# Existence and Decay Estimate of Global Solutions to Systems of Nonlinear Wave Equations with Damping and Source Terms

Authors: Yaojun Ye
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The initial-boundary value problem for a class of nonlinear wave equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set and obtain the asymptotic stability of global solutions through the use of a difference inequality.


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## Body

## 1. Introduction

In this paper, we are concerned with the global solvability and decay stabilization for the following nonlinear wave equations system:(1)utt-div(|∇u|p-2∇u)+|ut|q-2ut-Δut=|v|r+2|u|ru,(x,t)∈Ω×R+,(2)vtt-div(|∇v|p-2∇v)+|vt|q-2vt-Δvt=|u|r+2|v|rv,(x,t)∈Ω×R+
with the initial-boundary value conditions
(3)u(x,0)=u0(x)∈W01,p(Ω),ut(x,0)=u1(x)∈L2(Ω)x∈Ω,(4)v(x,0)=v0(x)∈W01,p(Ω),vt(x,0)=v1(x)∈L2(Ω)x∈Ω,(5)u(x,t)=0,v(x,t)=0,(x,t)∈∂Ω×R+,
where Ω is a bounded open domain in Rn with a smooth boundary ∂Ω, p,q≥2, r>0 and p<2(r+2)≤np/(n-p) for n≥p and p<2(r+2)<+∞ for n<p.Whenp=2, Medeiros and Miranda [1] proved the existence and uniqueness of global weak solutions. Cavalcanti et al. in [2–4] considered the asymptotic behavior for wave equation and an analogous hyperbolic-parabolic system with boundary damping and boundary source term. In paper [5, 6], the authors dealt with the existence, uniform decay rates, and blowup for solutions of systems of nonlinear wave equations with damping and source terms.Rammaha and Wilstein [7] and Yang [8] are concerned with the initial boundary value problem for a class of quasilinear evolution equations with nonlinear damping and source terms. Under appropriate conditions, by a Galerkin approximation scheme combined with the potential well method, they proved the existence and asymptotic behavior of global weak solutions when m<p, where m≥0 and p are, respectively, the growth orders of the nonlinear strain terms and the source term.Ono [9] considers the following initial-boundary value problem for nonlinear wave equations with nonlinear dissipative terms:
(6)utt-Δu+δ1ut+δ2|ut|βut-δ3Δut=|u|αu,(x,t)∈Ω×R+,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(x,t)=0,x∈∂Ω,t≥0,
where δi≥0, i=1,2,3, and α,β>0 are constants. The author mainly investigates on the blowup phenomenon to problem (6). On the other hand, in the case of δ1+δ2+δ3>0, he shows that the problem (6) admits a unique global solution, and its energy has some decay properties under some assumptions on u0 and initial energy E(0)≡E(u0,u1). In particular, when δ2>0 and δ1+δ3>0 in (6), the energy E(t)≡E(u(t),ut(t)) has some polynomial and exponential decay rates, respectively.For the following strongly damped nonlinear wave equation(7)utt-Δut-Δu+f(ut)+g(u)=h,
Dell’Oro and Pata [10] obtain the long-time behavior of the related solution semigroup, which is shown to possess the global attractor in the natural weak energy space. In addition, the existence of global and local solutions, decay estimates, and blowup for solutions of nonlinear wave equation with source and damping terms and exponential nonlinearities are studied in [11–14].In this paper, we prove the global existence for the problem (1)–(5) by applying the potential well theory introduced by Sattinger [15] and Payne and Sattinger [16]. Meanwhile, we obtain the asymptotic stabilization of global solutions by using a difference inequality [17].For simplicity of notations, hereafter we denote by∥·∥p the norm of Lp(Ω); ∥·∥ denotes L2(Ω) norm, and we write equivalent norm ∥·∇∥p instead of W01,p(Ω) norm ∥·∥W01,p(Ω). Moreover, C denotes various positive constants depending on the known constants and may be different at each appearance.


## 2. Local Existence

In this section, we investigate the local existence and uniqueness of the solutions of the problem (1)–(5). For this purpose, we list up two useful lemmas which will be used later and give the definition of weak solutions.Lemma 1.
Letu∈W01,p(Ω), then u∈Ls(Ω); and the inequality ∥u∥s≤C∥u∥W01,p(Ω) holds with a constant C>0 depending on Ω, p, and s, provided that 2≤s<+∞, 2≤n≤p and 2≤s≤np/(n-p), 2<p<n.Lemma 2 (Young inequality).
Leta,b≥0 and 1/p+1/q=1 for 1<p, q<+∞; then one has the inequality
(8)ab≤δap+C(δ)bq,
where δ>0 is an arbitrary constant, and C(δ) is a positive constant depending on δ.Definition 3.
A pair of functions(u,v) is said to be a weak solution of (1)–(5) on [0,T] if u,v∈C([0,T],W01,p(Ω)), ut,vt∈C([0,T],L2(Ω)), [u(0),v(0)]=[u0,v0]∈W01,p(Ω)×W01,p(Ω), [ut(0),vt(0)]=[u1,v1]∈L2(Ω)×L2(Ω), and [u,v] satisfies
(9)〈ut(t),ϕ〉L2(Ω)-〈u1,ϕ〉L2(Ω)+∫0t‍〈(|∇u|p-2∇u),∇ϕ〉L2(Ω)dτ+∫0t‍〈|ut|q-2ut,ϕ〉L2(Ω)dτ+∫0t‍〈∇ut,∇ϕ〉L2(Ω)=∫0t‍〈|v|r+2|u|ru,ϕ〉L2(Ω)dτ,〈vt(t),ψ〉L2(Ω)-〈v1,ψ〉L2(Ω)+∫0t‍〈(|∇v|p-2∇v),∇ψ〉L2(Ω)dτ+∫0t‍〈|vt|q-2vt,ψ〉L2(Ω)dτ+∫0t〈∇vt,∇ψ〉L2(Ω)‍=∫0t〈|u|r+2|v|rv,ψ〉L2(Ω)‍dτ,
for all test functions ϕ,ψ∈W01,p(Ω) and for almost all t∈[0,T].The local existence and uniqueness of solutions for problem (1)–(5) can be proved through the use of Galerkin method. The result reads as follows.Theorem 4 (local solution).
Supposed that[u0,v0]∈W01,p(Ω)×W01,p(Ω), [u1,v1]∈L2(Ω)×L2(Ω), and p<2(r+2)≤np/(n-p) if n≥p and p<2(r+2)<+∞ for n<p, then there exists T>0 such that the problem (1)–(5) has a unique local solution [u(t),v(t)] satisfying
(10)[u,v]∈L∞([0,T);W01,p(Ω)×W01,p(Ω));[ut,vt]∈L∞([0,T);L2(Ω)×L2(Ω)),(11)E(t)+∫0t‍(∥∇uτ(τ)∥2+∥∇vτ(τ)∥2+∥u(τ)∥qq+∥v(τ)∥qq)dτ=E(0),
where
(12)E(t)=12(∥ut∥2+∥vt∥2)+1p(∥∇u∥pp+∥∇v∥pp)-1r+2∥uv∥r+2r+2.Proof.
Let{ωi}i=1∞ be a basis for W01,p(Ω). Supposed that Vk is the subspace of W01,p(Ω) generated by {ω1,ω2,…,ωk},k∈N. We are going to look for the approximate solution
(13)uk(t)=∑i=1k‍gik(t)ωi,vk(t)=∑i=1k‍hik(t)ωi
which satisfies the following Cauchy problem:
(14)∫Ω‍(uk′′-div(|∇uk|p-2∇uk)+|uk′|q-2uk′-Δuk′)ωidx=∫Ω‍|vk|r+2|uk|rukωidx,(15)∫Ω‍(vk′′-div(|∇vk|p-2∇vk)+|vk′|q-2vk′-Δvk′)ωidx=∫Ω‍|uk|r+2|vk|rvkωidx,(16)uk(0)=u0k=∑i=1k‍(u0,ωi)ωi⟶u0,inW01,p(Ω),k⟶∞,(17)vk(0)=v0k=∑i=1k‍(v0,ωi)ωi⟶v0inW01,p(Ω),k⟶∞,(18)uk′(0)=u1k=∑i=1k‍(u1,ωi)ωi⟶u1inL2(Ω),k⟶∞,(19)vk′(0)=v1k=∑i=1k‍(v1,ωi)ωi⟶v1inL2(Ω),k⟶∞.
Note that, we can solve the problem (14)–(19) by a Picard’s iteration method in ordinary differential equations. Hence, there exists a solution in [0,Tk) for some Tk>0, and we can extend this solution to the whole interval [0,T] for any given T>0 by making use of the a priori estimates below.
Multiplying (14) by gik′(t) and (15) by hik′(t) and summing over i from 1 to k, we obtain
(20)12ddt(∥uk′(t)∥2+∥∇uk∥pp)+∥uk′(t)∥qq+∥∇uk′(t)∥2=∫Ω‍|vk|r+2|uk|rukuk′dx,(21)12ddt(∥vk′(t)∥2+∥∇vk∥pp)+∥vk′(t)∥qq+∥∇vk′(t)∥2=∫Ω‍|uk|r+2|vk|rvkvk′dx.
By summing (20) and (21) and integrating the resulting identity over [0,t], we have
(22)12(∥uk′(t)∥2+∥vk′(t)∥2+∥∇uk∥pp+∥∇vk∥pp)+∫0t‍(∥∇uk′(t)∥2+∥∇vk′(t)∥2+∥uk′(τ)∥qq+∥vk′(τ)∥qq)dτ≤C0+∫0t‍∫Ω‍(|vk|r+2|uk|rukuk′+|uk|r+2|vk|rvkvk′)dxdτ.
We estimate the right-hand terms of (22) as follows: we get from Hölder inequality and Lemmas 1 and 2 that
(23)|∫0t‍∫Ω‍(|vk|r+2|uk|rukuk′+|uk|r+2|vk|rvkvk′)dxdτ|≤∫0t‍(∥uk′(τ)∥2+∥vk′(τ)∥2)dτ+∫0t‍∫Ω‍|ukvk|2(r+1)(|uk|2+|vk|2)dxdτ≤∫0t‍(∥uk′(τ)∥2+∥vk′(τ)∥2)dτ+C∫0t‍(∥uk∥2(r+2)2(r+2)+∥vk∥2(r+2)2(r+2))dτ≤C∫0t‍(∥∇vk∥p2(r+2)∥uk′(τ)∥2+∥vk′(τ)∥2+∥∇uk∥p2(r+2)+∥∇vk∥p2(r+2))dτ≤C∫0t‍(∥∇vk∥pp∥∇uk∥pp∥uk′(τ)∥2+∥vk′(τ)∥2+∥∇uk∥pp+∥∇vk∥pp∥uk′(τ)∥2)2(r+2)/pdτ.
It follows from (22) and (23) that
(24)∥uk′(t)∥2+∥vk′(t)∥2+∥∇uk∥pp+∥∇vk∥pp+2∫0t‍(∥uk′(τ)∥qq∥+∥vk′(τ)∥qq+∥∇uk′(t)∥2+∥∇vk′(t)∥2)dτ≤2C0+C∫0t(∥∇vk∥pp∥uk′(τ)∥2+∥vk′(τ)∥2+∥∇uk∥pp+∥∇vk∥pp∥uk′(τ)∥2)2(r+2)/pdτ,
which implies that
(25)∥uk′(t)∥2+∥vk′(t)∥2+∥∇uk∥pp+∥∇vk∥pp≤2C0+C∫0t‍(∥∇vk∥pp∥∇uk∥pp∥uk′(τ)∥2+∥vk′(τ)∥2+∥∇uk∥pp+∥∇vk∥pp∥uk′(τ)∥2)2(r+2)/pdτ.
We get from (25) and Gronwall type inequality that
(26)∥uk′(t)∥2+∥vk′(t)∥2+∥∇uk∥pp+∥∇vk∥pp≤[2C0-2(r+2)-ppCt]-p/(2(r+2)-p).
Thus, we deduce from (26) that there exists a time T>0 such that
(27)∥uk′(t)∥2+∥vk′(t)∥2+∥∇uk∥pp+∥∇vk∥pp≤C1,∀t∈[0,T],
where C1 is a positive constant independent of k.
We have from (24) and (26) that
(28)2∫0t‍(∥uk′(τ)∥qq+∥vk′(τ)∥qq+∥∇uk′(τ)∥2+∥∇vk′(τ)∥2∥uk′(τ)∥qq)dτ≤C2,∀t∈[0,T].
It follows from (27) and (28) that
(29)∥uk′(t)∥2≤C1,∥vk′(t)∥2≤C1,∥∇uk∥pp≤C1,∥∇vk∥pp≤C1.uk′(t)andvk′(t)areboundedinL2([0,T];Lq(Ω))andL2([0,T];H01(Ω)).
Using the same process as the proof of Theorem 2.1 in paper [18], we derive that [u(t),v(t)] is a local solution of the problem (1)–(5). By (20) and (21), we conclude that (11) is valid.


## 3. Global Existence

In order to state our main results, we first introduce the following functionals:(30)J([u,v])=1p(∥∇u∥pp+∥∇v∥pp)-1r+2∥uv∥r+2r+2,(31)K([u,v])=(∥∇u∥pp+∥∇v∥pp)-2∥uv∥r+2r+2
for [u,v]∈W01,p(Ω)×W01,p(Ω).We put that(32)d=inf{supλ≥0J(λ[u,v]):[u,v]∈W01,p(Ω)×W01,p(Ω)/{[0,0]}supλ≥0}.
Then, we are able to define the stable set as follows for problem (1)–(5):
(33)W={[u,v]∈W01,p(Ω)×W01,p(Ω)∣K([u,v])>0,J([u,v])<dW01,p}∪{[0,0]}.
We denote the total energy related to (1) and (2) by (12), and
(34)E(0)=12(∥u1∥2+∥v1∥2)+1p(∥∇u0∥pp+∥∇v0∥pp)-1r+2∥u0v0∥r+2r+2
is the total energy of the initial data.Lemma 5.
Let[u,v] be a solution to problem (1)–(5); then, E(t) is a nonincreasing function for t>0 and
(35)ddtE(t)=-(∥ut∥qq+∥vt∥qq+∥∇ut∥22+∥∇vt∥22).We have from (11) that E(t) is the primitive of an integrable function. Therefore, E(t) is absolutely continuous, and equality (35) is satisfied.Lemma 6.
Supposed that[u,v]∈W01,p(Ω)×W01,p(Ω), and p<2(r+2)≤np/(n-p) if n≥p; p<2(r+2)<+∞ if n<p, then d>0.Proof.
Since(36)J(λ[u,v])=λpp(∥∇u∥pp+∥∇v∥pp)-λ2(r+2)r+2∥uv∥r+2r+2,
so we get
(37)ddλJ(λ[u,v])=λp-1(∥∇u∥pp+∥∇v∥pp)-2λ2r+3∥uv∥r+2r+2.
In case uv≠0, let (d/dλ)J(λ[u,v])=0, which implies that
(38)λ1=(∥∇u∥pp+∥∇v∥pp2∥uv∥r+2r+2)1/(2r-p+4).
Asλ=λ1, an elementary calculation shows that (d2/dλ2)J(λ[u,v])|λ=λ1<0. Therefore, we have that
(39)supλ≥0J(λ[u,v])=J(λ1[u,v])=2(r+2)-p2p(r+2)(∥∇u∥pp+∥∇v∥pp2p/(2r+4)∥uv∥r+2p/2)(2r+4)/(2r-p+4).
It follows from Hölder inequality and Lemma 1 that
(40)∥uv∥r+2p/2≤∥u∥2(r+2)p/2∥v∥2(r+2)p/2≤12(∥u∥2(r+2)p+∥v∥2(r+2)p)≤C(∥∇u∥pp+∥∇v∥pp).
We get from (39) and (40) that
(41)supλ≥0J(λ[u,v])≥2(r+2)-p2p(r+2)(2p/(2r+4)C)-(2r+4)/(2r-p+4)>0.
In case uv=0 and u=0 or v=0, then
(42)J(λ[u,v])=λpp(∥∇u∥pp+∥∇v∥pp).
Therefore, we have
(43)J(λ[u,v])=+∞.
We conclude from (41) and (43) that
(44)d≥2(r+2)-p2p(r+2)(2p/(2r+4)C)-(2r+4)/(2r-p+4)>0.
Thus, we complete the proof of Lemma 6.Lemma 7.
Supposed thatp<2(r+2)≤np/(n-p) for n≥p and p<2(r+2)<+∞ for n<p, if [u0,v0]∈W, [u1,v1]∈L2(Ω)×L2(Ω) and E(0)<d, then [u,v]∈W for ∀t∈[0,T).Proof.
Assume that there exists a numbert*∈(0,T) such that [u(t),v(t)]∈W on [0,t*) and u(t*)∉W. Then, in virtue of the continuity of u(t), we see u(t*)∈∂W, where ∂W denotes the boundary of domain W. From the definition of W and the continuity of J([u(t),v(t)]) and K([u(t),v(t)]) in t, we have either
(45)J([u(t*),v(t*)])=d
or
(46)K([u(t*),v(t*)])=0.
It follows from (12) and (30) that
(47)J([u(t*),v(t*)])≤E(t*)≤E(0)<d.
So, case (45) is impossible.
Assume that (46) holds; then, we get that
(48)ddλJ(λ[u(t*),v(t*)])=λp-1(1-λ2r-p+4)(∥∇u∥pp+∥∇v∥pp).
We obtain from (d/dλ)J(λ[u(t*),v(t*)])=0 that λ=1. Since
(49)d2dλ2J(λ[u(t*),v(t*)])|λ=1=-[(2(r+2)-p)+(2r+3)]<0.
Consequently, we get from (47) that
(50)supλ≥0J(λ[u(t*),v(t*)])=J([u(t*),v(t*)])<d,
which contradicts the definition of d. Hence, case (46) is impossible as well. Thus we conclude that [u(t),v(t)]∈W on [0,T).Theorem 8 (global solution).
Supposed thatp<2(r+2)≤np/(n-p) as n≥p and p<2(r+2)<+∞ as n<p, and [u(t),v(t)] is a local solution of problem (1)–(5) on [0,T). If [u0,v0]∈W, [u1,v1]∈L2(Ω)×L2(Ω) and E(0)<d, then [u(t),v(t)] is a global solution of problem (1)–(5).Proof.
It suffices to show that∥ut∥2+∥vt∥2+∥∇u∥pp+∥∇v∥pp is bounded uniformly with respect to t. Under the hypotheses in Theorem 8, we get from Lemma 7 that [u,v]∈W on [0,T). So the following formula holds on [0,T):
(51)J([u,v])=1p(∥∇u∥pp+∥∇v∥pp)-1r+2∥uv∥r+2r+2≥2(r+2)-p2p(r+2)(∥∇u∥pp+∥∇v∥pp).
We have from (51) that
(52)12(∥ut∥2+∥vt∥2)+2(r+2)-p2p(r+2)(∥∇u∥pp+∥∇v∥pp)≤12(∥ut∥2+∥vt∥2)+J([u(t),v(t)])=E(t)≤E(0)<d.
Hence, we get
(53)(∥ut∥2+∥vt∥2)+(∥∇u∥pp+∥∇v∥pp)≤max(2,2p(r+2)2(r+2)-p)d<+∞.
The above inequality and the continuation principle lead to the global existence of the solution [u,v] for problem (1)–(5).


## 4. Asymptotic Behavior of Global Solutions

The following lemma plays an important role in studying the decay estimate of global solutions for the problem (1)–(5).Lemma 9 (see [9]).
Suppose thatφ(t) is a nonincreasing nonnegative function on [0,+∞) and satisfies
(54)φ(t)r+1≤k(φ(t)-φ(t+1)),∀t≥0.
Then, φ(t) has the decay property
(55)φ(t)≤[rk(t-1)+M-r]-1/r,∀t≥1,
where k,r>0 are constants and M=maxt∈[0,1]φ(t).Lemma 10.
Under the assumptions of Theorem8, if initial value [u0,v0]∈W and [u1,v1]∈L2(Ω)×L2(Ω) are sufficiently small such that
(56)C2(r+2)(2p(r+2)2p(r+2)-pE(0))(2(r+2)-p)/p<1,
then
(57)(∥∇u∥pp+∥∇v∥pp)≤1θK([u,v]),
where θ=1-C2(r+2)((2p(r+2)/(2p(r+2)-p))E(0))(2(r+2)-p)/p>0 is a positive constant and C is the optimal Sobolev’s constant from W01,p(Ω) to L2(r+2)(Ω).Proof.
We have from Lemma1 and (52) that
(58)2∥uv∥r+2r+2≤2∥u∥2(r+2)r+2∥v∥2(r+2)r+2≤∥u∥2(r+2)2(r+2)+∥v∥2(r+2)2(r+2)≤C2(r+2)(∥∇u∥p2(r+2)+∥∇v∥p2(r+2))≤C2(r+2)(∥∇u∥p2(r+2)-p∥∇u∥pp+∥∇v∥p2(r+2)-p∥∇v∥pp)≤C2(r+2)(2p(r+2)2p(r+2)-pE(0))(2(r+2)-p)/p×(∥∇u∥pp+∥∇v∥pp).
Therefore, we get from (58) and (31) that
(59)[1-C2(r+2)(2p(r+2)2p(r+2)-pE(0))(2(r+2)-p)/p]×(∥∇u∥pp+∥∇v∥pp)≤K([u,v]).
Let
(60)θ=1-C2(r+2)(2p(r+2)2p(r+2)-pE(0))(2(r+2)-p)/p>0;
then, we have from (59) that
(61)∥∇u∥pp+∥∇v∥pp≤1θK([u,v]).Theorem 11.
Under the assumptions of Theorem8, if p<q<r+2 and (56) hold, then the global solution [u,v] in W of the problem (1)–(5) has the following decay property:
(62)E(t)≤[p-2pC(t-1)+M(p+q-pq)/p]p/(p+q-pq),∀t>1,
where M=maxt∈[0,1]E(t)>0 is some constant depending only on [u0,v0] and [u1,v1].Proof.
Multiplying (1) by ut and (2) by vt and integrating over Ω×[t,t+1], and summing up together, we get
(63)∫tt+1‍(∥ut(s)∥qq+∥vt(s)∥qq+∥∇ut(s)∥22+∥∇vt(s)∥22)ds=E(t)-E(t+1).
Thus, there exists t1∈[t,t+1/4], t2∈[t+3/4,t+1] such that
(64)4(∥ut(ti)∥qq+∥vt(ti)∥qq+∥∇ut(ti)∥22+∥∇vt(ti)∥22)=E(t)-E(t+1),i=1,2.
On the other hand, we multiply (1) by u and (2) by v and integrate over Ω×[t1,t2]. Adding them together, we obtain
(65)∫t1t2‍K([u,v])ds=∫t1t2‍∥ut∥2ds+∫t1t2‍∥vt∥2ds+(ut(t1),u(t1))-(ut(t2),u(t2))+(vt(t1),v(t2))-(vt(t2)v(t2))-(∫t1t2‍∫Ω‍|ut|q-2utudxds+∫t1t2‍∫Ω‍|vt|q-2vtvdxds)-∫t1t2∫Ω∇ut∇udxds-∫t1t2∫Ω‍∇vt∇vdxds.
From (63), Sobolev inequality, and Hölder inequality, we have
(66)∫t1t2‍∥ut∥2ds≤C∫t1t2‍∥∇ut∥2ds≤C(E(t)-E(t+1)),∫t1t2∥vt∥2‍ds≤C∫t1t2‍∥∇vt∥2ds≤C(E(t)-E(t+1)).
We get from (52), (64), and Lemmas 1 and 2 that
(67)|ut(ti),u(ti)|≤∥ut(ti)∥·∥u(ti)∥≤C∥∇ut(ti)∥·∥∇u(ti)∥p≤C(E(t)-E(t+1))1/2supt≤s≤t+1E(s)1/p≤C(ε)(E(t)-E(t+1))p/2(p-1)+εsupt≤s≤t+1E(s),i=1,2,|(vt(ti),v(ti))|≤∥vt(ti)∥·∥v(ti)∥≤C∥∇vt(ti)∥·∥∇v(ti)∥p≤C(E(t)-E(t+1))1/2supt≤s≤t+1E(s)1/p≤C(ε)(E(t)-E(t+1))p/2(p-1)+εsupt≤s≤t+1E(s),i=1,2.
From Hölder inequality and Lemma 2,we get
(68)|∫t1t2‍∫Ω‍|ut|q-2utudxds|≤∫t1t2‍∥ut∥qq-1∥u∥qds≤(∫t1t2∥ut∥qq‍ds)(q-1)/q(∫t1t2‍∥u∥qqds)1/q≤C(ε)∫t1t2‍∥ut∥qqds+ε∫t1t2‍∥u∥qqds,(69)|∫t1t2‍∫Ω‍|vt|q-2vtvdxds|≤∫t1t2‍∥vt∥qq-1∥v∥qds≤(∫t1t2‍∥vt∥qqds)(q-1)/q(∫t1t2‍∥v∥qqds)1/q≤C(ε)∫t1t2‍∥vt∥qqds+ε∫t1t2‍∥v∥qqds.
Sincep<q<r+2 and the property of the function f(x)=αx/x, α≥0, x>0, we obtain
(70)∥u∥qqq≤C∥u∥ppp+C∥u∥r+2r+2r+2,∥v∥qqq≤C∥v∥ppp+C∥v∥r+2r+2r+2.
We conclude from (69), (70), [u,v]∈W, and Lemma 1 that
(71)∥u∥qq+∥v∥qq≤C(∥u∥pp+∥u∥r+2r+2+∥v∥pp+∥v∥r+2r+2)≤C(∥u∥pp+∥∇u∥pp+∥v∥pp+∥∇v∥pp)≤C(∥∇u∥pp+∥∇v∥pp)≤CE(t).
It follows from (63), (68), (69), and (71) that
(72)|-(∫t1t2‍∫Ω‍|ut|q-2utudxds+∫t1t2‍∫Ω‍|vt|q-2vtvdxds)|≤C(ε)(E(t)-E(t+1))+εC∫t1t2‍E(s)ds,
and we obtain from (63), Sobolev inequality, Hölder inequality, and Lemma 2 that
(73)|-∫t1t2‍∫Ω‍∇ut∇uds|≤∫t1t2‍∥∇ut∥·∥∇u∥ds≤(∫t1t2‍∥∇ut∥2ds)1/2(∫t1t2‍∥∇u∥2ds)1/2≤C(E(t)-E(t+1))1/2(∫t1t2‍∥∇u∥p2ds)1/2≤C(E(t)-E(t+1))1/2(∫t1t2∥∇u∥pp‍ds)1/p≤C(E(t)-E(t+1))p/2(p-1)+ε∫t1t2‍∥∇u∥ppds.
Similarly, we have the following formula:
(74)|-∫t1t2‍∫Ω‍∇vt∇vds|≤∫t1t2‍∥∇vt∥·∥∇v∥ds≤(∫t1t2‍∥∇vt∥2ds)1/2(∫t1t2‍∥∇v∥2ds)1/2≤C(E(t)-E(t+1))1/2(∫t1t2‍∥∇v∥p2ds)1/2≤C(E(t)-E(t+1))1/2(∫t1t2∥∇v∥pp‍ds)1/p≤C(E(t)-E(t+1))p/2(p-1)+ε∫t1t2‍∥∇v∥ppds.
We get from (57), (73), and (74) that
(75)|∫t1t2‍∫Ω‍(∇ut∇u+∇vt∇v)ds|≤C(E(t)-E(t+1))p/2(p-1)+ε∫t1t2‍(∥∇u∥pp+∥∇v∥pp)ds≤C(E(t)-E(t+1))p/2(p-1)+εθ∫t1t2‍K([u,v])ds.
Choosing small enoughε, we have from (65), (66), (67), (72), and (75) that
(76)∫t1t2‍K([u,v])ds≤C[(E(t)-E(t+1))p/(2(p-1))(E(t)-E(t+1))+(E(t)-E(t+1))p/2(p-1)]+εsupt≤s≤t+1E(s)+ε∫t1t2‍E(s)ds.
It follows from (30) and (31) that
(77)J([u,v])=2(r+2)-p2p(r+2)(∥∇u∥pp+∥∇v∥pp)+12(r+2)K([u,v]).
On the other hand, from (12) and using (57) and (77), we deduce that
(78)E(t)=12(∥ut∥2+∥vt∥2)+J([u,v])=12(∥ut∥2+∥vt∥2)+2(r+2)-p2p(r+2)×(∥∇u∥pp+∥∇v∥pp)+12(r+2)K([u,v])≤12(∥ut∥2+∥vt∥2)+(2(r+2)-p2θp(r+2)+12(r+2))×K([u,v]).
By integrating (78) over [t1,t2], we obtain
(79)∫t1t2‍E(s)ds≤12∫t1t2‍(∥ut∥2+∥vt∥2)ds+(2(r+2)-p2θp(r+2)+12(r+2))∫t1t2‍K([u,v])ds.
For small enough ε, we have from (76) and (79) that
(80)∫t1t2‍E(s)ds≤C[(E(t)-E(t+1))+(E(t)-E(t+1))p/(2(p-1))]+εsupt≤s≤t+1E(s).
Thus, there exists t*∈[t1,t2], such that
(81)E(t*)≤C[(E(t)-E(t+1))+(E(t)-E(t+1))p/2(p-1)]+εsupt≤s≤t+1E(s).
Multiplying (1) by ut and (2) by vt and integrating over Ω×[t*,t2], and summing up, we get
(82)E(t2)=E(t*)-∫t*t2‍(∥ut∥qq+∥vt∥qq+∥∇ut∥2+∥∇vt∥2)ds.
Therefore, we obtain from (63), (81), and (82) that
(83)supt≤s≤t+1E(s)≤C[(E(t)-E(t+1))p/(2(p-1))(E(t)-E(t+1))+(E(t)-E(t+1))p/2(p-1)]+εsupt≤s≤t+1E(s).
Choosing small enough ε, we have from (83) that
(84)supt≤s≤t+1E(s)≤C[(E(t)-E(t+1))p/(2(p-1))(E(t)-E(t+1))+(E(t)-E(t+1))p/2(p-1)].
Sincep>2 and E(t)<E(0), we get
(85)supt≤s≤t+1E(s)≤C(E(t)-E(t+1))p/2(p-1).
Consequently,
(86)supt≤s≤t+1E(s)(2(p-1))/p≤C(E(t)-E(t+1)).
Thus, applying Lemma 9 to (86), we get
(87)E(t)≤[p-2pC(t-1)+M(p-2)/p]p/(2-p),∀t>1,
where M=maxt∈[0,1]E(t)>0 is some constant depending only on [u0,v0] and [u1,v1].



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*Source: 903625-2013-10-28.xml*