# Global Existence and General Decay of Solutions for a Quasilinear System with Degenerate Damping Terms

Authors: Fatma Ekinci; Erhan Pișkin; Salah Mahmoud Boulaaras; Ibrahim Mekawy
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

In this work, we consider a quasilinear system of viscoelastic equations with degenerate damping, dispersion, and source terms under Dirichlet boundary condition. Under some restrictions on the initial datum and standard conditions on relaxation functions, we study global existence and general decay of solutions. The results obtained here are generalization of the previous recent work.


---

## Body

## 1. Introduction

LetΩ be a bounded domain with a sufficiently smooth boundary in Rnn≥1. We investigate a quasilinear system of two viscoelastic equations in the presence of degenerate damping, dispersion, and source terms, namely,
(1)utηutt−Δu+∫0th1t−sΔusds−Δutt+uk+vlutj−1ut=f1u,v,x,t∈Ω×0,T,vtηvtt−Δv+0th2t−sΔvsds−Δvtt+vθ+uϱvts−1vt=f2u,v,x,t∈Ω×0,T,ux,t=vx,t=0,x,t∈∂Ω×0,T,ux,0=u0x,utx,0=u1x,x∈Ω,vx,0=v0x,vtx,0=v1x,x∈Ω,where, s≥1,η>0,k,l,θ,ϱ≥0, and hi.:R+⟶R+i=1,2 are positive relaxation functions which will be specified later. .a+.b.tτ−1.t and −Δ.tt are the degenerate damping term and the dispersion term, respectively.By taking(2)f1u,v=au+v2κ+1u+v+buκuvκ+2,f2u,v=au+v2κ+1u+v+bvκvuκ+2,in which a>0,b>0, and
(3)1<κ<+∞ifn=1,2and1<κ≤3−nn−2ifn≥3.It is simple to show that(4)uf1u,v+vf2u,v=2κ+2Fu,v,∀u,v∈R2,where
(5)Fu,v=12κ+2au+v2κ+2+2buvκ+2.To motivate our problem (1), it can trace back to the initial boundary value problem for the single viscoelastic equation of the form
(6)utηutt−Δu+∫0tht−sΔusds−Δutt+gu,ut=fu.This type problem appears a variety of mathematical models in applied science. For instance, in the theory of viscoelasticity, physics, and material science, problem (5) has been studied by various authors, and several results concerning blow-up and energy decay have been studied case (η≥0). For example, Liu [1] studied a general decay of solutions case gu,ut=0. Messaoudi and Tatar [2] applied the potential well method to indicate the global existence and uniform decay of solutions (gu,ut=0 instead of Δut). Furthermore, the authors obtained a blow-up result for positive initial energy. Wu [3] studied a general decay of solution case (gu,ut=utmut). Later, Wu [4] studied the same problem case gu,ut=ut and discussed the decay rate of solution energy. Recently, Yang et al. [5] proved the existence of global solution and asymptotic stability result without restrictive conditions on the relaxation function at infinity case (fu=σx,tWtt,x).In casegu,ut=0 and without dispersion term, problem (5) has been investigated by Song [6], and the blow-up result for positive initial energy has been proved.For a coupled system, He [7] investigated the following problem
(7)utηutt−Δu+∫0th1t−sΔusds−Δutt+utj−2ut=f1u,v,vtηvtt−Δv+∫0th2t−sΔvsds−Δvtt+vts−2vt=f2u,v,where η>0,j,s≥2. The author proved general and optimal decay of solutions. Then, in [8], the author investigated the same problem without damping term and established a general decay of solutions. Furthermore, the author obtained a blow-up of solutions for negative initial energy. In addition, problem (1) with in case η=0 and without dispersion term, Wu [9] proved a general decay of solutions. Later, Pișkin and Ekinci [10] studied a general decay and blow-up of solutions with nonpositive initial energy for problem (1) case (Kirchhoff-type instead of Δu and without dispersion term). In recent years, some other authors investigate the hyperbolic type system with degenerate damping term (see [11–14]).The rest of the paper is arranged as follows: in Section2, as preliminaries, we give necessary assumptions and lemmas that will be used later and local existence theorem without proof. In Section 3, we prove the global existence of solution. In the last section, we studied the general decay of solutions.


## 2. Preliminaries

We begin this section with some assumptions, notations, lemmas, and theorems. Denote the standartL2Ω norm by .=.L2Ω and LpΩ norm by .p=.LpΩ.To state and prove our result, we need some assumptions:(A1) Regardinghi:0,∞⟶0,∞,i=1,2 is C1 functions and satisfies
(8)hiα>0,hi′α≤0,1−0∞hiαdα=li>0,α≥0and nonincreasing differentiable positive C1 functions ς1 and ς2 such that
(9)hi′t≤−ςithit,t≥0,i=1,2.(A2) For the nonlinearity, we assume that(10)1≤j,sifn=1,2,1≤j,s≤n+2n−2ifn≥3.(A3) Assume thatη satisfies
(11)0<ηifn=1,2,0<η≤2n−2ifn≥3.In addition, we present some notations:(12)α⋄∇wt=∫0tαt−s∇wt−∇ws2ds,l=minl1,l2.Lemma 1 (Sobolev-Poincare inequality) [15].
Letq be a number with 2≤q<∞n=1,2 or 2≤q≤2n/n−2n≥3, and then there is a constant C∗=C∗Ω,q such that
(13)uq≤C∗∇uforu∈H01Ω.
Now, we state the local existence theorem that can be established by combining arguments of [7, 10].Theorem 2.
Assume that (A1)-(A3) and (2) hold. Let u0,v0∈H01Ω and u1,v1∈L2Ω be given. Then, for some T>0, problem (1) has a unique local weak solution in the following class:
(14)u,v∈C0,T;H2Ω∩H01Ω,ut∈C0,T;H01Ω∩Lj+1Ω,vt∈C0,T;H01Ω∩Ls+1Ω.We define the energy function as follows:(15)Et=1η+2utη+2η+2+vtη+2η+2+12h1⋄∇ut+h2⋄∇vt+∇ut2+∇vt2+121−∫0th1sds∇ut2+1−∫0th2sds∇vt2−∫ΩFu,vdx.Also, we define(16)It=∇ut2+∇vt2+1−∫0th1sds∇ut2+1−∫0th2sds∇vt2+h1⋄∇ut+h2⋄∇vt−2κ+2∫ΩFu,vdx,Jt=121−∫0th1sds∇ut2+1∫0th2sds∇vt2+12h1⋄∇ut+h2⋄∇vt+∇ut2+∇vt2−∫ΩFu,vdx.By computation, we get(17)ddtEt≤12h1′⋄∇ut+h2′⋄∇vt−12h1t∇u2+h2t∇v2−∫Ωuk+vlutj+1dx−∫Ωvθ+uϱvts+1dx≤0.


## 3. Global Existence

In this part, in order to state and prove the global existence of solution (1), we firstly give two lemmas.Lemma 3 [16].
Assume that (4) holds. Then, there exist ρ>0 such that for the solution u,v,
(18)u+v2κ+22κ+2+2uvκ+2κ+2≤ρl1∇u2+l2∇v2κ+2.Lemma 4.
Letu0,v0∈H01Ω, u1,v1∈L2Ω. Suppose that (A1)-(A3) hold. If
(19)I0>0andβ=ρ2κ+2κ+1E0κ+1<1,then
(20)It>0,∀t>0.Proof.
We haveI0>0 and by continuity of It about t, there exist a maximal time tm>0 such that
(21)It≥0,ont∈0,tm.
Lett0 be as follows:
(22)It0=0andIt>0,forall0≤t<t0.
By using (8), (9), and (A1), we get
(23)Jt=κ+12κ+21−∫0th1sds∇ut2+1−∫0th2sds∇vt2+h1⋄∇ut+h2⋄∇vt+∇ut2+∇vt2+12κ+2It≥κ+12κ+2l1∇ut2+l2∇vt2+h1⋄∇ut+h2⋄∇vt+∇ut2+∇vt2.
From (7) and (10), we have
(24)l1∇ut2+l2∇vt2≤2κ+2κ+1Jt≤2κ+2κ+1Et≤2κ+2κ+1E0,∀t∈0,t0.
By (11) and (12), we infer that
(25)2κ+2∫ΩFut0,vt0dx≤ρl1∇ut02+l2∇vt02κ+2≤ρ2κ+2κ+1E0κ+1l1∇ut02+l2∇vt02≤ρl1∇ut02+l2∇vt02≤1−∫0th1sds∇ut2+1−∫0th2sds∇vt2.
Thus, from (8), we obtain
(26)It0>0,which contradicts to (13). Thus, It>0 on 0,T.Theorem 5.
Suppose that the conditions of Lemma4 hold, then the solution (1) is bounded and global in time.Proof.
We have(27)E0≥Et=Jt+1η+2utη+2η+2+vtη+2η+2≥κ+12κ+2l1∇ut2+l2∇vt2+∇ut2+∇vt2+h1⋄∇ut+h2⋄∇vt+1η+2utη+2η+2+vtη+2η+2.
Thus,(28)∇ut2+∇vt2+∇ut2+∇vt2≤CE0,where positive constant C depends only on κ,l1,l2. This implies that the solution of problem (1) is global in time.


## 4. General Decay of Solutions

This section is devoted to show the decay of solution (1). Set
(29)Γt≔MEt+εΦt+Ϝt,where M and ε are positive constants and
(30)Φt=δ1t1η+1∫Ωutηutudx+∫Ω∇ut∇udx+δ2t1η+1∫Ωvtηvtvdx+∫Ω∇vt∇vdx,Ϝt=δ1t∫ΩΔut−utηutη+1∫0th1t−sut−usdsdx+δ2t∫ΩΔvt−vtηvtη+1∫0th2t−svt−vsdsdx.Lemma 6.
Forε which is small enough while M is large enough, the relation
(31)α1Γt≤Et≤α2Γt,∀t≥0holds for two positive constants α1 and α2.Proof.
As references [1, 10], it can be show easily that Γt and Et are equivalent in the sense that α1 and α2 are positive constants, depending on ε and M.Lemma 7 [3].
Assume that (12) holds. Let u,v be the solution of problem (1). Then, for σ≥0, we get
(32)∫Ω∫0th1t−sut−usdsσ+2dx≤1−l1σ+1c∗σ+22κ+2E0l1κ+1σ/2h1⋄∇ut,∫Ω∫0th2t−svt−vsdsσ+2dx≤1−l1σ+1c∗σ+22κ+2E0l2κ+1σ/2h2⋄∇vt.Lemma 8 [16].
Let (A1)-(A3) hold. Assume that u0,v0∈H01Ω, u1,v1∈L2Ω, be given and satisfying (12). Then, throughout the solution u,v of (1), there exist two positive constants β1 and β2 such that for any δ>0 and for all t≥0,(33)∫Ωf1u,v∫0th1t−sut−usdsdx≤β1δl1∇u2+l2∇v2+1−l1c∗2h1⋄∇ut4δ,∫Ωf2u,v∫0th2t−svt−vsdsdx≤β2δl1∇u2+l2∇v2+1−l2c∗2h2⋄∇ut4δ.Lemma 9.
Letu0,v0∈H01Ω, u1,v1∈L2Ω, be given and satisfying (12). Suppose that (A1)-(A3) hold. Then, for each t0>0, the functional Γt verifies, throughout the solution of (1)
(34)Γ′t≤−ξ1Et+ξ2h1⋄∇ut+h2⋄∇vt,t≥t0,where ξi>0,i=1,2.Proof.
By applying (18) and Eq.(1) and getting δi≡1i=1,2 in (18), we have
(35)Φ′t≤1η+1utη+2η+2+vtη+2η+2−∇u2−∇v2+∇ut2+∇vt2∫Ω∇ut∫0th1t−s∇usdsdx+∫Ω∇vt∫0th2t−s∇vsdsdx+2κ+2∫ΩFu,vdx−∫Ωuuk+vlututj−1dx−∫Ωvvθ+uρvtvts−1dx.
For estimating the seventh term in the right side of (22) as follows (see [17]):
(36)∫Ω∇ut∫0th1t−s∇usdsdx≤12∇u2+12∫Ω∫0th1t−s∇us−∇ut+∇utds2dx.
By exploiting Young’s inequality and the assumption that∫0th1sds≤∫0∞h1sds≤1−l1, for γ1>0,(37)∫Ω∇ut∫0th1t−s∇usdsdx≤12∇u2+121+γ1∫Ω∫0th1t−s∇usds2dx+121+1γ1∫Ω∫0th1t−s∇us−∇utds2dx≤12∇u2+121+γ1∫0th1sds2∇u2+121+1γ1∫0th1sdsh1⋄∇ut≤1+1+γ11−l122∇u2+1+1/γ11−l12h1⋄∇ut.
Similarly withγ2>0,(38)∫Ω∇vt∫0th2t−s∇vsdsdx≤1+1+γ21−l222∇v2+1+1/γ21−l22h2⋄∇vt.
By estimating the following terms in (22), we have
(39)∫Ωuuk+vlututj−1dx≤Ωuk+1utjdx+∫Ωuvlutjdx.
Exploiting Young’s inequality, Hölder’s inequality, Sobolev-Poincare inequality, (A3), and (15) for β1>0, one has
(40)∫Ωuk+1utjdx≤∫Ωukutj+1dxj/j+1∫Ωuk+j+1dxj/j+1≤jβ1−j+1/jj+1∫Ωukutj+1dx+β1j+1C∗k+j+1j+1∇uk+j+1≤jβ1−j+1/jj+1∫Ωuk+vlutj+1dx+β1j+1C∗k+j+1χ1k+j−1/2j+1∇u2,∫Ωuvlutjdx≤∫Ωvlutj+1dxj/j+1∫Ωvluj+1dxj/j+1≤jβ1−j+1/jj+1∫Ωuk+vlutj+1dx+β1j+12j+1v2l2l+u2j+12j+1≤jβ1−j+1/jj+1∫Ωuk+vlutj+1dx+β1j+1C∗2lχ2l−12j+1∇v2+β1j+1C∗2j+2χ1j2j+1∇u2,where
(41)χ1=2κ+2E0l1κ+1andχ2=2κ+2E0l2κ+1.
By inserting (27) and (28) into (26), we have
(42)∫Ωuuk+vlututj−1dx≤2jβ1−j+1/jj+1∫Ωuk+vlutj+1dx+β1j+1C∗2lχ2l−12j+1∇v2+β1j+1j+1C∗k+j+1χ1k+j−1/2+C∗2j+2χ1j2∇u2.
Similarly, forβ2>0, we have
(43)∫Ωvvθ+uρvtvts−1dx≤2sβ2−s+1/ss+1∫Ωvθ+uρvts+1dx+β2s+1C∗2ρχ1ρ−12s+1∇u2+β2s+1s+1C∗θ+s+1χ2θ+s−1/2+C∗2s+2χ2s2∇v2.
Thus, inserting (24) and (25) and (29) and (30) into (22), we obtain
(44)Φ′t≤1η+1utη+2η+2+vtη+2η+2−a1∇u2−a2∇v2+2κ+2∫ΩFu,vdx+∇ut2+∇vt2+1+1/γ11−l12h1⋄∇ut+1+1/γ21−l22h2⋄∇vt+2jβ1−j+1/jj+1∫Ωuk+vlutj+1dx+2sβ2−s+1/ss+1∫Ωvθ+uρvts+1dx,where
(45)a1=1−1+γ11−l122−β1j+1j+1C∗k+j+1χ1k+j−1/2+C∗2j+2χ1j2−β2s+1C∗2ρχ1ρ−12s+1,a2=1−1+γ21−l222−β2s+1s+1C∗θ+s+1χ2θ+s−1/2+C∗2s+2χ2s2−β1j+1C∗2lχ2l−12j+1.
At this moment, choosingγ1=l1/1−l1,γ2=l2/1−l2, and picking β1 and β2 small enough such that
(46)β1j+1j+1C∗k+j+1χ1k+j−1/2+C∗2j+2χ1j2+β2s+1C∗2ρχ1ρ−12s+1≤l14,β2s+1s+1C∗θ+s+1χ2θ+s−1/2+C∗2s+2χ2s2+β1j+1C∗2lχ2l−12j+1≤l24.
Consequently, (31) yields
(47)Φ′t≤1η+1utη+2η+2+vtη+2η+2−l14∇u2−l24∇v2+2κ+2∫ΩFu,vdx+1−l12l1h1⋄∇ut+1−l22l2h2⋄∇vt+∇ut2+∇vt2+2jβ1−j+1/jj+1∫Ωuk+vlutj+1dx+2sβ2−s+1/ss+1∫Ωvθ+uρvts+1dx.
In order to estimate theϜ′t, we set
(48)Ϝ1t=∫ΩΔut−1η+1utηut0th1t−sut−usdsdx,Then, by using equations (1), we have
(49)Ϝ1′t=−∫Ω∫0th1t−s∇usds∫0th1t−s∇ut−∇usdsdx+∫Ωuk+vlutj−1ut∫0th1t−sut−usdsdx+∫Ω∇ut∫0th1t−s∇ut−∇usdsdx−∫Ω∇utt∫0th1′t−s∇ut−∇usdsdx−1η+1∫Ωutηut∫0th1′t−sut−usdsdx−∫Ωf1u,v∫0th1t−sut−usdsdx−1η+1∫0th1sutη+2η+2−∫0th1s∇ut2.
For the first term of (33), by applying (A1), Hölder’s inequality, and Young’s inequality, we deduce
(50)−∫Ω∫0th1t−s∇usds∫0th1t−s∇ut−∇usdsdx≤δ∫Ω0th1t−s∇usds2dx+14δ∫Ω∫0th1t−s∇ut−∇usds2dx≤δ∫Ω∫0th1t−s∇us−∇ut+∇utds2dx+14δ∫0th1sdsh1⋄∇ut≤2δ1−l12∇u2+2δ+14δ1−l1h1⋄∇ut,∀δ>0.
Then, in order to estimate the following term, we seperate such that(51)∫Ωuk+vlutj−1ut∫0th1t−sut−usdsdx=I1+I2,where
(52)I1=∫Ωukutj−1ut∫0th1t−sut−usdsdx,I2=∫Ωvlutj−1ut∫0th1t−sut−usdsdx.
By Hölder’s inequality, Young’s inequality, (15), and (21), we get
(53)I1≤∫Ωukutj+1dxj/j+1∫Ωuk∫0th1t−sut−usdsj+1dx1/j+1≤jδ−j+1/jj+1∫Ωuk+vlutj+1dx+δj+1j+112u2k2k+12∫Ω∫0th1t−sut−usds2j+1dx≤jδ−j+1/jj+1∫Ωuk+vlutj+1dx+δj+1j+1c∗2k2χ1k−1∇u2+c∗2j+22χ1j1−l12j+12h1⋄∇utI2≤jδ−j+1/jj+1∫Ωuk+vlutj+1dx+δj+1j+1c∗2l2χ2l−1∇v2+c∗2j+22χ1j1−l12j+12h1⋄∇ut.
From (A1) assumption, Hölder’s inequality, and Young’s inequality, we get(54)∫Ω∇ut∫0th1t−s∇ut−∇usdsdx≤δ∇u2+1−l14δh1⋄∇ut.
In order to estimate the forth term, we use Young’s inequality, Sobolev-Poincare inequality, Hölder’s inequality, and (A1) assumption(55)−∫Ω∇utt∫0th1′t−s∇ut−∇usdsdx≤δ∇ut2−h104δh1′⋄∇ut.−1η+1∫Ωutηut∫0th1′t−sut−usdsdx≤1η+1δut2η+12η+1+14δ∫Ω∫0th1′t−sut−usds2dx≤1η+1δut2η+12η+1−h10c∗24δ∫Ω∫0th1′t−s∇ut−∇us2dsdx≤δc∗2η+1η+12κ+2E0κ+1η∇ut2−h10c∗24δη+1h1′⋄∇ut.
Combining these estimates (34)-(40) and (33) becomes
(56)Ϝ1′t≤2δ+12δ1−l1+δj+1c∗2j+22χ1j1−l12j+1j+1h1⋄∇ut+δ+2δ1−l12+δj+1j+1c∗2k2χ1k−1∇u2+δj+1j+1c∗2l2χ2l−1∇v2−h104δc∗2η+1+1h1′⋄∇ut+δ+δc∗2η+1η+12κ+2E0κ+1η−∫0th1sds∇ut2+2jδ−j+1/jj+1∫Ωuk+vlutj+1dx−1η+1∫0th1sdsutη+2η+2−∫Ωf1u,v∫0th1t−sut−usdsdx.
Similarly, let(57)Ϝ2t=ΩΔvt−1η+1vtηvt0th2t−svt−vsdsdx,then
(58)Ϝ2′t≤2δ+12δ1−l2+δs+1c∗2s+22χ2s1−l22s+1s+1h2⋄∇vt+δ+2δ1−l22+δs+1s+1c∗2θ2δ2θ−1∇v2+δs+1s+1c∗2ϱ2χ1ϱ−1∇u2−h204δ1+c∗2η+1h2′⋄∇vt+δ+δc∗2η+1η+12κ+2E0κ+1η−∫0th2sds∇vt2+2sδ−s+1/ss+1∫Ωvθ+uϱvts+1dx−∫Ωf2u,v∫0th2t−svt−vsdsdx−1η+1∫0th2sdsvtη+2η+2.
Since the functionhii=1,2 is positive, then for any t0>0,
(59)∫0thisds≥∫0t0hisds=hi0≥h3>0,∀t≥t0.
Hence, we conclude from (17), (10), (32), (41), and (42) that
(60)Γ′t≤−1η+1h3−εutη+2η+2+vtη+2η+2−h3−ε−δ−δc∗2η+1η+12κ+2E0κ+1η∇ut2+∇vt2−εl14−c2∇u2−εl24−c3∇v2+ε1−l12l1+c4h1⋄∇ut+ε1−l22l2+c5h2⋄∇vt+2εκ+2∫ΩFu,vdx+M2−h104δc∗2η+1+1h1′⋄∇ut+M2−h204δc∗2η+1+1h2′⋄∇vt+2jβ1−j+1/jj+1ε+1−M∫Ωuk+vlutj+1dx+2sβ2−s+1/ss+1ε+1−M∫Ωvθ+uρvts+1dx−∫Ωf1u,v∫0th1t−sut−usdsdx−∫Ωf2u,v∫0th2t−svt−vsdsdx,where
(61)c2=2δ1−l12+δj+1c∗2kχ1k−12j+1+δs+1c∗2ρχ1ρ−12s+1,c3=2δ1−l22+δs+1c∗2θχ2θ−12s+1+δj+1c∗2lχ2l−12j+1,c4=2δ+12δ1−l1+δj+1c∗2j+22χ1j1−l12j+1j+1,c5=2δ+12δ1−l2+δs+1c∗2s+22χ2s1−l22s+1s+1.
By using Lemma8 and (15) for the last two terms of (43), we obtain
(62)Γ′t≤−1η+1ε−h3utη+2η+2+vtη+2η+2−ε+δ+δc∗2η+1η+12κ+2E0κ+1η−h3∇ut2+∇vt2−εl14−c2+β1+β2δl1∇u2−εl24−εc3+β1+β2δl2∇v2+ε2l1+c∗24δ1−l1+c4h1⋄∇ut+ε2l2+c∗24δ1−l2+c5h2⋄∇vt+M2−h104δc∗2η+1+1h1′⋄∇ut+M2−h204δc∗2η+1+1h2′⋄∇ut+2εκ+2∫ΩFu,vdx+2jβ1−j+1/jj+1ε+1−M∫Ωuk+vlutj+1dx+2sβ2−s+1/ss+1ε+1−M∫Ωvθ+uρvts+1dx.
At this point, we chooseε and δ which are small enough, and we have
(63)1η+1h3−ε>0,h3−ε−δ−δc∗2η+1η+12κ+2E0κ+1η>0,εl14−c2+β1+β2δl1>0,εl24−c2+β1+β2δl2>0.
Further, we pickε so small and
(64)2jβ1−j+1/jj+1ε+1−M<0,2sβ2−s+1/ss+1ε+1−M<0.
Onceδ is fixed, we choose M that is sufficently large so that
(65)M2−h104δc∗2η+1+1≥0,M2−h204δc∗2η+1+1≥0.
Consequently, for allt≥t0, we reach at
(66)Γ′t≤−ξ1Et+ξ2h1⋄∇ut+h2⋄∇vt,where there are positive constants ξi,i=1,2.Now, we are ready to state our stability result.Theorem 10.
Suppose that (4) and (A1)-(A3) hold, and that u0,u1∈H01Ω×L2Ω and v0,v1∈H01Ω×L2Ω satisfy E0<E1 and
(67)l1∇u02+l2∇v021/2<α∗.
Then for each, the energy of (1) satisfies
(68)Et≤Ke−kt0tδsds,t≥t0,where δt≔minδ1t,δ2t and K and k are positive constants.Proof.
Multiplying (46) by δt, we get
(69)δtΓ′t≤−ξ1δtEt+ξ2δth1⋄∇ut+h2⋄∇vt.
Applying (A2) andδt≔minδ1t,δ2t and since −h1′⋄∇ut+h2′⋄∇vt≤−2E′t by (10), we obtain
(70)δtΓ′t≤−ξ1δtEt−ξ2δth1′⋄∇ut+h2′⋄∇vt≤−ξ1δtEt−2ξ2E′t,∀t≥t0.
That is(71)G′t≤−c∗δtEt≤−kδtGt,∀t≥t0.And here, Gt=δtΓt+CEt is equivalent to Et due to (20), and k is a positive constant. A simple integration of (50) leads to
(72)Gt≤Gt0e−kt0tδsds,∀t≥t0.
This completes the proof.


## 5. Conclusion

As far as we know, there have not been any global existences and general decay results in the literature known for quasilinear viscoelastic equations with degenerate damping terms. Our work extends the works for some quasilinear viscoelastic equations treated in the literature to the quasilinear viscoelastic equation with degenerate damping terms.



---

*Source: 4316238-2021-06-30.xml*