# New Estimates of Solution to Coupled System of Damped Wave Equations with Logarithmic External Forces

Authors: Loay Alkhalifa; Khaled Zennir
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In the paper, we consider new stability results of solution to class of coupled damped wave equations with logarithmic sources inℝn. We prove a new scenario of stability estimates by introducing a suitable Lyapunov functional combined with some estimates.


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## Body

## 1. Introduction

In the present paper, we consider an initial boundary value problem with damping terms and logarithmic sources, forx∈ℝn,t>0(1)∂ttv1+bv2=ϕxΔv1−∫0tϖ1t−pv1pdp+kv1lnv1,∂ttv2+bv1=ϕxΔv2−∫0tϖ2t−pv2pdp+kv2lnv2,v1x,0=v10x,v2x,0=v20x,∂tv1x,0=v11x,∂tv2x,0=v21x,where b>0, n≥3, and k is a small positive real number. The density function ρx>0, for all x∈ℝn, where ϕx−1=1/ϕx≡ρx, under homogeneous Drichlet boundary conditions.A related initial boundary value problem was considered by Han in [1]:
(2)utt+ut−Δu+u+u2u=ulnu2,x∈Ω,t∈0,T,ux,0=u0xutx,0=u1x,x∈Ω,ux,t=0,x∈∂Ω,t∈0,T,and the global existence of weak solutions was proved, for all u0,u1∈H01×L2 in ℝ3. The weak and strong damping terms in logarithmic wave equation
(3)utt+μut−Δu−ωΔut=ulnu,x∈Ω,t∈0,∞,ux,0=u0xutx,0=u1x,x∈Ω,ux,t=0,x∈∂Ω,t∈0,∞were introduced by Lian and Xu [2]. The global existence, asymptotic behavior, and blowup at three different initial energy levels (subcritical energy E0<d, critical initial energy E0=d, and the arbitrary high initial energy E0>0ω=0) were proved. In [3], Al-Gharabli established explicit and general energy decay results for the problem
(4)utt+Δ2u+u−∫0tgt−sΔ2uds=kulnu,x∈Ω,t∈0,∞,ux,0=u0xutx,0=u1x,x∈Ω,ux,t=∂u∂υ=0,x∈∂Ω,t∈0,∞.When the densityϕx≠1, Papadopoulos and Stavrakakis [4] considered the following semilinear hyperbolic initial value problem:
(5)utt+ϕxΔu+δut+λfu=ηx,x,t∈ℝn×ℝ+.The authors proved local existence of solutions and established the existence of a global attractor in the energy spaceD1,2ℝn×Lg2ℝn, where ϕx−1≔gx. Miyasita and Zennir [5] proved the global existence of the following viscoelastic wave equation:
(6)utt+aut−ϕxΔu+ωΔut−∫0tgt−sΔusds=uup−1,x∈ℝn,t>0,ux,0=u0x,x∈ℝn,utx,0=u1x,x∈ℝn.The novelty of our work lies primarily in the use of a new condition between the weights of damping the external forces, where we outline the effects of the damping term with less conditions on the viscoelastic terms. We also propose logarithmic nonlinearities in sources and used classical arguments to estimate them. These nonlinearities make the problem very interesting in the application point of view. In order to compensate for the lack of classical Poincaré’s inequality inℝn, we use the weighted function to use generalized Poincaré’s one. The main contribution of this paper is introduced in Theorem 8, where we obtain decay estimates with positive initial energy under a general assumption on the kernel. The rest of the paper is outline as follows. In Section 2, we give some preliminaries and our main results. In Section 3, we will prove the general decay of energy to the problem.


## 2. Preliminaries and Main Results

We state some assumptions and definitions that will be useful in this paper. With respect to the relaxation functionsϖ1,ϖ2, we assume for i=1,2.(H1)ϖ1,ϖ2∈C1ℝ+,ℝ+ satisfy for any t≥0,
(7)ϖi0>0,∫0∞ϖipdp=li0<∞,1−∫0tϖipdp=li>0(H2) There exist nonincreasing differentiable functionsζ1,ζ2:ℝ+⟶ℝ+ that satisfy
(8)ζit>0,ϖi′t≤−ζitϖitfort≥0(H3) The functionρ:ℝn⟶ℝ+∗,ρx∈C0,γℝn with γ∈0,1 and ρ∈Lsℝn∩L∞ℝn, where s=2n/2n−qn+2qDefinition 1 (see [4]).
We define the function spaces of our problem and their norms as follows:(9)H=v∈L2n/n−2ℝn∣∇v∈L2ℝnn.Let the function spacesH as the closure of C0∞ℝn with respect to the norm vH=v,vH1/2 for the inner product:
(10)v,wH=∫ℝn∇v·∇wdx,and Lρ2ℝn be defined with the norm vLρ2=v,vLρ21/2 for
(11)v,wLρ2=∫ℝnρvwdx.For generalq∈1,+∞, Lρqℝn is the weighted Lq space under a weighted norm
(12)vLρq=∫ℝnρvqdx1/q.To distinguish the usualLq space from the weighted one, we denote the standard Lq norm by
(13)vq=∫ℝnvqdx1/q.We denote an eigenpairλj,wjj∈ℕ⊂ℝ×H by
(14)−ϕxΔwj=λjwjx∈ℝnfor any j∈ℕ. Then, according to [4],
(15)0<λ1≤λ2≤⋯≤λj≤⋯↑+∞holds and wj is a complete orthonormal system in H.Now, we introduce Sobolev embedding and generalized Poincaré’s inequalities.Lemma 2.
Letρ satisfy (H3). Then, there are positive constants CS>0 and CP>0 that depend only on n and ρ such that
(16)v2n/n−2≤CSvH,vLρ2≤CPvHfor v∈H.Lemma 3 (see Lemma 2.2 in [6]).
Letρ satisfy (H3). Then, we have
(17)vLρq≤CqvH,Cq=CSρs1/qfor v∈H, where s=2n/2n−qn+2q for 1≤q≤2n/n−2.The energy functional associated to problem (1) is given by
(18)Et=12∑i=12∂tvitLρ22+12∑i=121−∫0tϖipdp∇vit2+bv1tv2t2+12∑i=12ϖi∘∇vit−k2∑i=12∫ℝnρxvi2lnvidx+k4∑i=12viLρ22≥12∑i=12∥∂tvit∥Lρ22+12∑i=121−∫0tϖipdp∇vit2+12∑i=12ϖi∘∇vit−k2∑i=12∫ℝnρxvi2lnvidx+k4∑i=12viLρ22,where
(19)ϖ∘v=∫0tϖt−pvt−vp22dp.With direct differentiation of (18), using (1), we obtain
(20)∂tEt=−12∑i=12ϖitviH2−ϖi′∘∇vi≤0,which let our system dissipative.Lemma 4 (see [7]) (logarithmic Sobolev inequality).
Letsu be any function in H01Ω and a>0 be any number. Then,
(21)∫Ωv2lnvdx≤12v22lnv22+a22π∇v22−1+lnav22.Lemma 5 (see [8]) (logarithmic Gronwall inequality).
Letc>0,γ∈L10,T;ℝ+, and assume that the function ω:0,T⟶1,∞ satisfies
(22)ωt≤c1+∫0tγpωplnωpdp,0≤t≤T,then
(23)ωt≤cexpc∫0tγpdp,0≤t≤T.We define the following functionals(24)Jt=12∑i=121−∫0tϖipdp∇vit2+12∑i=12ϖi∘∇vit−k2∑i=12∫ℝnρxvi2lnvidx+k4∑i=12viLρ22,It=12∑i=121−∫0tϖipdp∇vit2+12∑i=12ϖi∘∇vit−k2∑i=12∫ℝnρxvi2lnvidx.Then, we introduce(25)W=v1,v2:v1,v2∈H:It>0,Jt<d∪0.∑i=12vi2<4dforallt∈0,T.Lemma 6.
Letv10,v11,v20,v21∈H×Lρ2ℝn such that 0<E0<d and It0>0. Then, we have
(26)v1,v2∈W,Theorem 7 (see [5]).
Letv10,v11,v20,v21∈H×Lρ2ℝn. Under the assumptions (H1)–(H3). Then, problem (1) has a global weak solution u in the space
(27)v1,v2∈C0,+∞;H∩C10,+∞;Lρ2ℝn2.Then, the main result in this paper is the general decay of energy to problem (1) which is given in the following theorem.Theorem 8.
Assume the assumptions (H1)–(H3) hold and0<E0<d. Let v1,v2 be the weak solution of problem (1) with the initial data v10,v11,v20,v21∈Hℝn×Lρ2ℝn. Then, there exist constant β>0 such that the energy Et defined by (18) satisfies for all t>0,
(28)Et≤β1+∫t0tξε0+1pdp−1/ε0,ε0∈0,1.


## 3. Asymptotic Behavior forE0<d

The following technical lemmas are useful to prove the general decay of energy to problem (1).Lemma 9.
Under the assumptions in Theorem8, then the functional Φt defined by
(29)Φt=∫ℝnρxv1t∂tv1+v2t∂tv2tdxdxsatisfies for any t≥0,
(30)Φ′t≤∑i=12∂tvitLρ22−12∑i=12li∇vit2+∑i=121−li4εϖi∘∇vit+k∑i=12∫ℝnρxvi2lnvidx.Proof.
We differentiateΦt, using (1), we can get
(31)Φ′t=∑i=12∂tviLρ22−∑i=12∇vi2+∑i=12∫ℝn∇vit·∫0tϖit−p∇vipdpdx−2b∫ℝnρxv1v2dx+k∑i=12∫ℝnρxvi2lnvidx.
It follows from Young and Poincaré’s inequality that for anyε>0,
(32)∫ℝn∇vit·∫0tϖit−p∇vipdpdx=∫ℝn∇vit·∫0tϖit−p∇vip−∇vitdpdx+∫0tϖipdp∇vit22≤1−li∇vi2+ε∇vi22+14ε∫ℝn∫0tϖit−p∇vip−∇vtdp2dx≤1−li+ε∇vi22+1−li4εϖi∘∇vit.
Exploit Young and Poincaré’s inequalities to estimate(33)2b∫ℝnρxv1v2dx≤εc∗∇viLρ22+c∗4ε∇v2Lρ22.
Inserting (32)–(33) into (31) yields for any ε>0,
(34)Φ′t≤∑i=12∂tvitLρ22−∑i=12l−ε−εc∗∇vit2+∑i=121−li4εϖi∘∇vit+k∑i=12∫ℝnρxvi2lnvidx.
Takingε>0 small enough in (34) such that
(35)li−ε−εc∗>l2.
The proof is hence complete.Lemma 10.
Under the assumptions in Theorem8, then the functional ψt defined by
(36)ψt=−∫ℝnρx∂tv1t∫0tϖ1t−pv1t−v1pdpdx−∫ℝnρx∂tv2t∫0tϖ2t−pv2t−v2pdpdxsatisfies for any δ>0,
(37)ψ′t≤∑i=12δ1−li2+1+bc∗∇vit2−∑i=12∫0tϖisdp−2δ∂tvitLρ22+C∑i=12∫0tϖisdpϖi∘∇vit−ϖi0c∗4δ∑i=12ϖi′∘Δvit+cε0∑i=12ϖi∘∇vi1/1+ε0.Proof.
Taking the derivative ofψt and using (1), we conclude that
(38)ψ′t=∑i=12∫ℝn∇vit∫0tϖit−p∇vit−∇vipdpdx−∫ℝn∫0tϖ1t−p∇v1pdp∫0tϖ1t−p∇v1t−∇v1pdpdx−∫ℝn∫0tϖ2t−p∇v2pdp∫0tϖ2t−p∇v2t−∇v2pdpdx+b∫ℝnρxv2∫0tϖ1t−pv1t−v1pdpdx+b∫ℝnρxv1∫0tϖ2t−pv2t−v2pdpdx−k∑i=12∫ℝnρxvilnvi∫0tϖit−pvit−vipdpdx−∑i=12∫0tϖipdp∂tviLρ22−∑i=12∫ℝnρx∂tvi∫0tϖi′t−pvit−vipdpdx.
We then use Young and Poincaré’s inequalities; we can get for anyδ>0,
(39)∫ℝn∇vit∫0tϖit−p∇vit−∇vipdpdx≤δ∇vi2+14δ∫0tϖipdpϖi∘∇vit.
The second and third terms can be treated as(40)∫ℝn∫0tϖit−p∇vipdp∫0tϖit−p∇vit−∇vipdpdx≤δ1−li2∇vi2+1+14δ∫0tϖipdpϖi∘∇vit.
The fourth and fifth terms will be estimated by(41)∫ℝnρxv2∫0tϖ1t−pv1t−v1pdpdx≤δc∗∇v22+c∗4δ∫0tϖ1pdpϖ1∘∇v1t,∫ℝnρxv1∫0tϖ2t−pv2t−v2pdpdx≤δc∗∇v12+c∗4δ∫0tϖ2pdpϖ2∘∇v2t,respectively.
For the last term, we have(42)∫ℝnρx∂tvi∫0tϖi′t−pvit−vipdpdx≤δ∂tviLρ22+c∗4δ∫0t−ϖi′pdpϖi′∘∇vit≤δ∂tviLρ22−ϖi0c∗4δϖi′∘∇vit.
Letε0∈0,1 and gs=sε0lns−s. Notice that g is continous on 0,∞, its limit at 0 is 0, and its limit at ∞ is −∞. Then, g has a maximum mε0 on 0,∞, so the following inequality holds
(43)slns≤s2+mε0s1−ε0,foralls>0.
Using the Cauchy-Schwartz’s inequality and applying (43), yields, for any δ>0,
(44)k∫ℝnρxviln∣vi∣∫0tϖit−pvit−vipdpdx≤k∫ℝnρxvi2+mε0vi1−ε0∫0tϖit−pvit−vipdpdx≤c∫ℝnρxvi2∫0tϖit−pvit−vipdpdx+δ∥vi∥Lρ22+∫ℝn∫0tϖit−pvit−vipdpdx2/1+ε0≤δc∗∥∇vi∥22+14δϖi∘∇vit+cε0ϖi∘∇vi1/1+ε0.
Combining (39)–(44) with (39) gives us (37) with
(45)C=bc∗+24δ+2δ.
Therefore, the proof is complete.Now, we define a Lyapunov functionalLt by
(46)Lt=MEt+ε1Φt+ε2ψt,where M, ε1, and ε2 are positive constants which will be taken later.It is easy to see thatLt and Et are equivalent in the sense that there exist two positive constants β1 and β2 such that
(47)β1Et≤Lt≤β2Et.Remark 11 (see [3]).
Sinceζi is nonincreasing, we have
(48)ζitϖi∘∇vi1/1+ε0≤C−E′t1/1+ε0.Proof of Theorem8.
For any fixedt0>0, we have for any t≥t0,
(49)∫0tϖipdp≥∫0t0ϖipdp≔ϖi0.
It follows from (37), (30), and (20) that
(50)L′t=ME′t+ε1Φ′t+ψ′t≤−∑i=12ϖi0−2δ−ε1∂tvitLρ22−∑i=12li2ε1−δ1−li2+1+bc∗∇vit22+∑i=12C1ε1+Cliϖi∘∇vit−M2∑i=12ϖitvit2+ε1k∑i=12∫ℝnρxvi2lnvidx+ε1cε0∑i=12ϖi∘∇vi1/1+ε0+C3∑i=12ϖi′∘∇vit.
Using the logarithmic Sobolev inequality, we have(51)L′t≤−∑i=12ϖi0−2δ−ε1∂tvtiLρ22+C3∑i=12ϖi′∘∇vit−∑i=12li2ε1−δ1−li2+1+bc∗−ε1kα22π∇vit22+∑i=12C1ε1+Cliϖi∘∇vit−M2∑i=12ϖitvit2+ε1k12∑i=12vi22lnvi22−ε1k1+lnα∑i=12vi22+ε1cε0∑i=12ϖi∘∇vi1/1+ε0.
Recalling (18) and Et≤E0<d, we get
(52)lnv22<ln4kEt<ln4kE0<ln4kd.
Now, we takeε1>0 small enough so that
(53)ϖi0−2δ−ε1>0.
For any fixedε1>0, we pick δ>0 so small that
(54)li2ε1−δ1−li2+1>li4ε1.
On the other hand, we chooseM>0 large enough so that (47) holds, and further
(55)C3=M2−ϖi04δ>0.
We can conclude that there exist two positive constantm and C′ such that
(56)L′t≤−mEt+C′∑i=12ϖi∘∇vit+ε1cε0∑i=12ϖi∘∇vi1/1+ε0.
Multiplying (56) by ζt=minζ1,ζ2 by (H2) and use the fact that
(57)ϖi∘∇vit≤cϖi∘∇vi1/1+ε0t,and (48), we get
(58)ζtL′t≤−mζtEt+c−E′t1/1+ε0.
Multiply (58) by ζε0tEε0t and recall that ζ′t≤0 to obtain
(59)ζε0+1tEε0tL′t≤−mζε0tEε0+1t+cζEε0t−E′t1/1+ε0.
Using Young’s inequality, for anyδ>0,
(60)ζε0+1tEε0tL′t≤−mζε0+1tEε0+1t+cδζε0+1tEε0+1t−cδE′t≤−m−δcζε0+1tEε0+1t−cE′t,which implies
(61)ζε0+1Eε0L+cEt≤−m−δcvε0+1tEε0+1t.
It is clear that to get(62)L1t=ζε0+1Eε0L+cE~Et.
By using (61) and ζ′t≤0, we arrive at
(63)L1′t=ζε0+1Eε0L+cE≤−m′ζε0tEε0+1t.
Integration overt0,t leads to for some constant m′>0 such that
(64)L1t≤m′1+∫t0tζε0+1pdp−1/ε0.
The equivalence ofL1t and E completes Proof of Theorem 8.Remark 12.
(1)
We mention here that we have coupled our system without the classical way, i.e., our idea is not to couple equations in the logarithmic nonlinear terms(2)
Most contribution here is to obtain our nonexistence result with less conditions on the viscoelastic terms



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