# Weakly Coupled Systems of Semilinear Damped Waves with Different Scale-Invariant Time-Dependent Dissipation Terms

Authors: Abdelhamid Mohammed Djaouti
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, our interest is the global existence of small data solutions to the Cauchy problem for weakly coupled systems of semilinear damped waves with different scale-invariant dissipation terms, where the data are supposed to belong to different classes of regularity and different power nonlinearities.


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## Body

## 1. Introduction

In this paper, we investigate the following weakly coupled system:(1)utt−Δu+μ11+tut=vp,u0,x=u0x,ut0,x=u1x,vtt−Δv+μ21+tvt=uq,v0,x=v0x,vt0,x=v1x,where t,x∈0,∞×ℝn and μ1,μ2>1 are real constants.Let us first present some history and previous results for problems strongly related to our model. In the last decades, the Cauchy problem for wave equation(2)utt−Δu=ft,u,u0,x=u0x,ut0,x=u1x,where t,x∈0,∞×ℝn, was investigated by a lot of researchers. For ft,u=up,p>1, the critical exponent describes the threshold between global (in time) existence of small data weak solutions and blow-up of local (in time) small data weak solutions (for more details, the readers can see [1–16]). Let us turn to the study of semilinear damped wave equation with power nonlinearity(3)utt−Δu+ut=ft,u,u0,x=u0x,ut0,x=u1x,where t,x∈0,∞×ℝn. Matsumura in [17] proved the estimates for the homogeneous problem. Having these estimates, Nakao and Ono in [18] proved the local (in time) existence of energy solutions for p≤pGN≔n/n−2,n≥3, and the global (in time) existence using special technique.Problem (3) was studied also in [19–24] where the Fujita exponent pFuj≔1+2/n plays a pivotal role as critical exponent. It has been proved in [25] that the Fujita exponent remains critical if we consider in (3) the wave equation with a time-dependent effective damping btut satisfying suitable assumptions. So, the model we have in mind is(4)utt−Δu+btut=ft,u,u0,x=u0x,ut0,x=u1x.The dissipation term is effective with respect to classification of Wirth [26] which means that the linear estimates have the same decay rate of the corresponding heat equation btut−Δu=0 (for more details about the classification, the readers can see [23, 27–30]). The set of coefficients of effective damping can be presented by functions satisfying the following assumptions:(i)
b is a positive and monotonic function with tbt⟶∞ as t⟶∞.(ii)
1+t2bt−1∈L10,∞.(iii)
b∈C30,∞ and bkt≲bt/1+tkfork=1,2,3.(iv)
1/b∉L10,∞ and there exists a constant a∈0,1 such that tb′t≤abt.For the special casebt=μ/1+tr,μ>0,r<1, the global existence has been obtained in [31, 32]. If bt is a sufficiently smooth function satisfying limsupt⟶∞tbt<1, then the dissipation is noneffective [33].Many papers treated the limit wherer=1 which means the following semilinear Cauchy problem:(5)utt−Δu+μ1+tut=up,u0,x=u0x,ut0,x=u1x.In [34–36], the authors showed that the situation depends strongly on the value of μ; in other words, the transition of μ from 1 to ∞ describes the change from a hyperbolic to a parabolic-like model from the point of decay estimates for solutions. Furthermore, they proved that the decay rate of solutions for large μ is the same as that obtained for solutions of the classical damped wave equation. A particular case of the Cauchy problem (5) with μ=2 was studied in [37].Some papers are devoted to the weakly coupled systems of semilinear classical damped wave equations with power nonlinearities. The model we have in mind is(6)utt−Δu+ut=up,u0,x=u0x,ut0,x=u1x,utt−Δv+vt=up,v0,x=v0x,vt0,x=v1x,where t,x∈0,∞×ℝn. In 2007, Sun and Wang proved in [38] that if(7)maxp;q+1pq−1<n2,for n=1 or 3, then the solution exists globally in time for small initial data, while if maxp;q+1/pq−1≥n/2, then every solution having positive average value does not exist globally. In [39], the author generalized the previous results to the case where n=1,2,3 and improved time decay estimates for n=2. In 2014, using the weighted energy method, Nishihara determined in [32] the critical exponent for any space dimension.In [40–43], the authors studied the system of weakly coupled semilinear damped waves with time-dependent coefficients in the dissipation terms(8)utt−Δu+btut=vp,vtt−Δv+btvt=uq,u0,x=u0x,ut0,x=u1x,v0,x=v0x,vt0,x=v1x.Recently, in [43], the author treated problem (8) with modified nonlinearities.Let us now turn to our main problem described in (1). In this paper, we will show how the interaction between parameters μ1 and μ2 influence the results obtained for the effective case, that is, the change from effective to noneffective.The paper is organized as follows. We start by some background and previous results for single equation. After that, we will show our main results of global (in time) existence; moreover, we summarize all cases appearing in Section2 in a table. Next, in Section 3, we prove existence of solution by applying Banach’s fixed point. Section 4 concludes the paper.We introduce fors>0 and m∈1,2 the function space(9)Am,s≔Hs∩Lm×Hmaxs−1;0∩Lm,with the norm(10)u,vAm,s≔uHs+uLm+vHmaxs−1;0+vLm.In [41], the estimates for the solution to the Cauchy problem(11)utt−Δu+μ1+tut=0,u0,x=u0x,ut0,x=u1xwere proved for different classes of regularity of the data, low regular data, data from energy space, data from Sobolev spaces with suitable regularity, and large regular data. We summarize these results in the following theorems.Theorem 1.
Let us assume the datau0,u1∈Am,s with s>0. Then, the solution u to the Cauchy problem (11) satisfies for μ>1 the following decay estimates.
Fors≥0,(12)Dsut,·L2ℝn≲u0,u1Am,s×1+t−2−m/2mn−sifμ>2−mmn+2s,1+t−μ/21+log1+t2−m/2mifμ=2−mmn+2s,1+t−μ/2ifμ<2−mmn+2s,and for s≥1:(13)Ds−1utt,·L2ℝn≲u0,u1Am,s×1+t−2−m/2mn−sifμ>2−mmn+2s,1+t−μ/21+log1+t2−m/2mifμ=2−mmn+2s,1+t−μ/2ifμ<2−mmn+2s.In order to use Duhamel’s principle in the next sections, we consider the family of parameter-dependent Cauchy problems(14)vtt−Δv+μ1+tvt=0,vτ,x=0,vtτ,x=v1τ,x,0≤τ≤t.Theorem 2.
Let us assumev1∈Hmaxs−1;0ℝn∩Lmℝn with s>0. Then, the solution v to the Cauchy problem (14) satisfies for μ>1 the following decay estimates for s≥0:(15)Ds−1vtt,·L2ℝn≲v1Lmℝn+1+τ2−m/2mn+s−1+v1H˙s−1+ℝn1+τ×1+t−2−m/2mn−sifμ>2−mmn+2s,1+t−μ/21+log1+t2−m/2mifμ=2−mmn+2s,1+t−μ/2ifμ<2−mmn+2s,and for s≥1:(16)Ds−1vtt,·L2ℝn≲v1Lmℝn+1+τ2−m/2mn+s−1+v1H˙s−1+ℝn+1+τ2−m/2mn+s−2+v1H˙s−2+ℝn1+τ×1+t−2−m/2mn−s,ifμ>2−mmn+2s,1+t−μ/21+log1+t2−m/2m,ifμ=2−mmn+2s,1+t−μ/21+τ−2−m/2mn+μ/2−s,ifμ<2−mmn+2s,where H˙s is the homogeneous Sobolev space.


## 2. Main Results

### 2.1. Low Regular Data

In this section, we are interested in Cauchy problem (1), where the initial data are supposed to have low regularity; or in other words, the data belong to the Sobolev space Hsℝn for s∈0,1, with additional regularity Lmℝn for m∈1,2.From the estimates of Theorem1 and further considerations, we remark the existence of five cases corresponding to the value of μ. These cases are as follows:(17)μ>2−mmn+2s,μ=2−mmn+2s,2−mmn+2s>μ>2−mmn,μ=2−mmn,μ<2−mmn.These cases generate for the system (1) a lot of cases corresponding to the values of μ1 and μ2. In Theorem 3, we restrict ourselves to three cases which are from our point of view are more interesting and important. The remaining cases will be treated in Remark 1.Lemma 1.
Letp satisfy the conditions(18)2m≤p,ifn=1ands∈12,1,2m≤p≤pGN,sn,otherwise.
Then, the following statements are valid:(i)
If(19)Mτ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+sDsuτ,·L2ℝn,then(20)uτ,xpLmℝn≲1+τ−n/mp−1Mτ,up,uτ,xpLmℝn≲1+τ−n/mp+n/2Mτ,up.These estimates imply(21)uτ,xpLmℝn+1+τ2−m/2mnuτ,xpL2ℝn≲1+τ−n/mp−1Mτ,up.(ii)
If(22)Mτ,u=1+τμ/2uτ,·L2ℝn+Dsuτ,·L2ℝn,then(23)uτ,xpLmℝn+uτ,xpL2ℝn≲1+τ−μ/2pMτ,up.These estimates imply(24)uτ,x|pLmℝn+1+τ2−m/2mnuτ,xpL2ℝn≲1+τ−μ/2p+2−m/2mnMτ,up.The proof of Lemma1 is basically concluded after using the Gagliardo–Nirenberg inequality from Proposition A.1 and the definition of the space Mτ,u.Theorem 3.
Letn≤4s/2−m. The data u0,u1,v0,v1 are supposed to belong to Am,s×Am,s, s∈0,1, m∈1,2, and minμ1;μ2>1. Moreover, let the exponents p and q of the power nonlinearities satisfy condition (18) and(25)minp;q>pFuj,mnifminμ1;μ2>2−mmn+2s,(26)p>4μ2+2−mμ2mn,q>2mn+m2+nμ22mifμ1>2−mmn+2s,μ2<2−mmn,(27)p>μ1μ2+4μ2,q>μ2μ1+4μ1ifmaxμ1;μ2<2−mmn.Then, there exists a uniquely determined global (in time) Sobolev solution to (1) in(28)C0,∞,Hsℝn2.Furthermore, the solution satisfies fors∈0,1 the following decay estimates:(29)Dsut,·L2ℝn≲u0,u1Am,s+v0,v1Am,s×1+t−2−m/2mn−sifμ1>2−mmn+2s,1+t−μ1/2ifμ1<2−mmn,Dsvt,·L2ℝn≲u0,u1Am,s+v0,v1Am,s×1+t−2−m/2mn−sifμ2>2−mmn+2s,1+t−μ1/2ifμ2<2−mmn.Example 1.
If we consider model (1) for n=2, m=11/10, and s=9/10, then by using (25)–(27) from Theorem 3, we obtain the following statements:(30)ifμ1=4,μ2=3,thenminp;q>pFuj,11/102=2110,ifμ1=4,μ2=2,thenp>5322∼2.4,q>763220∼3.46,ifμ1=2,μ2=32,thenp>4,q>114=2.75.Remark 1.
In the following table, we will show all conditions generated by the different cases for the values ofμ1 and μ2 (see Table 1).Table 1
Cases of the interplay betweenμ1 and μ2.
caseAdmissible range forpAdmissible range forq(1)μ1>2−m/mn+2sμ2=2−m/mn+2sp>pFuj,mnq>pFuj,mn(2)μ1>2−m/mn+2s2−m/mn<μ2<2−m/mn+2sp>min4m/2−mn+1;2+n/m2/μ2−2s+nq>pFuj,mn(3)μ1>2−m/mn+2sμ2=2−m/mnp>1+4/μ2q>pFuj,mn(4)μ1=μ2=2−m/mn+2sp>pFuj,mnq>pFuj,mn(5)μ1=2−m/mn+2s2−m/mn<μ2<2−m/mn+2sp>min4m/2−mn+1;2+n/m2/μ2−2s+nq>pFuj,mn(6)μ1=2−m/mn+2sμ2=2−m/mnp>1+4/μ2q>pFuj,mn(7)μ1=2−m/mn+2sμ2=2−m/mnp>4/μ2+2−m/μ2mnq>2m/n+m/2+nμ2/2m(8)2−m/mn<μ1<2−m/mn+2s2−m/mn<μ2<2−m/mn+2sp>min4m/2−mn+1;2+n/m2/μ2−2s+nq>min4m/2−mn+1;2+n/m2/μ1−2s+n(9)2−m/mn<μ1<2−m/mn+2sμ2=2−m/mnp>1+4/μ2q>min4m/2−mn+1;2+n/m2/μ1−2s+n(10)2−m/mn<μ1<2−m/mn+2sμ2=2−m/mnp>4/μ2+2−m/μ2mnq>min4m/2−mn+1;2+n/m2/μ1−2s+n(11)μ1=μ2=2−m/mnp>1+4/μ2q>1+4/μ1(12)μ1=2−m/mnμ2=2−m/mnp>1+4/μ2q>μ2/μ1+4/μ1


### 2.2. Data from Energy Space

If the data are in the energy space, then we get fors=1 a similar case to the case of the previous section because the estimates for Ds=1ut,·L2ℝn and utt,·L2ℝn coincide with those of the previous section. Moreover, we obtain the global existence in time of energy solutions. Consequently, we have the following result.Theorem 4.
Letn≤4/2−m,u0,u1,v0,v1∈Am,1×Am,1,m∈1,2, and minμ1;μ2>1. Moreover, let the exponents p and q of power nonlinearities satisfy condition (18) and(31)minp;q>pFuj,mnifminμ1;μ2>2−mmn+2,p>4μ2+2−mμ2mn,q>2mn+m2+nμ22mifμ1>2−mmn+2,μ2<2−mmn,p>μ1μ2+4μ2,q>μ2μ1+4μ1ifmaxμ1;μ2<2−mmn.
Then, there exists a small constantϵ0 such that if(32)u0,u1Am,1+v0,v1Am,1≤ε0,then there exists a uniquely determined global (in time) energy solution to (1) in(33)C0,∞,H1ℝn∩C10,∞,L2ℝn2.
Furthermore, the solution satisfies the following decay estimates:(34)utt,·L2ℝn+∇ut,·L2ℝn≲u0,u1Am,1+v0,v1Am,1×1+t−2−m/2mn−1,ifμ1>2−mmn+2,1+t−μ1/2,ifμ1<2−mmn,utt,·L2ℝn≲u0,u1Am,1+v0,v1Am,1×1+t−2−m/2mn−1,ifμ1>2−mmn+2,1+t−μ1/2,ifμ1<2−mmn,vtt,·L2ℝn+∇vt,·L2ℝn≲u0,u1Am,1+v0,v1Am,1×1+t−2−m/2mn−1,ifμ2>2−mmn+2,1+t−μ1/2,ifμ2<2−mmn,vtt,·L2ℝn≲u0,u1Am,1+v0,v1Am,1×1+t−2−m/2mn−1,ifμ2>2−mmn+2,1+t−μ1/2,ifμ2<2−mmn.Example 2.
If we consider system (1) for n=2 and m=2, then for given μ1=μ2=21/10, we get after using the last case the following admissible ranges for p and q:(35)minp;q>3.


### 2.3. Data from Sobolev Spaces with Suitable Regularity

This section is devoted to the case where the data are from Sobolev spaces with suitable regularity. We will treat the same cases of the previous sections corresponding to the values ofμ1 and μ2. In the following lemma, we will provide some estimates which are important tools in the proofs of our main results.Lemma 2.
Letp>ssatisfy the following condition:(36)2<p,ifs∈n2,n2+1,2<p,ifs∈2,n2.
Then, the following statements are valid.(i)
If(37)Mτ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+1utτ,·L2ℝn+1+τ2−m/2mn+sDsuτ,·L2ℝn+Ds−1utτ,·L2ℝn,then(38)uτ,xpLmℝn≲1+τ−n/mp−1Mτ,up,uτ,xpL2ℝn≲1+τ−n/mp+n/2Mτ,up,uτ,xpH˙s−1ℝn≲1+τ−n/mp+n/2−s−1Mτ,up,(39)uτ,xp−u˜τ,xpH˙s−1ℝn≲1+Bτ,0−n/mp+n/2−s−1×Mτ,v−v˜Mτ,up−1+Mτ,u˜p−1.These estimates imply(40)upLmℝn+1+τ2−m/2mn+s−1upH˙s−1ℝn+1+τ2−m/2mn+s−1upH˙s−2ℝn×1+τ≲1+τ−n/mp−1+1Mτ,up.(ii)
If(41)Mτ,u=1+τμ/2uτ,·L2ℝn+utτ,·L2ℝn+Dsuτ,·L2ℝn+Ds−1utτ,·L2ℝn,then(42)uτ,xpLmℝn+uτ,xpL2ℝn+uτ,xpH˙sℝn≲1+τ−μ/2pMτ,up,(43)uτ,xp−u˜τ,xpH˙s−1ℝn≲1+Bτ,0−μ/2pMτ,v−v˜Mτ,up−1+Mτ,u˜p−1.These estimates imply(44)upLmℝn+1+τ2−m/2mn+s−1upH˙s−1ℝn+1+τ2−m/2mn+s−2upH˙s−2ℝn×1+τ≲1+τ−μ/2p+2−m/2mn+sMτ,up.Using thefractional chain rule from the Appendix and the definition of space Mτ,u, one can prove the desired statements.Theorem 5.
Letn≥4. The regularity parameterss1ands2satisfy the following conditions:(45)s1,s2∈2,n2+1,0<s2−s1<1ands1≠s2.
The datau0,u1andv0,v1are supposed to belong toAm,s1×Am,s2 with m∈1,2. Furthermore, we assume for the exponents p and q the following conditions:(46)s1<p,s2<q,ifn≤2s1,s1<p,s2<q≤1+2n−2s1,if2s1<n≤2s2,s1<p≤1+2n−2s2,s2<q≤1+2n−2s1,ifn>2s2,minp;q>pFuj,mn,ifμ1>2−mmn+2s1,μ2>2−mmn+2s2,p>2+2s2μ2+2−mμ2mn,q>2mn+m2+nμ22m,ifμ1>2−mmn+2s1,μ2<2−mmn,p>μ1μ2+2+2s2μ2,q>μ2μ1+2+2s1μ1,ifmaxμ1;μ2<2−mmn.Then, there exists a uniquely determined global (in time) energy solution to (1) in(47)C0,∞,Hs1ℝn∩C10,∞,Hs1−1ℝn×C0,∞,Hs2ℝn∩C10,∞,Hs2−1ℝn.Furthermore, the solution satisfies the following decay estimates:(48)ut,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mnifμ1>2−mmn+2s1,1+t−μ1/2ifμ1<2−mmn,utt,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mnifμ1>2−mmn+2s1,1+t−μ1/2ifμ1<2−mmn,Ds1uτ,·L2ℝn+Ds1−1utτ,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn−s1ifμ1>2−mmn+2s1,1+t−μ1/2ifμ1<2−mmn,vt,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mnifμ2>2−mmn+2s2,1+t−μ2/2ifμ2<2−mmn,vtt,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mnifμ2>2−mmn+2s2,1+t−μ2/2ifμ2<2−mmn,Ds2vτ,·L2ℝn+Ds2−1vtτ,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn−s1ifμ2>2−mmn+2s2,1+t−μ2/2ifμ2<2−mmn.


### 2.4. Large Regular Data

Comparing this section, where the data are supposed to have large regularity, with the previous section, we feel differences in the treatment only ifμ1 and μ2 are sufficiently large. For this reason, we restrict ourselves to formulate the results without giving a proof. The proof is very similar to the proof of Theorem 5.Lemma 3.
Letp>s. Then, using the rule for fractional powers from Proposition D.1 in the Appendix, the following estimates hold: if(49)Mτ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+1utτ,·L2ℝn+1+τ2−m/2mn+sDsuτ,·L2ℝn+Ds−1utτ,·L2ℝn,then(50)uτ,xpLmℝn≲1+τ−n/mp−1Mτ,up,uτ,xpL2ℝn≲1+τ−n/mp+m/2Mτ,up,uτ,xpH˙sℝn≲1+τ−2−m/2mnp−s−s∗p−1Mτ,up,uτ,xp−u˜τ,xpH˙s−1ℝn≲1+Bτ,0−2−m/2mnp−s−1−s∗p−1×Mτ,v−v˜Mτ,up−1+Mτ,u˜p−1,where we assume s∗<n/2.
We remark that(51)−n21m−12p−s−12−ε2−s∗2p−1≤−n2mp+n4.
These estimates imply the estimate(52)upLmℝn+1+τ2−m/2mn+s−1upH˙s−1ℝn+1+τ2−m/2mn+s−2upH˙s−2ℝn1+τ≲1+τ−n/mp−1+sMτ,up.
If(53)Mτ,u=1+τμ/2uτ,·L2ℝn+utτ,·L2ℝn+Dsuτ,·L2ℝn+Ds−1utτ,·L2ℝn,then(54)uτ,xpLmℝn+uτ,xpL2ℝn+uτ,xpH˙sℝn≲1+τ−μ/2pMτ,up,uτ,xp−u˜τ,xpH˙s−1ℝn≲1+Bτ,0−μ/2p×Mτ,v−v˜Mτ,up−1+Mτ,u˜p−1.
These estimates imply the estimate(55)upLmℝn+1+τ2−m/2mn+s−1upH˙s−1ℝn+1+τ2−m/2mn+s−2upH˙s−2ℝn1+τ≲1+τ−μ/2p+2−m/2mn+sMτ,up.Theorem 6.
Letn≥4. The datau0,u1andv0,v1are supposed to belong toAm,s1×Am,s2withm∈1,2ands2>s1>n/2+1. Moreover, we assume(56)p>s1,q>s˜2,where s˜2∈s1,s1+1s and s˜2≤s2. Furthermore, we assume for the exponents p and q the following conditions:(57)p>1+s˜2+1mn,q>1+s1+1mn,ifμ1>2−mmn+2s1,μ2>2−mmn+2s˜2,p>2+2s˜2μ2+2−mμ2mn,q>2+2s˜2mn+m2+nμ22m,ifμ1>2−mmn+2s1,μ2<2−mmn,p>μ1μ2+2+2s˜2μ2,q>μ2μ1+2+2s1μ1,ifmaxμ1;μ2<2−mmn.Then, there exists a uniquely determined global (in time) energy solution to (1) in(58)C0,∞∩C10,∞,Hs1−1ℝn×C10,∞,Hs˜2ℝn∩C10,∞,Hs˜2−1ℝn.Furthermore, the solution satisfies the following decay estimates:(59)ut,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn,ifμ1>2−mmn+2s1,1+t−μ1/2,ifμ1<2−mmn,utt,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn−1,ifμ1>2−mmn+2s1,1+t−μ1/2,ifμ1<2−mmn,Ds1uτ,·L2ℝn+Ds1−1uτ,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn−s1,ifμ1>2−mmn+2s1,1+t−μ1/2,ifμ1<2−mmn,vt,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn,ifμ2>2−mmn+2s˜2,1+t−μ1/2,ifμ2<2−mmn,vtt,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn−1,ifμ2>2−mmn+2s˜2,1+t−μ1/2,ifμ2<2−mmn,Ds˜2uτ,·L2ℝn+Ds˜2−1uτ,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn−s˜2,ifμ2>2−mmn+2s˜2,1+t−μ1/2,ifμ2<2−mmn.


## 2.1. Low Regular Data

In this section, we are interested in Cauchy problem (1), where the initial data are supposed to have low regularity; or in other words, the data belong to the Sobolev space Hsℝn for s∈0,1, with additional regularity Lmℝn for m∈1,2.From the estimates of Theorem1 and further considerations, we remark the existence of five cases corresponding to the value of μ. These cases are as follows:(17)μ>2−mmn+2s,μ=2−mmn+2s,2−mmn+2s>μ>2−mmn,μ=2−mmn,μ<2−mmn.These cases generate for the system (1) a lot of cases corresponding to the values of μ1 and μ2. In Theorem 3, we restrict ourselves to three cases which are from our point of view are more interesting and important. The remaining cases will be treated in Remark 1.Lemma 1.
Letp satisfy the conditions(18)2m≤p,ifn=1ands∈12,1,2m≤p≤pGN,sn,otherwise.
Then, the following statements are valid:(i)
If(19)Mτ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+sDsuτ,·L2ℝn,then(20)uτ,xpLmℝn≲1+τ−n/mp−1Mτ,up,uτ,xpLmℝn≲1+τ−n/mp+n/2Mτ,up.These estimates imply(21)uτ,xpLmℝn+1+τ2−m/2mnuτ,xpL2ℝn≲1+τ−n/mp−1Mτ,up.(ii)
If(22)Mτ,u=1+τμ/2uτ,·L2ℝn+Dsuτ,·L2ℝn,then(23)uτ,xpLmℝn+uτ,xpL2ℝn≲1+τ−μ/2pMτ,up.These estimates imply(24)uτ,x|pLmℝn+1+τ2−m/2mnuτ,xpL2ℝn≲1+τ−μ/2p+2−m/2mnMτ,up.The proof of Lemma1 is basically concluded after using the Gagliardo–Nirenberg inequality from Proposition A.1 and the definition of the space Mτ,u.Theorem 3.
Letn≤4s/2−m. The data u0,u1,v0,v1 are supposed to belong to Am,s×Am,s, s∈0,1, m∈1,2, and minμ1;μ2>1. Moreover, let the exponents p and q of the power nonlinearities satisfy condition (18) and(25)minp;q>pFuj,mnifminμ1;μ2>2−mmn+2s,(26)p>4μ2+2−mμ2mn,q>2mn+m2+nμ22mifμ1>2−mmn+2s,μ2<2−mmn,(27)p>μ1μ2+4μ2,q>μ2μ1+4μ1ifmaxμ1;μ2<2−mmn.Then, there exists a uniquely determined global (in time) Sobolev solution to (1) in(28)C0,∞,Hsℝn2.Furthermore, the solution satisfies fors∈0,1 the following decay estimates:(29)Dsut,·L2ℝn≲u0,u1Am,s+v0,v1Am,s×1+t−2−m/2mn−sifμ1>2−mmn+2s,1+t−μ1/2ifμ1<2−mmn,Dsvt,·L2ℝn≲u0,u1Am,s+v0,v1Am,s×1+t−2−m/2mn−sifμ2>2−mmn+2s,1+t−μ1/2ifμ2<2−mmn.Example 1.
If we consider model (1) for n=2, m=11/10, and s=9/10, then by using (25)–(27) from Theorem 3, we obtain the following statements:(30)ifμ1=4,μ2=3,thenminp;q>pFuj,11/102=2110,ifμ1=4,μ2=2,thenp>5322∼2.4,q>763220∼3.46,ifμ1=2,μ2=32,thenp>4,q>114=2.75.Remark 1.
In the following table, we will show all conditions generated by the different cases for the values ofμ1 and μ2 (see Table 1).Table 1
Cases of the interplay betweenμ1 and μ2.
caseAdmissible range forpAdmissible range forq(1)μ1>2−m/mn+2sμ2=2−m/mn+2sp>pFuj,mnq>pFuj,mn(2)μ1>2−m/mn+2s2−m/mn<μ2<2−m/mn+2sp>min4m/2−mn+1;2+n/m2/μ2−2s+nq>pFuj,mn(3)μ1>2−m/mn+2sμ2=2−m/mnp>1+4/μ2q>pFuj,mn(4)μ1=μ2=2−m/mn+2sp>pFuj,mnq>pFuj,mn(5)μ1=2−m/mn+2s2−m/mn<μ2<2−m/mn+2sp>min4m/2−mn+1;2+n/m2/μ2−2s+nq>pFuj,mn(6)μ1=2−m/mn+2sμ2=2−m/mnp>1+4/μ2q>pFuj,mn(7)μ1=2−m/mn+2sμ2=2−m/mnp>4/μ2+2−m/μ2mnq>2m/n+m/2+nμ2/2m(8)2−m/mn<μ1<2−m/mn+2s2−m/mn<μ2<2−m/mn+2sp>min4m/2−mn+1;2+n/m2/μ2−2s+nq>min4m/2−mn+1;2+n/m2/μ1−2s+n(9)2−m/mn<μ1<2−m/mn+2sμ2=2−m/mnp>1+4/μ2q>min4m/2−mn+1;2+n/m2/μ1−2s+n(10)2−m/mn<μ1<2−m/mn+2sμ2=2−m/mnp>4/μ2+2−m/μ2mnq>min4m/2−mn+1;2+n/m2/μ1−2s+n(11)μ1=μ2=2−m/mnp>1+4/μ2q>1+4/μ1(12)μ1=2−m/mnμ2=2−m/mnp>1+4/μ2q>μ2/μ1+4/μ1


## 2.2. Data from Energy Space

If the data are in the energy space, then we get fors=1 a similar case to the case of the previous section because the estimates for Ds=1ut,·L2ℝn and utt,·L2ℝn coincide with those of the previous section. Moreover, we obtain the global existence in time of energy solutions. Consequently, we have the following result.Theorem 4.
Letn≤4/2−m,u0,u1,v0,v1∈Am,1×Am,1,m∈1,2, and minμ1;μ2>1. Moreover, let the exponents p and q of power nonlinearities satisfy condition (18) and(31)minp;q>pFuj,mnifminμ1;μ2>2−mmn+2,p>4μ2+2−mμ2mn,q>2mn+m2+nμ22mifμ1>2−mmn+2,μ2<2−mmn,p>μ1μ2+4μ2,q>μ2μ1+4μ1ifmaxμ1;μ2<2−mmn.
Then, there exists a small constantϵ0 such that if(32)u0,u1Am,1+v0,v1Am,1≤ε0,then there exists a uniquely determined global (in time) energy solution to (1) in(33)C0,∞,H1ℝn∩C10,∞,L2ℝn2.
Furthermore, the solution satisfies the following decay estimates:(34)utt,·L2ℝn+∇ut,·L2ℝn≲u0,u1Am,1+v0,v1Am,1×1+t−2−m/2mn−1,ifμ1>2−mmn+2,1+t−μ1/2,ifμ1<2−mmn,utt,·L2ℝn≲u0,u1Am,1+v0,v1Am,1×1+t−2−m/2mn−1,ifμ1>2−mmn+2,1+t−μ1/2,ifμ1<2−mmn,vtt,·L2ℝn+∇vt,·L2ℝn≲u0,u1Am,1+v0,v1Am,1×1+t−2−m/2mn−1,ifμ2>2−mmn+2,1+t−μ1/2,ifμ2<2−mmn,vtt,·L2ℝn≲u0,u1Am,1+v0,v1Am,1×1+t−2−m/2mn−1,ifμ2>2−mmn+2,1+t−μ1/2,ifμ2<2−mmn.Example 2.
If we consider system (1) for n=2 and m=2, then for given μ1=μ2=21/10, we get after using the last case the following admissible ranges for p and q:(35)minp;q>3.


## 2.3. Data from Sobolev Spaces with Suitable Regularity

This section is devoted to the case where the data are from Sobolev spaces with suitable regularity. We will treat the same cases of the previous sections corresponding to the values ofμ1 and μ2. In the following lemma, we will provide some estimates which are important tools in the proofs of our main results.Lemma 2.
Letp>ssatisfy the following condition:(36)2<p,ifs∈n2,n2+1,2<p,ifs∈2,n2.
Then, the following statements are valid.(i)
If(37)Mτ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+1utτ,·L2ℝn+1+τ2−m/2mn+sDsuτ,·L2ℝn+Ds−1utτ,·L2ℝn,then(38)uτ,xpLmℝn≲1+τ−n/mp−1Mτ,up,uτ,xpL2ℝn≲1+τ−n/mp+n/2Mτ,up,uτ,xpH˙s−1ℝn≲1+τ−n/mp+n/2−s−1Mτ,up,(39)uτ,xp−u˜τ,xpH˙s−1ℝn≲1+Bτ,0−n/mp+n/2−s−1×Mτ,v−v˜Mτ,up−1+Mτ,u˜p−1.These estimates imply(40)upLmℝn+1+τ2−m/2mn+s−1upH˙s−1ℝn+1+τ2−m/2mn+s−1upH˙s−2ℝn×1+τ≲1+τ−n/mp−1+1Mτ,up.(ii)
If(41)Mτ,u=1+τμ/2uτ,·L2ℝn+utτ,·L2ℝn+Dsuτ,·L2ℝn+Ds−1utτ,·L2ℝn,then(42)uτ,xpLmℝn+uτ,xpL2ℝn+uτ,xpH˙sℝn≲1+τ−μ/2pMτ,up,(43)uτ,xp−u˜τ,xpH˙s−1ℝn≲1+Bτ,0−μ/2pMτ,v−v˜Mτ,up−1+Mτ,u˜p−1.These estimates imply(44)upLmℝn+1+τ2−m/2mn+s−1upH˙s−1ℝn+1+τ2−m/2mn+s−2upH˙s−2ℝn×1+τ≲1+τ−μ/2p+2−m/2mn+sMτ,up.Using thefractional chain rule from the Appendix and the definition of space Mτ,u, one can prove the desired statements.Theorem 5.
Letn≥4. The regularity parameterss1ands2satisfy the following conditions:(45)s1,s2∈2,n2+1,0<s2−s1<1ands1≠s2.
The datau0,u1andv0,v1are supposed to belong toAm,s1×Am,s2 with m∈1,2. Furthermore, we assume for the exponents p and q the following conditions:(46)s1<p,s2<q,ifn≤2s1,s1<p,s2<q≤1+2n−2s1,if2s1<n≤2s2,s1<p≤1+2n−2s2,s2<q≤1+2n−2s1,ifn>2s2,minp;q>pFuj,mn,ifμ1>2−mmn+2s1,μ2>2−mmn+2s2,p>2+2s2μ2+2−mμ2mn,q>2mn+m2+nμ22m,ifμ1>2−mmn+2s1,μ2<2−mmn,p>μ1μ2+2+2s2μ2,q>μ2μ1+2+2s1μ1,ifmaxμ1;μ2<2−mmn.Then, there exists a uniquely determined global (in time) energy solution to (1) in(47)C0,∞,Hs1ℝn∩C10,∞,Hs1−1ℝn×C0,∞,Hs2ℝn∩C10,∞,Hs2−1ℝn.Furthermore, the solution satisfies the following decay estimates:(48)ut,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mnifμ1>2−mmn+2s1,1+t−μ1/2ifμ1<2−mmn,utt,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mnifμ1>2−mmn+2s1,1+t−μ1/2ifμ1<2−mmn,Ds1uτ,·L2ℝn+Ds1−1utτ,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn−s1ifμ1>2−mmn+2s1,1+t−μ1/2ifμ1<2−mmn,vt,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mnifμ2>2−mmn+2s2,1+t−μ2/2ifμ2<2−mmn,vtt,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mnifμ2>2−mmn+2s2,1+t−μ2/2ifμ2<2−mmn,Ds2vτ,·L2ℝn+Ds2−1vtτ,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn−s1ifμ2>2−mmn+2s2,1+t−μ2/2ifμ2<2−mmn.


## 2.4. Large Regular Data

Comparing this section, where the data are supposed to have large regularity, with the previous section, we feel differences in the treatment only ifμ1 and μ2 are sufficiently large. For this reason, we restrict ourselves to formulate the results without giving a proof. The proof is very similar to the proof of Theorem 5.Lemma 3.
Letp>s. Then, using the rule for fractional powers from Proposition D.1 in the Appendix, the following estimates hold: if(49)Mτ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+1utτ,·L2ℝn+1+τ2−m/2mn+sDsuτ,·L2ℝn+Ds−1utτ,·L2ℝn,then(50)uτ,xpLmℝn≲1+τ−n/mp−1Mτ,up,uτ,xpL2ℝn≲1+τ−n/mp+m/2Mτ,up,uτ,xpH˙sℝn≲1+τ−2−m/2mnp−s−s∗p−1Mτ,up,uτ,xp−u˜τ,xpH˙s−1ℝn≲1+Bτ,0−2−m/2mnp−s−1−s∗p−1×Mτ,v−v˜Mτ,up−1+Mτ,u˜p−1,where we assume s∗<n/2.
We remark that(51)−n21m−12p−s−12−ε2−s∗2p−1≤−n2mp+n4.
These estimates imply the estimate(52)upLmℝn+1+τ2−m/2mn+s−1upH˙s−1ℝn+1+τ2−m/2mn+s−2upH˙s−2ℝn1+τ≲1+τ−n/mp−1+sMτ,up.
If(53)Mτ,u=1+τμ/2uτ,·L2ℝn+utτ,·L2ℝn+Dsuτ,·L2ℝn+Ds−1utτ,·L2ℝn,then(54)uτ,xpLmℝn+uτ,xpL2ℝn+uτ,xpH˙sℝn≲1+τ−μ/2pMτ,up,uτ,xp−u˜τ,xpH˙s−1ℝn≲1+Bτ,0−μ/2p×Mτ,v−v˜Mτ,up−1+Mτ,u˜p−1.
These estimates imply the estimate(55)upLmℝn+1+τ2−m/2mn+s−1upH˙s−1ℝn+1+τ2−m/2mn+s−2upH˙s−2ℝn1+τ≲1+τ−μ/2p+2−m/2mn+sMτ,up.Theorem 6.
Letn≥4. The datau0,u1andv0,v1are supposed to belong toAm,s1×Am,s2withm∈1,2ands2>s1>n/2+1. Moreover, we assume(56)p>s1,q>s˜2,where s˜2∈s1,s1+1s and s˜2≤s2. Furthermore, we assume for the exponents p and q the following conditions:(57)p>1+s˜2+1mn,q>1+s1+1mn,ifμ1>2−mmn+2s1,μ2>2−mmn+2s˜2,p>2+2s˜2μ2+2−mμ2mn,q>2+2s˜2mn+m2+nμ22m,ifμ1>2−mmn+2s1,μ2<2−mmn,p>μ1μ2+2+2s˜2μ2,q>μ2μ1+2+2s1μ1,ifmaxμ1;μ2<2−mmn.Then, there exists a uniquely determined global (in time) energy solution to (1) in(58)C0,∞∩C10,∞,Hs1−1ℝn×C10,∞,Hs˜2ℝn∩C10,∞,Hs˜2−1ℝn.Furthermore, the solution satisfies the following decay estimates:(59)ut,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn,ifμ1>2−mmn+2s1,1+t−μ1/2,ifμ1<2−mmn,utt,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn−1,ifμ1>2−mmn+2s1,1+t−μ1/2,ifμ1<2−mmn,Ds1uτ,·L2ℝn+Ds1−1uτ,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn−s1,ifμ1>2−mmn+2s1,1+t−μ1/2,ifμ1<2−mmn,vt,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn,ifμ2>2−mmn+2s˜2,1+t−μ1/2,ifμ2<2−mmn,vtt,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn−1,ifμ2>2−mmn+2s˜2,1+t−μ1/2,ifμ2<2−mmn,Ds˜2uτ,·L2ℝn+Ds˜2−1uτ,·L2ℝn≲u0,u1Am,s1+v0,v1Am,s2×1+t−2−m/2mn−s˜2,ifμ2>2−mmn+2s˜2,1+t−μ1/2,ifμ2<2−mmn.


## 3. Philosophy of Our Approach and Proofs

### 3.1. Proof of Theorem3

Let us define the space of solutionsXt as follows:(60)Xt=u,v∈C0,t,Hsℝn2,with the norm(61)u,vXt=supτ∈0,tM1τ,u+M2τ,v,where M1τ,u and M2τ,u will be defined in the treatment of each case. Let N be the mapping on Xt which is defined by(62)N:u,v∈Xt⟶Nu,v=uln+unl,vln+vnl,where(63)ulnt,x≔E1,0t,0,x∗xu0x+E1,1t,0,x∗xu1x,unlt,x≔∫0tE1,1t,τ,x∗xfτ,vdτ,vlnt,x≔E2,0t,0,x∗xv0x+E2,1t,0,x∗xv1x,vnlt,x≔∫0tE2,1t,τ,x∗xgτ,udτ.We denote byE1,0=E1,0t,0,x and E1,1=E1,1t,0,x the fundamental solutions to the Cauchy problem(64)utt−Δu+μ11+tut=0,u0,x=u0x,ut0,x=u1x,and by E2,0=E2,0t,0,x and E2,1=E2,1t,0,x the fundamental solutions to the Cauchy problem(65)vtt−Δv+μ21+tvt=0,v0,x=v0x,vt0,x=v1x.From PropositionE.1, the goal is to prove the following estimates:(66)Nu,vXt≲u0,u1Am,s+v0,v1Am,s+u,vXtp+u,vXtq,(67)Nu,v−Nu˜,v˜Xt≲u,v−u˜,v˜Xt×u,vXtp−1+u˜,v˜Xtp−1+u,vXtq−1+u˜,v˜Xtq−1.From the definition of the norm of the solution spaceXt, which we will define in each case in correspondence with the main goals, we can immediately obtain(68)uln,vlnXt≲u0,u1Am,s+v0,v1Am,s.We complete the proof of all results separately by showing the inequality(69)unl,vnlXt≲u,vXtp+u,vXtq,which leads to (66).(i)
minμ1;μ2>2−m/mn+2s.We choose(70)M1τ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+sDsuτ,·L2ℝn,M2τ,v=1+τ2−m/2mnvτ,·L2ℝn+1+τ2−m/2mn+sDsvτ,·L2ℝn.For the first componentunl, we use estimate (15) for μ1>2−m/mn+2s and estimate (21) to obtain(71)Dsunlt,·L2ℝn≲∫0tvτ,xpLmℝn+1+τ2−m/2mnvτ,xpL2ℝn1+τ1+t−2−m/2mn−sdτ≲1+t−2−m/2mn−s∫0tvτ,xpLmℝn+1+τ2−m/2mnvτ,xpL2ℝn1+τdτ≲u,vXtp1+t−2−m/2mn−s∫0t1+τ−n/mp−1+1dτ≲u,vXtp1+t−2−m/2mn−s,where we usedp>pFuj,mn. Then, we get(72)Dsunlt,·L2ℝn≲u,vXtp1+t−2−m/2mn−s.In thesame way, we prove(73)unlt,·L2ℝn≲u,vXtp1+t−2−m/2mn−s.Forvnl, we use estimate (15) for μ2>2−m/mn+2s and estimate (21) to obtain(74)Dsvnlt,·L2ℝn≲∫0tvτ,xqLmℝn+1+τ2−m/2mnvτ,xqL2ℝn1+τ1+t−2−m/2mn−sdτ≲1+t−2−m/2mn−s∫0tvτ,xqLmℝn+1+τ2−m/2mnvτ,xqL2ℝn1+τdτ≲u,vXtq1+t−2−m/2mn−s∫0t1+τ−n/mq−1+1dτ≲u,vXtq1+t−2−m/2mn−s,where we usedq>pFuj,mn. Then, we get(75)Dsvnlt,·L2ℝn≲u,vXtq1+t−2−m/2mn−s.In the same way, we prove(76)vnlt,·L2ℝn≲u,vXtq1+t−2−m/2mn−s.From (72) to (76), we complete the proof of (69).(ii)
μ1>2−m/mn+2s,μ2<2−m/mn.We choose(77)M1τ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+sDsuτ,·L2ℝn,M2τ,v=1+τμ2/2vτ,·L2ℝn+Dsvτ,·L2ℝn.For the first componentunl, we use estimate (15) for μ1>2−m/mn+2s and estimate (24) to obtain(78)Dsunlt,·L2ℝn≲∫0tvτ,xpLmℝn+1+τ2−m/2mnvτ,xpL2ℝn1+τ1+t−2−m/2mn−sdτ≲1+t−2−m/2mn−s∫0tvτ,xpLmℝn+1+τ2−m/2mnvτ,xpL2ℝn1+τdτ≲u,vXtp1+t−2−m/2mn−s∫0t1+τ−μ2/2p+2−m/2mn+1dτ≲u,vXtp1+t−2−m/2mn−s,where weused (26) for the exponent p. Then, we get(79)Dsunlt,·L2ℝn≲u,vXtp1+t−2−m/2mn−s.In the same way, we prove(80)unlt,·L2ℝn≲u,vXtp1+t−2−m/2mn.Forvnl, we use estimate (15) for μ2<2−m/mn and estimate (21) to obtain(81)Dsvnlt,·L2ℝn≲∫0tuτ,·qLmℝn+1+τ2−m/2mnuτ,·qL2ℝn×1+t−μ2/21+τ−2−m/2mn+μ2/2−s+1dτ≲u,vXtq1+t−μ2/2∫0t1+τ−n/mq−1−2−m/2mn+μ2/2−s+1dτ≲u,vXtq1+t−μ2/2,where we used (26) for the exponent q. Then, we get(82)Dsvnlt,·L2ℝn≲u,vXtq1+t−μ2/2.In the sameway, we prove(83)vnlt,·L2ℝn≲u,vXtq1+t−μ2/2.From (79) to (83), we complete the proof of (69).(iii)
maxμ1;μ2<2−m/mn.We choose(84)M1τ,u=1+τμ1/2uτ,·L2ℝn+Dsuτ,·L2ℝn,M2τ,v=1+τμ1/2vτ,·L2ℝn+Dsvτ,·L2ℝn.Forunl, we use estimate (15) for μ1<2−m/mn and estimate (24) to obtain(85)Dsunlt,·L2ℝn≲∫0tvτ,xpLmℝn+1+τ2−m/2mnvτ,xpL2ℝn×1+t−μ1/21+τ−2−m/2mn+μ1/2−s+1dτ≲u,vXtp1+t−μ1/2∫0t1+τ−2−m/2mn+μ1/2−s+1−μ2/2p+2−m/2mndτ≲u,vXtp1+t−μ1/2,where we usedp>4−2s/μ2+μ1/μ1 which is included in (27). Then, we get(86)Dsunlt,·L2ℝn≲u,vXtp1+t−μ1/2.In thesame way, for s=0, we can prove(87)unlt,·L2ℝn≲u,vXtp1+t−μ1/2,where we use condition (27).Analogously, using (27) for q, we can prove(88)Dsvnlt,·L2ℝn≲u,vXtq1+t−μ2/2,(89)vnlt,·L2ℝn≲u,vXtq1+t−μ2/2.From (86) to (89), we complete the proof of (69).To prove (67), we assume that u,v and u˜,v˜ are two elements from the function space Xt. Then, we have(90)Nu,v−Nu˜,v˜=unlt,x−u˜nlt,x,vnlt,x−v˜nlt,x=∫0tE1t,τ,x∗xvτ,xp−v˜τ,xpdτ,∫0tE1t,τ,x∗xuτ,xq−u˜τ,xqdτ.Followingthe same steps and cases used to prove the first inequality (66), we can estimate(91)Ds∫0tE1t,τ,x∗xvτ,xp−v˜τ,xpdτL2,∫0tE1t,τ,x∗xvτ,xp−v˜τ,xpdτL2,Ds∫0tE1t,τ,x∗xuτ,xq−u˜τ,xqdτL2,∫0tE1t,τ,x∗xuτ,xq−u˜τ,xqdτL2,whichcomplete the proof.


### 3.2. Proof of Theorem5

To prove this theorem, we follow the same steps of the proof of Theorem3. Then, our main goal is to prove (69) which implies (66). To prove (67), we use estimates (39) or (43) and follow the same steps used to prove previous theorem, in particular, inequality (66). We split the proof into three cases.(i)
μ1>2−m/mn+2s1,μ2>2−m/mn+2s2.We choose(92)M1τ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+1utτ,·L2ℝn+1+τ2−m/2mn+s1Ds1uτ,·L2ℝn+D|s1−1utτ,·L2ℝn,M2τ,v=1+τ2−m/2mnvτ,·L2ℝn+1+τ2−m/2mn+1vtτ,·L2ℝn+1+τ2−m/2mn+s2Ds2vτ,·L2ℝn+Ds2−1vtτ,·L2ℝn.For the first componentunl, we estimate the most complicate norm (from our point of view) which is Ds1−1utnlτ,·L2ℝn. After using estimate (16) for μ1>2−m/mn+2s1 and estimate (40), we obtain(93)Ds1−1utnlt,·L2ℝn≲∫0tvτ,xpLmℝn+1+τ2−m/2mn+s1−1vτ,xpH˙s1−1ℝn+1+τ2−m/2mn+s1−2vτ,x|pH˙s1−2ℝn1+τ1+t−2−m/2mn+s1dτ≲u,vXtp1+t−2−m/2mn+s1,where weused the condition p>pFuj,mn. Then, we get(94)Ds1−1utnlt,·L2ℝn≲u,vXtp1+t−2−m/2mn−s1.In the sameway, we get(95)unlt,·L2ℝn≲u,vXtp1+t−2−m/2mn,(96)unlt,·L2ℝn≲u,vXtp1+t−2−m/2mn−1,(97)Ds1unlt,·L2ℝn≲u,vXtp1+t−2−m/2mn−s1.Analogously, forthe second component vnl, we use q>pFuj,mn to derive the estimates(98)Ds2−1vtnlt,·L2ℝn≲u,vXtq1+t−2−m/2mn−s2,(99)vnlt,·L2ℝn≲u,vXtq1+t−2−m/2mn,(100)vtnlt,·L2ℝn≲u,vXtq1+t−2−m/2mn−1,(101)Ds2vnlt,·L2ℝn≲u,vXtq1+t−2−m/2mn−s2.From (94)–(101), we get (69).(ii)
μ1>2−m/mn+2s,μ2<2−m/mn.We choose(102)M1τ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+1utτ,·L2ℝn+1+τ2−m/2mn+s1Ds1uτ,·L2ℝn+Ds1−1utτ,·L2ℝn,M2τ,v=1+τμ2/2vτ,·L2ℝn+vtτ,·L2ℝn+Ds2vτ,·L2ℝn+Ds2−1vtτ,·L2ℝn,Webegin to estimate the norm Ds1−1utnlτ,·L2ℝn. After using estimate (16) for μ1>2−m/mn+2s1 and estimate (44), we have(103)Ds1−1utnlt,·L2ℝn≲∫0tvτ,xpLmℝn+1+τ2−m/2mn+s1−1vτ,xpH˙s1−1ℝn+1+τ2−m/2mn+s1−2vτ,xpH˙s1−2ℝn1+τ1+t−2−m/2mn−s1dτ≲u,vXtp1+t−2−m/2mn+s1∫0t1+τ−μ2/2p+2−m/2mn+s1dτ≲u,vXtp1+t−2−m/2mn+s1,where weused the condition p>2+2s2/μ2+2−m/μ2mn. Then, we get (94), and in a similar way, one can prove (95)–(97).  To estimate the norm Ds2−1vtnlτ,·L2ℝn, we use estimate (16) for μ2<2−m/mn and estimate (40). Then, it follows(104)Ds2−1vtnlt,·L2ℝn≲∫0tuτ,xqLmℝn+1+τ2−m/2mn+s2−1vτ,xpH˙s1−1ℝn+1+τ2−m/2mn+s2−2vτ,xpH˙s2−2ℝn1+τ1+t−2−m/2mn−s2dτ≲u,vXtq1+t−μ2/2∫0t1+τ−n/mq−1−2−m/2mn+μ2/2−s2dτ≲u,vXtq1+t−μ2/2−s2,where we used the conditionq>2−2s2m/n+m/2+nμ2/2m. We may conclude(105)Ds2−1vtnlt,·L2ℝn≲u,vXtq1+t−μ2/2.Analogously, by using the conditionq>2m/n+m/2+nμ2/2m, we can prove(106)vnlt,·L2ℝn+vtnlt,·L2ℝn+Ds2vnlt,·L2ℝn≲u,vXtq1+t−μ2/2.In this way, we complete the proof of the second case.(iii)
maxμ1;μ2<2−m/mn.Wechoose(107)M1τ,u=1+τμ1/2uτ,·L2ℝn+utτ,·L2ℝn+Ds1uτ,·L2ℝn+Ds1−1utτ,·L2ℝnM2τ,v=1+τμ2/2vτ,·L2ℝn+utτ,·L2ℝn+Ds2uτ,·L2ℝn+Ds2−1utτ,·L2ℝn.Forunl, we use estimate (16) for μ1<2−m/mn and estimate (44) to obtain(108)Ds1−1utnlt,·L2ℝn≲∫0tvτ,xpLmℝn+1+τ2−m/2mn+s1−1vτ,xpH˙s1−1ℝn+1+τ2−m/2mn+s1−2vτ,xpH˙s1−2ℝn1+τ1+t−2−m/2mn−s1dτ≲u,vXtp1+t−μ1/2∫0t1+τ−2−m/mn+μ1/2−s1−μ/2p+2−m/mn+s2dτ≲u,vXtq1+t−μ2/2,where weused the condition p>μ1/μ2+2+2s2−2s1/μ2. Then,(109)Ds1−1utnlt,·L2ℝn≲u,vXtp1+t−μ1/2.In the same way, we can prove forp>μ1/μ2+2+2s2−2s1/μ2 the estimate(110)unlt,·L2ℝn+utnlt,·L2ℝn+Ds1unlt,·L2ℝn≲u,vXtp1+t−μ1/2.Analogously,under the condition q>μ2/μ1+2+2s1/μ1, it follows(111)Ds2−1vtt,·L2ℝn+vt,·L2ℝn+vtt,·L2ℝn+Ds2vtt,·L2ℝn≲u,vXtq1+t−μ2/2.From (109)–(111), we get (69).Thiscompletes the proof.


### 3.3. Proof of Theorem6

The proof can be obtained by following the same steps of the proof of Theorem5, but using the estimates of Lemma 3 instead of the estimates of Lemma 2. In other words, we use the rules for fractional powers instead of the fractional chain rule.


## 3.1. Proof of Theorem3

Let us define the space of solutionsXt as follows:(60)Xt=u,v∈C0,t,Hsℝn2,with the norm(61)u,vXt=supτ∈0,tM1τ,u+M2τ,v,where M1τ,u and M2τ,u will be defined in the treatment of each case. Let N be the mapping on Xt which is defined by(62)N:u,v∈Xt⟶Nu,v=uln+unl,vln+vnl,where(63)ulnt,x≔E1,0t,0,x∗xu0x+E1,1t,0,x∗xu1x,unlt,x≔∫0tE1,1t,τ,x∗xfτ,vdτ,vlnt,x≔E2,0t,0,x∗xv0x+E2,1t,0,x∗xv1x,vnlt,x≔∫0tE2,1t,τ,x∗xgτ,udτ.We denote byE1,0=E1,0t,0,x and E1,1=E1,1t,0,x the fundamental solutions to the Cauchy problem(64)utt−Δu+μ11+tut=0,u0,x=u0x,ut0,x=u1x,and by E2,0=E2,0t,0,x and E2,1=E2,1t,0,x the fundamental solutions to the Cauchy problem(65)vtt−Δv+μ21+tvt=0,v0,x=v0x,vt0,x=v1x.From PropositionE.1, the goal is to prove the following estimates:(66)Nu,vXt≲u0,u1Am,s+v0,v1Am,s+u,vXtp+u,vXtq,(67)Nu,v−Nu˜,v˜Xt≲u,v−u˜,v˜Xt×u,vXtp−1+u˜,v˜Xtp−1+u,vXtq−1+u˜,v˜Xtq−1.From the definition of the norm of the solution spaceXt, which we will define in each case in correspondence with the main goals, we can immediately obtain(68)uln,vlnXt≲u0,u1Am,s+v0,v1Am,s.We complete the proof of all results separately by showing the inequality(69)unl,vnlXt≲u,vXtp+u,vXtq,which leads to (66).(i)
minμ1;μ2>2−m/mn+2s.We choose(70)M1τ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+sDsuτ,·L2ℝn,M2τ,v=1+τ2−m/2mnvτ,·L2ℝn+1+τ2−m/2mn+sDsvτ,·L2ℝn.For the first componentunl, we use estimate (15) for μ1>2−m/mn+2s and estimate (21) to obtain(71)Dsunlt,·L2ℝn≲∫0tvτ,xpLmℝn+1+τ2−m/2mnvτ,xpL2ℝn1+τ1+t−2−m/2mn−sdτ≲1+t−2−m/2mn−s∫0tvτ,xpLmℝn+1+τ2−m/2mnvτ,xpL2ℝn1+τdτ≲u,vXtp1+t−2−m/2mn−s∫0t1+τ−n/mp−1+1dτ≲u,vXtp1+t−2−m/2mn−s,where we usedp>pFuj,mn. Then, we get(72)Dsunlt,·L2ℝn≲u,vXtp1+t−2−m/2mn−s.In thesame way, we prove(73)unlt,·L2ℝn≲u,vXtp1+t−2−m/2mn−s.Forvnl, we use estimate (15) for μ2>2−m/mn+2s and estimate (21) to obtain(74)Dsvnlt,·L2ℝn≲∫0tvτ,xqLmℝn+1+τ2−m/2mnvτ,xqL2ℝn1+τ1+t−2−m/2mn−sdτ≲1+t−2−m/2mn−s∫0tvτ,xqLmℝn+1+τ2−m/2mnvτ,xqL2ℝn1+τdτ≲u,vXtq1+t−2−m/2mn−s∫0t1+τ−n/mq−1+1dτ≲u,vXtq1+t−2−m/2mn−s,where we usedq>pFuj,mn. Then, we get(75)Dsvnlt,·L2ℝn≲u,vXtq1+t−2−m/2mn−s.In the same way, we prove(76)vnlt,·L2ℝn≲u,vXtq1+t−2−m/2mn−s.From (72) to (76), we complete the proof of (69).(ii)
μ1>2−m/mn+2s,μ2<2−m/mn.We choose(77)M1τ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+sDsuτ,·L2ℝn,M2τ,v=1+τμ2/2vτ,·L2ℝn+Dsvτ,·L2ℝn.For the first componentunl, we use estimate (15) for μ1>2−m/mn+2s and estimate (24) to obtain(78)Dsunlt,·L2ℝn≲∫0tvτ,xpLmℝn+1+τ2−m/2mnvτ,xpL2ℝn1+τ1+t−2−m/2mn−sdτ≲1+t−2−m/2mn−s∫0tvτ,xpLmℝn+1+τ2−m/2mnvτ,xpL2ℝn1+τdτ≲u,vXtp1+t−2−m/2mn−s∫0t1+τ−μ2/2p+2−m/2mn+1dτ≲u,vXtp1+t−2−m/2mn−s,where weused (26) for the exponent p. Then, we get(79)Dsunlt,·L2ℝn≲u,vXtp1+t−2−m/2mn−s.In the same way, we prove(80)unlt,·L2ℝn≲u,vXtp1+t−2−m/2mn.Forvnl, we use estimate (15) for μ2<2−m/mn and estimate (21) to obtain(81)Dsvnlt,·L2ℝn≲∫0tuτ,·qLmℝn+1+τ2−m/2mnuτ,·qL2ℝn×1+t−μ2/21+τ−2−m/2mn+μ2/2−s+1dτ≲u,vXtq1+t−μ2/2∫0t1+τ−n/mq−1−2−m/2mn+μ2/2−s+1dτ≲u,vXtq1+t−μ2/2,where we used (26) for the exponent q. Then, we get(82)Dsvnlt,·L2ℝn≲u,vXtq1+t−μ2/2.In the sameway, we prove(83)vnlt,·L2ℝn≲u,vXtq1+t−μ2/2.From (79) to (83), we complete the proof of (69).(iii)
maxμ1;μ2<2−m/mn.We choose(84)M1τ,u=1+τμ1/2uτ,·L2ℝn+Dsuτ,·L2ℝn,M2τ,v=1+τμ1/2vτ,·L2ℝn+Dsvτ,·L2ℝn.Forunl, we use estimate (15) for μ1<2−m/mn and estimate (24) to obtain(85)Dsunlt,·L2ℝn≲∫0tvτ,xpLmℝn+1+τ2−m/2mnvτ,xpL2ℝn×1+t−μ1/21+τ−2−m/2mn+μ1/2−s+1dτ≲u,vXtp1+t−μ1/2∫0t1+τ−2−m/2mn+μ1/2−s+1−μ2/2p+2−m/2mndτ≲u,vXtp1+t−μ1/2,where we usedp>4−2s/μ2+μ1/μ1 which is included in (27). Then, we get(86)Dsunlt,·L2ℝn≲u,vXtp1+t−μ1/2.In thesame way, for s=0, we can prove(87)unlt,·L2ℝn≲u,vXtp1+t−μ1/2,where we use condition (27).Analogously, using (27) for q, we can prove(88)Dsvnlt,·L2ℝn≲u,vXtq1+t−μ2/2,(89)vnlt,·L2ℝn≲u,vXtq1+t−μ2/2.From (86) to (89), we complete the proof of (69).To prove (67), we assume that u,v and u˜,v˜ are two elements from the function space Xt. Then, we have(90)Nu,v−Nu˜,v˜=unlt,x−u˜nlt,x,vnlt,x−v˜nlt,x=∫0tE1t,τ,x∗xvτ,xp−v˜τ,xpdτ,∫0tE1t,τ,x∗xuτ,xq−u˜τ,xqdτ.Followingthe same steps and cases used to prove the first inequality (66), we can estimate(91)Ds∫0tE1t,τ,x∗xvτ,xp−v˜τ,xpdτL2,∫0tE1t,τ,x∗xvτ,xp−v˜τ,xpdτL2,Ds∫0tE1t,τ,x∗xuτ,xq−u˜τ,xqdτL2,∫0tE1t,τ,x∗xuτ,xq−u˜τ,xqdτL2,whichcomplete the proof.


## 3.2. Proof of Theorem5

To prove this theorem, we follow the same steps of the proof of Theorem3. Then, our main goal is to prove (69) which implies (66). To prove (67), we use estimates (39) or (43) and follow the same steps used to prove previous theorem, in particular, inequality (66). We split the proof into three cases.(i)
μ1>2−m/mn+2s1,μ2>2−m/mn+2s2.We choose(92)M1τ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+1utτ,·L2ℝn+1+τ2−m/2mn+s1Ds1uτ,·L2ℝn+D|s1−1utτ,·L2ℝn,M2τ,v=1+τ2−m/2mnvτ,·L2ℝn+1+τ2−m/2mn+1vtτ,·L2ℝn+1+τ2−m/2mn+s2Ds2vτ,·L2ℝn+Ds2−1vtτ,·L2ℝn.For the first componentunl, we estimate the most complicate norm (from our point of view) which is Ds1−1utnlτ,·L2ℝn. After using estimate (16) for μ1>2−m/mn+2s1 and estimate (40), we obtain(93)Ds1−1utnlt,·L2ℝn≲∫0tvτ,xpLmℝn+1+τ2−m/2mn+s1−1vτ,xpH˙s1−1ℝn+1+τ2−m/2mn+s1−2vτ,x|pH˙s1−2ℝn1+τ1+t−2−m/2mn+s1dτ≲u,vXtp1+t−2−m/2mn+s1,where weused the condition p>pFuj,mn. Then, we get(94)Ds1−1utnlt,·L2ℝn≲u,vXtp1+t−2−m/2mn−s1.In the sameway, we get(95)unlt,·L2ℝn≲u,vXtp1+t−2−m/2mn,(96)unlt,·L2ℝn≲u,vXtp1+t−2−m/2mn−1,(97)Ds1unlt,·L2ℝn≲u,vXtp1+t−2−m/2mn−s1.Analogously, forthe second component vnl, we use q>pFuj,mn to derive the estimates(98)Ds2−1vtnlt,·L2ℝn≲u,vXtq1+t−2−m/2mn−s2,(99)vnlt,·L2ℝn≲u,vXtq1+t−2−m/2mn,(100)vtnlt,·L2ℝn≲u,vXtq1+t−2−m/2mn−1,(101)Ds2vnlt,·L2ℝn≲u,vXtq1+t−2−m/2mn−s2.From (94)–(101), we get (69).(ii)
μ1>2−m/mn+2s,μ2<2−m/mn.We choose(102)M1τ,u=1+τ2−m/2mnuτ,·L2ℝn+1+τ2−m/2mn+1utτ,·L2ℝn+1+τ2−m/2mn+s1Ds1uτ,·L2ℝn+Ds1−1utτ,·L2ℝn,M2τ,v=1+τμ2/2vτ,·L2ℝn+vtτ,·L2ℝn+Ds2vτ,·L2ℝn+Ds2−1vtτ,·L2ℝn,Webegin to estimate the norm Ds1−1utnlτ,·L2ℝn. After using estimate (16) for μ1>2−m/mn+2s1 and estimate (44), we have(103)Ds1−1utnlt,·L2ℝn≲∫0tvτ,xpLmℝn+1+τ2−m/2mn+s1−1vτ,xpH˙s1−1ℝn+1+τ2−m/2mn+s1−2vτ,xpH˙s1−2ℝn1+τ1+t−2−m/2mn−s1dτ≲u,vXtp1+t−2−m/2mn+s1∫0t1+τ−μ2/2p+2−m/2mn+s1dτ≲u,vXtp1+t−2−m/2mn+s1,where weused the condition p>2+2s2/μ2+2−m/μ2mn. Then, we get (94), and in a similar way, one can prove (95)–(97).  To estimate the norm Ds2−1vtnlτ,·L2ℝn, we use estimate (16) for μ2<2−m/mn and estimate (40). Then, it follows(104)Ds2−1vtnlt,·L2ℝn≲∫0tuτ,xqLmℝn+1+τ2−m/2mn+s2−1vτ,xpH˙s1−1ℝn+1+τ2−m/2mn+s2−2vτ,xpH˙s2−2ℝn1+τ1+t−2−m/2mn−s2dτ≲u,vXtq1+t−μ2/2∫0t1+τ−n/mq−1−2−m/2mn+μ2/2−s2dτ≲u,vXtq1+t−μ2/2−s2,where we used the conditionq>2−2s2m/n+m/2+nμ2/2m. We may conclude(105)Ds2−1vtnlt,·L2ℝn≲u,vXtq1+t−μ2/2.Analogously, by using the conditionq>2m/n+m/2+nμ2/2m, we can prove(106)vnlt,·L2ℝn+vtnlt,·L2ℝn+Ds2vnlt,·L2ℝn≲u,vXtq1+t−μ2/2.In this way, we complete the proof of the second case.(iii)
maxμ1;μ2<2−m/mn.Wechoose(107)M1τ,u=1+τμ1/2uτ,·L2ℝn+utτ,·L2ℝn+Ds1uτ,·L2ℝn+Ds1−1utτ,·L2ℝnM2τ,v=1+τμ2/2vτ,·L2ℝn+utτ,·L2ℝn+Ds2uτ,·L2ℝn+Ds2−1utτ,·L2ℝn.Forunl, we use estimate (16) for μ1<2−m/mn and estimate (44) to obtain(108)Ds1−1utnlt,·L2ℝn≲∫0tvτ,xpLmℝn+1+τ2−m/2mn+s1−1vτ,xpH˙s1−1ℝn+1+τ2−m/2mn+s1−2vτ,xpH˙s1−2ℝn1+τ1+t−2−m/2mn−s1dτ≲u,vXtp1+t−μ1/2∫0t1+τ−2−m/mn+μ1/2−s1−μ/2p+2−m/mn+s2dτ≲u,vXtq1+t−μ2/2,where weused the condition p>μ1/μ2+2+2s2−2s1/μ2. Then,(109)Ds1−1utnlt,·L2ℝn≲u,vXtp1+t−μ1/2.In the same way, we can prove forp>μ1/μ2+2+2s2−2s1/μ2 the estimate(110)unlt,·L2ℝn+utnlt,·L2ℝn+Ds1unlt,·L2ℝn≲u,vXtp1+t−μ1/2.Analogously,under the condition q>μ2/μ1+2+2s1/μ1, it follows(111)Ds2−1vtt,·L2ℝn+vt,·L2ℝn+vtt,·L2ℝn+Ds2vtt,·L2ℝn≲u,vXtq1+t−μ2/2.From (109)–(111), we get (69).Thiscompletes the proof.


## 3.3. Proof of Theorem6

The proof can be obtained by following the same steps of the proof of Theorem5, but using the estimates of Lemma 3 instead of the estimates of Lemma 2. In other words, we use the rules for fractional powers instead of the fractional chain rule.


## 4. Concluding Remarks

In this section, we compare the results obtained previously in [40–42] with our results. Then, we consider the following weakly coupled systems of semilinear damped wave equations (see Table 2).Table 2
Effective case versus scale-invariant case.
Effective caseScale-invariant caseutt−Δu+1/1+tr1ut=vpvtt−Δv+1/1+tr2vt=uqu0,x=u0x,ut0,x=u1xv0,x=v0x,vt0,x=v1xutt−Δu+u1/1+tut=vpvtt−Δv+u2/1+tvt=uqu0,x=u0x,ut0,x=u1xv0,x=v0x,vt0,x=v1xr1,r2∈−1,1 and μ1,μ2 are positive real numbers.(i)
If the values ofμ1 and μ2 are sufficiently large, we remark that the obtained results for both models coincide. For this reason, we may interpret the above system for large μ1 and μ2 as an effective one.(ii)
Ifμ1 and μ2 are small, then we obtain quite different results for the above models. In particular, the admissible ranges for the exponents p and q determined are p>μ1/μ2+4/μ2 and q>μ2/μ1+4/μ1 which are different to pFuj,mn for the effective case. We see that the results are more restrictive for the scale-invariant case. For this reason, we should interpret the above system for small μ1 and μ2 as a noneffective one.



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