to the dense and continuous inclusions(3.147)B⊂Lλ(Ω)⊂L2(Ω)⊂Lλ*(Ω)⊂B′,
where Lλ*(Ω)=(Lλ(Ω))′, λ*=λ/(λ-1), we extend the inner product in L2(Ω) to the duality product in B′×B. Now, we integrate the wave equation, and since ℒ:B→B′ is closed, we get, in the sense of B′,
(3.148)Δ2∫tt+ϵu(τ)dτ=v(t)-v(t+ϵ)+∫tt+ϵM(‖∇u(τ)‖22)Δu(τ)dτ-∫tt+ϵg(v(τ))dτ+∫tt+ϵf(u(τ))dτ.
By the Hölder inequality and (3.126),
(3.149)‖∫tt+ϵg(v(τ))dτ‖λ*≤δ∫tt+ϵ‖(v(τ))‖λλ-1dτ≤δ1/λ(E0-d)(λ-1)/λ.
Boundedness ofu(t) in B, since B⊂L2(r-1)(Ω), yields the estimate
(3.150)‖∫tt+ϵf(u(τ))dτ‖2≤μ∫tt+ϵ‖(u(τ))‖2(r-1)r-1dτ≤μC(Ω)sup⁡t≥0‖u(t)‖B≤μC(Ω)C(K).
Also,(3.151)‖∫tt+ϵM(‖∇u(τ)‖22)Δu(τ)dτ‖2≤sup⁡t≥0{M(‖∇u(t)‖22)‖u(t)‖B}≤C(K),
Therefore, (3.149)–(3.151) in (3.148) imply that
(3.152)‖Δ2∫tt+ϵu(τ)dτ‖λ*≤C,
for some constant C>0 and every t≥0.
B⊂Lλ(Ω) is compact by assumption. By Schauder Theorem, see for instance Brézis [27], B⊂Lλ(Ω) is compact if and only if Lλ*(Ω)⊂B′ is compact. Then, (3.152) implies that {Δ2∫tt+ϵu(τ)dτ}t≥0 is precompact in B′.
Necessity
Suppose that the solution converges, strongly inH, to the set ℰ∞. Since the energy is nonincreasing, E(t)≥E∞≡limt→∞E(t)≥d, for all t≥0, and the proof is complete.Remark 3.9.
By Theorem3.3 and (2.35), every H-neighborhood of the ground state, 𝒮¯∩𝒰¯=[(ue,0)∈H:ue∈𝒩*], is connected, through an orbit, with (0,0). Furthermore, if E∞=d in Theorem 3.8, the ground state attracts every solution such that (u(t),v(t))∈[E(u,v)≥d], for every t≥0.



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