# Blow-Up of Solutions with High Energies of a Coupled System of Hyperbolic Equations

Authors: Jorge A. Esquivel-Avila
Journal: Abstract and Applied Analysis
 (2019)

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## Abstract

We consider an abstract coupled evolution system of second order in time. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We compare our results with those in the literature and show how we improve them.


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## Body

## 1. Introduction

A coupled Klein-Gordon system, in electromagnetic theory, was first introduced in [1]. Posteriorly, further generalizations have been studied. In particular, the following system was analyzed in [2](1)KGutt-Δu+m12u+K1xu=a1p+1vq+1up-1u,vtt-Δv+m22v+K2xv=a2q+1up+1vq-1v,u0,x=u0x,ut0,x=u1x,v0,x=v0x,vt0,x=v1x,on R×RN, where ai>0,mi≠0,Ki(x)≥0,x∈RN,i=1,2, and p>1,q>1. The existence and uniqueness of weak solutions of (KG), as well as characterizations for blow-up and globality, by means of the potential well method for values of the initial energy smaller than the mountain pass level, were proved in [2]. In the same paper, sufficient conditions were given to obtain blow-up for arbitrary positive values of the initial energy. The purpose of our work is to study an abstract hyperbolic coupled system and improve some results about nonexistence of global solutions presented in the literature for some concrete systems. In particular, we shall improve some blow-up results presented in [2], for the problem (KG). Precisely, we consider the following abstract problem:(2)PPutt+Au=Fu,v,Qvtt+Bv=Gu,v,u0=u0,ut0=u1,v0=v0,vt0=v1,on R, where we assumed that the operators (3)P:WP⟶WP′,Q:WQ⟶WQ′,A:VA⟶VA′,B:VB⟶VB′are linear, continuous, positive, and symmetric, where (4)VA⊂WP⊂H,VB⊂WQ⊂Hare linear subspaces of the Hilbert space H with inner product (·,·) and norm ·. Here, H′,WP′,WQ′,VA′,VB′ are the corresponding dual spaces and we identify H=H′. Then, (5)H⊂WP′⊂VA′,H⊂WQ′⊂VB′.By means of the operators P, Q, A, and B, we define the following bilinear forms: (6)Pu,w≡Pu,wWP×WP′,uWP2≡Pu,u,∀u,w∈WP,Qv,w~≡Qv,w~WQ×WQ′,vWQ2≡Qv,v,∀v,w~∈WQ,Au,w≡Au,wVA×VA′,uVA2≡Au,u,∀u,w∈VA,Bv,w~≡Bv,w~VB×VB′,vVB2≡Bv,v,∀v,w~∈VB.We assume that there exists c>0, such that (7)H0uVA2+vVB2≥cuWP2+vWQ2,∀u,v∈VA×VB.The nonlinear source terms F:VA×VB→H and G:VA×VB→H, are such that F(0,0)=0=G(0,0), and (F,G):(VA×VB)×(VA×VB)→H×H is a potential operator with potential K:VA×VB→R, that is, F(u,v)=∂uK(u,v) and G(u,v)=∂vK(u,v), and they satisfy (8)H1Fu,v,u+Gu,v,v-rKu,v≥0,∀u,v∈VA×VB,where r>2 is a constant.


## 2. Functional Framework

We shall analyze qualitative properties for a set of solutions of problem(P). To this end, we define the phase space (9)H≡VA×WP×VB×WQ.We assume that the following local existence and uniqueness result is met.Theorem 1.
For every initial data((u0,u1),(v0,v1))∈H, there exists T>0, and a unique local solution ((u0,u1),(v0,v1))↦((u,u˙),(v,v˙))∈C(0,T;H),u˙(t)≡d/dtu(t),v˙(t)≡d/dtv(t), such that problem (P) is satisfied in the following sense (10)ddtPu˙t,w+Aut,w=Fut,vt,w,ddtQv˙t,w~+Bvt,w~=Gut,vt,w~,a. e. in (0,T) and for every (w,w~)∈VA×VB. Furthermore, the following energy equation holds for T>t≥t0≥0, (11)Eut0,u˙t0,vt0,v˙t0=Eut,u˙t,vt,v˙t≡12u˙tWP2+v˙tWQ2+Jut,vt,Jut,vt≡12utVA2+vtVB2-Kut,vt.Remark 2.
Problem(P) is invariant if we reverse the time direction: t↦-t. Indeed, the solution backwards ((u(t),u˙(t)),(v(t),v˙(t))),t<0, with initial data ((u0,u1),(v0,v1)) is the solution forwards ((u(-t),u˙(-t)),(v(-t),v˙(-t))), -t>0 with initial data ((u0,-u1),(v0,-v1)).An important set of solutions are the equilibria, that is, solutions independent of time:u˙=0=v˙. In this case, (u,v) satisfies (12)Au,w=Fu,v,w,Bv,w~=Gu,v,w~,for every (w,w~)∈VA×VB. In particular, for w=u,w~=v, (13)uVA2=Fu,v,u,vVB2=Gu,v,v,and then (14)Iu,v≡uVA2+vVB2-Fu,v,u-Gu,v,v=0.By(H0), (u,v)≡(0,0) is an equilibrium. The set of equilibria (u,v)≠(0,0), with minimal energy is called ground state, and the corresponding value of the energy is the mountain pass level denoted by d; see [3]. For the problem (KG), the sign of I(u0,v0) characterizes either blow-up in finite time or boundedness of solutions if E0≡E(u0,u1,v0,v1)<d. Indeed, blow-up and boundedness properties hold if, I(u0,v0)<0 and I(u0,v0)>0, respectively; see [2]. Similar analysis have been done to prove similar characterizations for coupled systems of wave equations with linear and nonlinear damping terms; see [4–8] and references therein, just to cite some works of the abundant literature in the field. The qualitative analysis of the solutions with high energies is almost unknown. There are some works that prove blow-up if I(u0,v0)<0 and some other conditions on E0 and the initial data are satisfied; see for instance [2]. Similar theorems have been proved in [6, 9–11], for damped systems of semilinear wave equations. The purpose of this work is to improve considerably the existing results for blow-up for systems(P), with high energies. We shall generalize the technique used in a previous work for a single equation; see [12].


## 3. Nonexistence of Global Solutions

We consider the following orthogonal decomposition of the velocities(15)u˙=Pu˙,uuWP2u+h,v˙=Qv˙,vvWQ2v+h~where P(u,h)=0 and Q(v,h~)=0. Then, we define the functionals (16)Ru,u˙≡Pu˙,u2uWP2,Sv,v˙≡Qv˙,v2vWQ2Consequently, (17)u˙WP2=hWP2+Ru,u˙,v˙WQ2=h~WQ2+Sv,v˙.Also, we define (18)Ψ1u≡uWP2,Ψ2v≡vWQ2,Ψu,v≡Ψ1u+Ψ2vΦu,u˙,v,v˙≡cΨu,v+Pu,u˙+Qv,v˙2Ψu,v.where c>0 is the constant in H(0). We also define the following functions (19)ηqu,u˙,v,v˙≡12Φu,u˙,v,v˙-crΨu,vcΨu,vΦu,u˙,v,v˙q,forq≥0,μλu,u˙,v,v˙≡12Φu,u˙,v,v˙-crΨu,vλcΨu,vΦu,u˙,v,v˙r-2/2,forλ∈0,1,σνu,u˙,v,v˙≡12Φu,u˙,v,v˙-cνrΨu,v,forν>1.If P(u,u˙)+Q(v,v˙)>0, we notice that q↦ηq(u,u˙,v,v˙) is strictly increasing, λ↦μλ is strictly decreasing, and ν↦σν is strictly decreasing. They have the following relations (20)limλ→1⁡μλu,u˙,v,v˙=ηr-2/2u,u˙,v,v˙,limν→1⁡σνu,u˙,v,v˙=η0u,u˙,v,v˙,and σν(u,u˙,v,v˙)<η0(u,u˙,v,v˙)<ηr-2/2(u,u˙,v,v˙)<μλ(u,u˙,v,v˙).For the(KG) system, a recent work [2] proved blow-up of solutions with initial energy: E0<c(r-4)/2rΨ(u0,v0). We observe that (c(r-4)/2r)Ψ(u0,v0)<c(r-2)/2rΨ(u0,v0)<η0(u0,u1,v0,v1). Here, we shall prove nonexistence of global solutions of the(P) system with initial energy σν∗(u0,u1,v0,v1)<E0<μλ∗(u0,u1,v0,v1), for some ν∗>1 and λ∗∈(r-2)/r,1. Furthermore, we shall prove that for any positive value of the initial energy there are initial data implying nonexistence of global solutions.Theorem 3.
Consider any solution of problem(P) in the sense of Theorem 1. Assume that hypotheses (H0) and (H1) are met and that (21)u0WP2+v0WQ2>0,Pu0,u1+Qv0,v1>0,are satisfied. Then, we construct the nonempty interval (22)I0≡α0,β0⊂0,12Φu0,u1,v0,v1,where (23)α0=σν∗u0,u1,v0,v1=r-22rcΨu0,v0ν∗2/r-2,β0=μλ∗u0,u1,v0,v1=r-22rΦu0,u1,v0,v1λ∗,for some r-2/r<λ∗<1 and ν∗>1, and we have the following assertions.
(i) If the initial energy is such thatE0∈I0, then the maximal time of existence of the solution is finite.
(ii) For fixedΨ(u0,v0), (24)Pu0,u1+Qv0,v1⟼I0=β0-α0is strictly increasing, and (25)limPu0,u1+Qv0,v1→∞α0=0=limPu0,u1+Qv0,v1→∞β0-12Φu0,u1,v0,v1,limPu0,u1+Qv0,v1→∞ν∗=∞,limPu0,u1+Qv0,v1→∞λ∗=r-2r.Corollary 4.
Assume that hypotheses of Theorem3 are met. For every number E>0, we can choose initial data with P(u0,u1)+Q(v0,v1) large enough, so that E∈I0, and then the corresponding solution with E0=E exists only up to a finite time.


## 4. Proofs

Proof (of Theorem3).
First, we will assume that the solution is global and then, by means of a differential inequality in terms ofΨ, we shall get a contradiction. Then, assume that Ψ(u(t),v(t)) exists for any t≥0. We observe that d/dtΨ(u(t),v(t))=2(P(u(t),u˙(t))+Q(v(t),v˙(t))). Now, we define G(t)≡Ψ-r-2/4(u(t),v(t)), and due to (21), we have that for t≥0, close to zero, the following inequality holds (26)ddtGt=-r-24Ψ-r+2/4ut,vtddtΨut,vt=-r-22Ψ-r+2/4ut,vtPut,u˙t+Qvt,v˙t<0.By energy equation and hypotheses (H0) and (H1), we obtain (27)d2dt2Ψut,vt=2u˙tWP2+v˙tWQ2-Iut,vt=2u˙tWP2+v˙tWQ2-Iut+2rE0-2rE0≥r+2Rut,u˙t+Svt,v˙t+r-2utVA2+vtVB2-2rE0.≥r+24d/dtΨ1ut2Ψ1ut+d/dtΨ2vt2Ψ2vt+cr-2Ψ1ut+Ψ2vt-2rE0.≥r+24d/dtΨut2Ψut,vt+cr-2Ψut,vv-2rE0,where we used the following (28)Ψ2vtddtΨ1ut-Ψ1utddtΨ2vt2≥0⟺ddtΨ1ut2Ψ22vt+ddtΨ2vt2Ψ12ut≥2Ψ1utΨ2vtddtΨ1utddtΨ2vt⟺ddtΨ1ut2Ψ2vtΨ1ut+ddtΨ2vt2Ψ1utΨ2vt≥2ddtΨ1utddtΨ2vt⟺ddtΨ1ut2Ψ2vtΨ1ut+1+ddtΨ2vt2Ψ1utΨ2vt+1≥ddtΨ1ut+ddtΨ2vt2⟺d/dtΨ1ut2Ψ1ut+d/dtΨ2vt2Ψ2vt≥d/dtΨ1ut+d/dtΨ2vt2Ψ1ut+Ψ2vt=d/dtΨut,vt2Ψut,vt.Then, (29)d2dt2Gt=r-24Ψ-r+2/4ut,vt×r+24d/dtΨut,vt2Ψut,vt-d2dt2Ψut,vt≤-cr-224Gt+E0rr-22Gtr+2/r-2,and since d/dtG(t)<0, we get (30)ddtGt2≥r-2242E0G2r/r-2t-cG2t+C0,where (31)C0≡ddtG02-r-2242E0G2r/r-20-cG20.We shall prove that there exists a constant ρ0>0 such that (32)ddtGt2≥ρ02>0,∀t≥0,and then (33)ddtGt≤-ρ0<0,∀t≥0.Hence, (34)0≤Gt≤-ρ0t+G0,∀t≥0which is impossible for any t>G(0)/ρ0. Then, the solution only exits up to a finite time.
Next, we prove that (32) is satisfied. To this end, we consider the right-hand side of (30) and define, for s≥0, (35)Fs≡r-2242E0sr/r-2-cs+C0,and we notice that (36)Fs≥Fs0,∀s≥0,with s0≡c(r-2)/2rE0(r-2)/2>0, and (37)Fs0=r-2242E0s0r/r-2-cs0+C0,=-r-2E0cr-22rE0r/2+C0,Also, (38)C0=r-224u0WP2+v0WQ2-r+2/2Pu0,u1+Qv0,v12-r-2242E0u0WP2+v0WQ2-r/2-cu0WP2+v0WQ2-r-2/2.We observe that (32) is satisfied if ρ02=F(s0)>0, which is characterized by (39)2crcr-22rE0r-2/2+2E0u0WP2+v0WQ2-r/2<u0WP2+v0WQ2-r+2/2Pu0,u1+Qv0,v12+cu0WP2+v0WQ2-r-2/2,and it is equivalent to (40)E0+cr-2Ψu0,v02rE0r-2/2cΨu0,v0r<12Φu0,u1,v0,v1,where (41)Φu0,u1,v0,v1≡cΨu0,v0+Pu0,u1+Qv0,v12Ψu0,v0.Now, in order to guarantee that (40) is satisfied, we define, for s≥0, (42)Ns≡s+cr-2Ψu0,v02rsr-2/2cΨu0,v0r.We observe that N(s)→∞ as either s→0 or s→∞, and (43)Ns≥Ns1=c2Ψu0,v0,∀s≥0,for s1≡r-2/2rcΨ(u0,v0). Moreover, by (21), there exist exactly two different roots of N(s)=1/2Φ(u0,u1,v0,v1), denoted by α0 and β0, such that (44)0<α0<s1<β0<12Φu0,u1,v0,v1,12Ψu0,v0<Ns<12Φu0,u1,v0,v1,∀s∈I0≡α0,β0,s≠s1.And since N is strictly monotone for s<s1 and s>s1, it follows that, for fixed Ψ(u0,v0), the interval I0 grows as P(u0,u1)+Q(v0,v1) grows. Precisely, (45)limPu0,u1+Qv0,v1→∞12Φu0,u1,v0,v1-β0=0=limPu0,u1+Qv0,v1→∞α0.Then, (32) holds if and only if the initial energy satisfies (46)NE0<12Φu0,u1,v0,v1,that is, if E0∈I0. This proves that the maximum time of existence must be finite if the initial energy is within this interval.
Next, we shall find the values ofα0 and β0. Remember that these are the roots of N(s)=1/2Φ(u0,u1,v0,v1). To find α0, we consider the function (47)σνu0,u1,v0,v1≡12Φu0,u1,v0,v1-cνrΨu0,v0,defined for ν>1, and the equation (48)Nσνu0,u1,v0,v1=12Φu0,u1,v0,v1which holds if and only if (49)1ν2/r-2=2rcr-2σνu0,u1,v0,v1Ψu0,v0which is equivalent to (50)2rν+r-2r1ν2/r-2=Φu0,u1,v0,v1cΨu0,v0.We notice that (51)fs≡2rs+r-2r1s2/r-2⟶∞as s→0 and s→∞. Also, min{s>0}f(s)=f(1)=1. Moreover, (52)Φu0,u1,v0,v1cΨu0,v0>1.Then, the equation for ν has two roots and only one is bigger than 1. Furthermore, at this root, ν∗>1, (53)α0=σν∗u0,u1,v0,v1=cr-2Ψu0,v02r1ν∗2/r-2,limPu0,u1+Qv0,v1→∞ν∗=∞.
Next, we consider the function(54)μλu0,u1,v0,v1≡12Φu0,u1,v0,v1-crΨu0,v0λcΨu0,v0Φu0,u1,v0,v1r-2/2,defined for (r-2)/r<λ<1, and the equation (55)Nμλu0,u1,v0,v1=12Φu0,u1,v0,v1which is equivalent to (56)crΨu0,v0cr-22rΨu0,v0μλu0,u1,v0,v1r-2/2=crΨu0,v0λcΨu0,v0Φu0,u1,v0,v1r-2/2,and it is characterized by (57)r-22r=λμλu0,u1,v0,v1Φu0,u1,v0,v1which holds if and only if (58)2rλcΨu0,v0Φu0,u1,v0,v1r/2=λ-r-2r.Notice that (59)gλ≡2rλcΨu0,v0Φu0,u1,v0,v1r/2,hλ≡λ-r-2r,are strictly monotone increasing, and (60)g0>h0,g1<h1.Then, there exists one and only one λ∗∈(r-2)/r,1 where g(λ∗)=h(λ∗). Moreover, (61)β0=μλ∗u0,u1,v0,v1=r-2Φu0,u1,v0,v12r1λ∗,limPu0,u1+Qv0,v1→∞λ∗=r-2r.Proof (of Corollary4).
Since(62)P+Q⟶∞⟹α0⟶0,β0⟶∞then, for every E>0, there exists M>0, such that (63)P+Q>M⟹E∈I0=α0,β0.Hence, the corresponding solution with initial energy E0=E satisfying (21) exists only up to a finite time.Remark 5.
For small energies,E0<d, the potential well method characterizes the qualitative behavior of any solution in terms of the sign of I(u0,v0); see [4–8]. In particular, blow-up is characterized if I(u0,v0)<0. Let us examine the situation for any positive value of the initial energy.
Assume that hypotheses of Theorem3 are met. From energy equation, (H0), (H1), (21), E0<β0, and since r-2/r<λ∗<1, we obtain (64)Iu0,v0=2E0-u1WP2-v1WQ2+2Ku0,v0-Fu0,v0,u0-Gu0,v0,v0≤2E0-Pu0,u12u0WP2+Qv0,v12v0WQ2-r-2Ku0,v0<2β0-Pu0,u1+Qv0,v12Ψu0,v0-r-2Ku0,v0=r-2Φu0,u1,v0,v1rλ∗-Pu0,u1+Qv0,v12Ψu0,v0-r-2Ku0,v0.Let us assume that (65)Ku0,v0>1r-2cΨu0,v0,and then (66)Iu0,v0<-1-r-2rλ∗Φu0,u1,v0,v1<0.Hence, if the source term is large enough at the initial data, the inequality I(u0,v0)<0 is a necessary condition for nonexistence of global solutions. However, it seems that the condition I(u0,v0)<0, alone, does not imply nonexistence of global solutions for high energies; see [2, 6, 9–11]. Moreover, we did not require the sign of I(u0,v0) in the proof of Theorem 3.
From Corollary4, global nonexistence for small positive energies is obtained if P(u0,u1)+Q(v0,v1) is large enough and, consequently, since (67)2E0+Ku0,v0-u0VA2+v0VB2Ψu0,v0≥Pu0,u1+Qv0,v12,then K(u0,v0) must be also sufficiently big. Then, by the previous discussion, I(u0,v0)<0 is implied. Apparently, only for energies E0<d, the condition I(u0,v0)<0 characterizes the nonexistence of global solutions of problem (P).Remark 6.
We shall prove the following lower bound forβ0(68)β0>12Pu0,u1+Qv0,v12Ψu0,v0.This inequality is equivalent to (69)rλ∗r-2-1Pu0,u1+Qv0,v12<cΨ2u0,v0.In order to prove it, let us define, for any s≡P(u0,u1)+Q(v0,v1)2>0, the positive function (70)ls≡rλ∗r-2-1s>0,and we remember that λ∗ is a function of s, defined implicitly by (71)2rλ∗cΨ0Φ0r/2=λ∗-r-2r,where (72)Φ0≡cΨ0+sΨ0,Ψ0≡Ψu0,v0,Φ0≡Φu0,u1,v0,v1.On the other hand, from Theorem 3 we know that (73)lims→∞⁡λ∗=r-2r,lims→0⁡λ∗=1.Then, from the definition of λ∗ and Φ0, (74)lims→∞ls=2r-2lims→∞sλ∗cΨ0Φ0r/2=2r-2lims→∞sλ∗cΨ02cΨ02+sr/2=0.Also, (75)lims→0⁡ls=0.Consequently, there is some s∗∈(0,∞), such that l(s∗)=maxs∈(0,∞)l(s). After some calculations, we find that (76)s∗=cΨ02r-21-λ∗rλ∗-r-21-λ∗,ls∗=cΨ02rλ∗-r-21-λ∗rλ∗-r-21-λ∗and consequently (77)ls≤ls∗<cΨ02,foranys>0.


## 5. Some Examples

### 5.1. Nonlinear Klein-Gordon System

Klein-Gordon systems like(KG) were studied in [2, 4–11], where blow-up results were proved. We shall illustrate how Theorem 3 is applied for each one of these systems.


#### 5.1.1. Wang [2]

We rewrite the system(KG), introduced at the beginning of this work, as follows: (78)KG∗αutt-αΔu+αm12u+αK1xu=a2′p+1vq+1up-1u,vtt-Δv+m22v+K2xv=a2′q+1up+1vq-1v,u0,x=u0x,ut0,x=u1x,v0,x=v0x,vt0,x=v1x,on R×RN, where α≡a2(p+1)/a1(q+1), a2′≡a2/q+1.Here,P=αId, Q=Id= the identity operator, H=WP=WQ=L2(RN), and (·,·)2=·22 are the inner product and the norm square in L2(RN), respectively. Also, (79)Au=-αΔu+αm12u+αK1xuBv=-Δv+αm22v+αK2xv,VA=u∈H1RN:∫RNK1xuxdx<∞,VB=v∈H1RN:∫RNK2xvxdx<∞.Finally, (80)Fu,v=a2′p+1vq+1up-1u,Gu,v=a2′q+1up+1vq-1v.Hypothesis (H0) holds with c=min⁡{m12,m22}, (81)uVA2+vVB2≥α∇u22+∇v22+cαu22+v22≥cαu22+v22=cuWP2+vWQ2.Hypothesis (H1) holds with r≡p+q+2>4, since the potential operator K of (F,G) is (82)Ku,v=a2′vq+1,up+12,and then, (83)Fu,v,u2+Gu,v,v2-rKu,v=0.Theorem 1 is true and nonexistence of global solutions is due to blow-up; see [2] for the details, where some bounds on p,q are required. Consequently, by Theorem 3, if the initial data satisfy (84)αu022+v022>0,αu0,u12+v0,v12>0,and the initial energy is such that (85)min⁡m12,m22p+q2p+q+2αu022+v022ν∗2/p+q<E0<p+q2p+q+2λ∗min⁡m12,m22αu022+v022+αu0,u12+v0,v122αu022+v022,then the solution blows up in finite time in the norm of H. We notice that as α(u0,u1)2+(v0,v1)2 grows, then ν∗ grows, and consequently the lower bound of E0 is close to zero, and also the upper bound grows. That is, by Corollary 4 for every positive initial energy E0, there exist initial data satisfying (84) such that the solution blows up. This result is new in the literature. Sufficient conditions have been given before. Indeed, blow-up is proved in [2] if (84) holds, I(u0,v0)<0, and the initial energy is such that (86)0<E0<min⁡m12,m22p+q2p+q+2αu022+v022.In Theorem 3 we did not assume any sign of I(u0,v0) and β0 is larger than the upper bound on E0 given in [2].


#### 5.1.2. Liu [6]

For the corresponding linear damped problem of(KG) blow-up was proved under the same sufficient conditions given in [2] for high energies, and for E0<d by means of the potential well method.


#### 5.1.3. Korpusov [10]

A Klein-Gordon system with linear damping was analyzed with source terms of the form(87)Fu,v=4u+av3+2buv2,Gu,v=4au+av+2bu2v,where a>0,b>0. Hypothesis (H1) holds with r=4, and the potential operator (88)Ku,v=u+av44+buv22,and then, (89)Fu,v,u2+Gu,v,v2-4Ku,v=0.Blow-up of solutions was proved if (84) holds with α=1, I(u0,v0)<0, and (90)E0<12Pu0,u1+Qv0,v1-μΨu0,v02Ψu0,v0,where μ is the coefficient of the linear damping term in the system studied in [13]. If μ=0, (91)E0<12Pu0,u1+Qv0,v12Ψu0,v0<β0,where the second inequality is the lower bound for β0, obtained in Remark 6. Hence, Theorem 3 improves the result presented in [10].


#### 5.1.4. Aliev and Yusifova [9]

A system of several equations with linear damping was studied. If we consider only two, without damping, these are like those in(KG)∗ with a2′=1, and hypothesis (H1) holds with r=p+q+2>2, because p>0 and q>0. The blow-up of solutions is proved in [9] if (84) holds, with α=1, I(u0,v0)<0, and (92)E0<p+q2p+q+2u0WP2+v0WQ2<β0,where u0WP2=(p+1)u022 and v0WQ2=(q+1)v022. Again, Theorem 3 improves the result presented in [9].


#### 5.1.5. Wu [11]

A system like(KG) with a linear damping was studied with no particular source terms. However, these are assumed to satisfy (H1) with r=2+4δ>2, some δ=r-2/4>0. Blow-up is proved for high energies, under the following conditions: I(u0,v0)<0 and (93)r-22u0,u12+v0,v12>u022+v022>r-2rE0>0.These conditions are more restrictive than the ones assumed in Theorem 3.


#### 5.1.6. Ye [8], Benaissa et al. [4]

Klein-Gordon system like(KG) was studied with nonlinear damping and degenerated nonlinear damping terms, respectively. In both papers, blow-up was showed for energies E0<d, by the potential well method and the following source terms were considered (94)Fu,v=a1u+v2p+1u+v+a2upvp+2u,Gu,v=a1u+v2p+1u+v+a2vpup+2v,with p>-1, a1>0 and a2>0. Hypothesis (H1) holds with r=2(p+2)>2, and the potential operator (95)Ku,v=12p+2a1u+v2p+22p+2+2a2uvp+2p+2,and then, (96)Fu,v,u2+Gu,v,v2-2p+2Ku,v=0.Hence, Theorem 3 and Corollary 4 are applied and blow-up is proved for the undamped case in [4, 8], for high energies.


#### 5.1.7. Wu [7], Gan and Zhang [5]

By means of the potential well method, blow-up was showed for solutions of systems like(KG), with E0<d, without damping, and with linear damping, respectively. In those papers, particular cases of the following source terms were considered (97)Fu,v=a1u2p+a2up-1vp+1u,Gu,v=a1v2p+a2vp-1up+1v,with p>0, a1>0, and a2>0. Hypothesis (H1) holds with r=2(p+1)>2, and the potential operator (98)Ku,v=12p+1a1u2p+12p+1+a1v2p+12p+1+2a2uvp+1p+1,and then, (99)Fu,v,u2+Gu,v,v2-2p+1Ku,v=0.Hence, Theorem 3 and Corollary 4 are applied and blow-up is proved for the undamped case in [5, 7], for high energies.


### 5.2. Generalized Boussinesq System

We consider the system(100)GButt-α1Δu-α2Δutt+α3Δ2u+m1u+ΔFu,v=0,vtt-β1Δv-β2Δvtt+β3Δ2v+m2v+ΔGu,v=0,u0,x=u0x,ut0,x=u1x,v0,x=v0x,vt0,x=v1x,on R×RN, where αi>0,βi>0,i=1,2,3,mj>0,j=1,2. For the physics of the problem we refer to [13, 14].Applying(-Δ)-1 to the system above, we get (101)GB∗-Δ-1+α2Idutt+-α3Δ+m1-Δ-1+α1Idu=Fu,v,-Δ-1+β2Idvtt+-β3Δ+m2-Δ-1+β1Idv=Gu,v,u0,x=u0x,ut0,x=u1x,v0,x=v0x,vt0,x=v1x,on R×RN. Then, (102)Pu˙=-Δ-1+α2Idu˙,Au=-α3Δ+m1-Δ-1+α1Idu,Qv˙=-Δ-1+β2Idv˙,Bu=-β3Δ+m2-Δ-1+β1Idv,and H=L2(RN), (103)WP=WQ=u∈L2RN:-Δ-1/2u∈L2RN,VA=VB=u∈H1RN:-Δ-1/2u∈L2RN.Moreover, if (104)u∗2=u,u∗≡-Δ-1/2u,-Δ-1/2u2,then (105)uWP2=u∗2+α2u22,uVA2=α3∇u22+m1u∗2+α1u22,vWQ2=v∗2+β2v22,vVB2=β3∇v22+m2v∗2+β1v22.Consequently, hypothesis (H0) holds with c=minm1,α1/α2,m2,β1/β2. We assume that the source terms (F,G) and the corresponding potential operator K do not have any particular form but they satisfy (H1). We do not know any reference proving an existence and uniqueness result for this system. We assume that Theorem 1 is true then nonexistence of global solutions is due to blow-up. By Theorem 3 and Corollary 4 for every positive initial energy E0, where (106)E0=12u1∗2+α2u122+α3∇u022+m1u0∗2+α1u022+12v1∗2+β2v122+β3∇v022+m2v0∗2+β1v022-Ku0,v0,there exists initial data such that (107)u0∗2+α2u022+v0∗2+β2v022>0,u0,u1∗+α2u0,u12+v0,v1∗+β2v0,v12>0imply the nonexistence of global solutions in the norm of H.


## 5.1. Nonlinear Klein-Gordon System

Klein-Gordon systems like(KG) were studied in [2, 4–11], where blow-up results were proved. We shall illustrate how Theorem 3 is applied for each one of these systems.


### 5.1.1. Wang [2]

We rewrite the system(KG), introduced at the beginning of this work, as follows: (78)KG∗αutt-αΔu+αm12u+αK1xu=a2′p+1vq+1up-1u,vtt-Δv+m22v+K2xv=a2′q+1up+1vq-1v,u0,x=u0x,ut0,x=u1x,v0,x=v0x,vt0,x=v1x,on R×RN, where α≡a2(p+1)/a1(q+1), a2′≡a2/q+1.Here,P=αId, Q=Id= the identity operator, H=WP=WQ=L2(RN), and (·,·)2=·22 are the inner product and the norm square in L2(RN), respectively. Also, (79)Au=-αΔu+αm12u+αK1xuBv=-Δv+αm22v+αK2xv,VA=u∈H1RN:∫RNK1xuxdx<∞,VB=v∈H1RN:∫RNK2xvxdx<∞.Finally, (80)Fu,v=a2′p+1vq+1up-1u,Gu,v=a2′q+1up+1vq-1v.Hypothesis (H0) holds with c=min⁡{m12,m22}, (81)uVA2+vVB2≥α∇u22+∇v22+cαu22+v22≥cαu22+v22=cuWP2+vWQ2.Hypothesis (H1) holds with r≡p+q+2>4, since the potential operator K of (F,G) is (82)Ku,v=a2′vq+1,up+12,and then, (83)Fu,v,u2+Gu,v,v2-rKu,v=0.Theorem 1 is true and nonexistence of global solutions is due to blow-up; see [2] for the details, where some bounds on p,q are required. Consequently, by Theorem 3, if the initial data satisfy (84)αu022+v022>0,αu0,u12+v0,v12>0,and the initial energy is such that (85)min⁡m12,m22p+q2p+q+2αu022+v022ν∗2/p+q<E0<p+q2p+q+2λ∗min⁡m12,m22αu022+v022+αu0,u12+v0,v122αu022+v022,then the solution blows up in finite time in the norm of H. We notice that as α(u0,u1)2+(v0,v1)2 grows, then ν∗ grows, and consequently the lower bound of E0 is close to zero, and also the upper bound grows. That is, by Corollary 4 for every positive initial energy E0, there exist initial data satisfying (84) such that the solution blows up. This result is new in the literature. Sufficient conditions have been given before. Indeed, blow-up is proved in [2] if (84) holds, I(u0,v0)<0, and the initial energy is such that (86)0<E0<min⁡m12,m22p+q2p+q+2αu022+v022.In Theorem 3 we did not assume any sign of I(u0,v0) and β0 is larger than the upper bound on E0 given in [2].


### 5.1.2. Liu [6]

For the corresponding linear damped problem of(KG) blow-up was proved under the same sufficient conditions given in [2] for high energies, and for E0<d by means of the potential well method.


### 5.1.3. Korpusov [10]

A Klein-Gordon system with linear damping was analyzed with source terms of the form(87)Fu,v=4u+av3+2buv2,Gu,v=4au+av+2bu2v,where a>0,b>0. Hypothesis (H1) holds with r=4, and the potential operator (88)Ku,v=u+av44+buv22,and then, (89)Fu,v,u2+Gu,v,v2-4Ku,v=0.Blow-up of solutions was proved if (84) holds with α=1, I(u0,v0)<0, and (90)E0<12Pu0,u1+Qv0,v1-μΨu0,v02Ψu0,v0,where μ is the coefficient of the linear damping term in the system studied in [13]. If μ=0, (91)E0<12Pu0,u1+Qv0,v12Ψu0,v0<β0,where the second inequality is the lower bound for β0, obtained in Remark 6. Hence, Theorem 3 improves the result presented in [10].


### 5.1.4. Aliev and Yusifova [9]

A system of several equations with linear damping was studied. If we consider only two, without damping, these are like those in(KG)∗ with a2′=1, and hypothesis (H1) holds with r=p+q+2>2, because p>0 and q>0. The blow-up of solutions is proved in [9] if (84) holds, with α=1, I(u0,v0)<0, and (92)E0<p+q2p+q+2u0WP2+v0WQ2<β0,where u0WP2=(p+1)u022 and v0WQ2=(q+1)v022. Again, Theorem 3 improves the result presented in [9].


### 5.1.5. Wu [11]

A system like(KG) with a linear damping was studied with no particular source terms. However, these are assumed to satisfy (H1) with r=2+4δ>2, some δ=r-2/4>0. Blow-up is proved for high energies, under the following conditions: I(u0,v0)<0 and (93)r-22u0,u12+v0,v12>u022+v022>r-2rE0>0.These conditions are more restrictive than the ones assumed in Theorem 3.


### 5.1.6. Ye [8], Benaissa et al. [4]

Klein-Gordon system like(KG) was studied with nonlinear damping and degenerated nonlinear damping terms, respectively. In both papers, blow-up was showed for energies E0<d, by the potential well method and the following source terms were considered (94)Fu,v=a1u+v2p+1u+v+a2upvp+2u,Gu,v=a1u+v2p+1u+v+a2vpup+2v,with p>-1, a1>0 and a2>0. Hypothesis (H1) holds with r=2(p+2)>2, and the potential operator (95)Ku,v=12p+2a1u+v2p+22p+2+2a2uvp+2p+2,and then, (96)Fu,v,u2+Gu,v,v2-2p+2Ku,v=0.Hence, Theorem 3 and Corollary 4 are applied and blow-up is proved for the undamped case in [4, 8], for high energies.


### 5.1.7. Wu [7], Gan and Zhang [5]

By means of the potential well method, blow-up was showed for solutions of systems like(KG), with E0<d, without damping, and with linear damping, respectively. In those papers, particular cases of the following source terms were considered (97)Fu,v=a1u2p+a2up-1vp+1u,Gu,v=a1v2p+a2vp-1up+1v,with p>0, a1>0, and a2>0. Hypothesis (H1) holds with r=2(p+1)>2, and the potential operator (98)Ku,v=12p+1a1u2p+12p+1+a1v2p+12p+1+2a2uvp+1p+1,and then, (99)Fu,v,u2+Gu,v,v2-2p+1Ku,v=0.Hence, Theorem 3 and Corollary 4 are applied and blow-up is proved for the undamped case in [5, 7], for high energies.


## 5.1.1. Wang [2]

We rewrite the system(KG), introduced at the beginning of this work, as follows: (78)KG∗αutt-αΔu+αm12u+αK1xu=a2′p+1vq+1up-1u,vtt-Δv+m22v+K2xv=a2′q+1up+1vq-1v,u0,x=u0x,ut0,x=u1x,v0,x=v0x,vt0,x=v1x,on R×RN, where α≡a2(p+1)/a1(q+1), a2′≡a2/q+1.Here,P=αId, Q=Id= the identity operator, H=WP=WQ=L2(RN), and (·,·)2=·22 are the inner product and the norm square in L2(RN), respectively. Also, (79)Au=-αΔu+αm12u+αK1xuBv=-Δv+αm22v+αK2xv,VA=u∈H1RN:∫RNK1xuxdx<∞,VB=v∈H1RN:∫RNK2xvxdx<∞.Finally, (80)Fu,v=a2′p+1vq+1up-1u,Gu,v=a2′q+1up+1vq-1v.Hypothesis (H0) holds with c=min⁡{m12,m22}, (81)uVA2+vVB2≥α∇u22+∇v22+cαu22+v22≥cαu22+v22=cuWP2+vWQ2.Hypothesis (H1) holds with r≡p+q+2>4, since the potential operator K of (F,G) is (82)Ku,v=a2′vq+1,up+12,and then, (83)Fu,v,u2+Gu,v,v2-rKu,v=0.Theorem 1 is true and nonexistence of global solutions is due to blow-up; see [2] for the details, where some bounds on p,q are required. Consequently, by Theorem 3, if the initial data satisfy (84)αu022+v022>0,αu0,u12+v0,v12>0,and the initial energy is such that (85)min⁡m12,m22p+q2p+q+2αu022+v022ν∗2/p+q<E0<p+q2p+q+2λ∗min⁡m12,m22αu022+v022+αu0,u12+v0,v122αu022+v022,then the solution blows up in finite time in the norm of H. We notice that as α(u0,u1)2+(v0,v1)2 grows, then ν∗ grows, and consequently the lower bound of E0 is close to zero, and also the upper bound grows. That is, by Corollary 4 for every positive initial energy E0, there exist initial data satisfying (84) such that the solution blows up. This result is new in the literature. Sufficient conditions have been given before. Indeed, blow-up is proved in [2] if (84) holds, I(u0,v0)<0, and the initial energy is such that (86)0<E0<min⁡m12,m22p+q2p+q+2αu022+v022.In Theorem 3 we did not assume any sign of I(u0,v0) and β0 is larger than the upper bound on E0 given in [2].


## 5.1.2. Liu [6]

For the corresponding linear damped problem of(KG) blow-up was proved under the same sufficient conditions given in [2] for high energies, and for E0<d by means of the potential well method.


## 5.1.3. Korpusov [10]

A Klein-Gordon system with linear damping was analyzed with source terms of the form(87)Fu,v=4u+av3+2buv2,Gu,v=4au+av+2bu2v,where a>0,b>0. Hypothesis (H1) holds with r=4, and the potential operator (88)Ku,v=u+av44+buv22,and then, (89)Fu,v,u2+Gu,v,v2-4Ku,v=0.Blow-up of solutions was proved if (84) holds with α=1, I(u0,v0)<0, and (90)E0<12Pu0,u1+Qv0,v1-μΨu0,v02Ψu0,v0,where μ is the coefficient of the linear damping term in the system studied in [13]. If μ=0, (91)E0<12Pu0,u1+Qv0,v12Ψu0,v0<β0,where the second inequality is the lower bound for β0, obtained in Remark 6. Hence, Theorem 3 improves the result presented in [10].


## 5.1.4. Aliev and Yusifova [9]

A system of several equations with linear damping was studied. If we consider only two, without damping, these are like those in(KG)∗ with a2′=1, and hypothesis (H1) holds with r=p+q+2>2, because p>0 and q>0. The blow-up of solutions is proved in [9] if (84) holds, with α=1, I(u0,v0)<0, and (92)E0<p+q2p+q+2u0WP2+v0WQ2<β0,where u0WP2=(p+1)u022 and v0WQ2=(q+1)v022. Again, Theorem 3 improves the result presented in [9].


## 5.1.5. Wu [11]

A system like(KG) with a linear damping was studied with no particular source terms. However, these are assumed to satisfy (H1) with r=2+4δ>2, some δ=r-2/4>0. Blow-up is proved for high energies, under the following conditions: I(u0,v0)<0 and (93)r-22u0,u12+v0,v12>u022+v022>r-2rE0>0.These conditions are more restrictive than the ones assumed in Theorem 3.


## 5.1.6. Ye [8], Benaissa et al. [4]

Klein-Gordon system like(KG) was studied with nonlinear damping and degenerated nonlinear damping terms, respectively. In both papers, blow-up was showed for energies E0<d, by the potential well method and the following source terms were considered (94)Fu,v=a1u+v2p+1u+v+a2upvp+2u,Gu,v=a1u+v2p+1u+v+a2vpup+2v,with p>-1, a1>0 and a2>0. Hypothesis (H1) holds with r=2(p+2)>2, and the potential operator (95)Ku,v=12p+2a1u+v2p+22p+2+2a2uvp+2p+2,and then, (96)Fu,v,u2+Gu,v,v2-2p+2Ku,v=0.Hence, Theorem 3 and Corollary 4 are applied and blow-up is proved for the undamped case in [4, 8], for high energies.


## 5.1.7. Wu [7], Gan and Zhang [5]

By means of the potential well method, blow-up was showed for solutions of systems like(KG), with E0<d, without damping, and with linear damping, respectively. In those papers, particular cases of the following source terms were considered (97)Fu,v=a1u2p+a2up-1vp+1u,Gu,v=a1v2p+a2vp-1up+1v,with p>0, a1>0, and a2>0. Hypothesis (H1) holds with r=2(p+1)>2, and the potential operator (98)Ku,v=12p+1a1u2p+12p+1+a1v2p+12p+1+2a2uvp+1p+1,and then, (99)Fu,v,u2+Gu,v,v2-2p+1Ku,v=0.Hence, Theorem 3 and Corollary 4 are applied and blow-up is proved for the undamped case in [5, 7], for high energies.


## 5.2. Generalized Boussinesq System

We consider the system(100)GButt-α1Δu-α2Δutt+α3Δ2u+m1u+ΔFu,v=0,vtt-β1Δv-β2Δvtt+β3Δ2v+m2v+ΔGu,v=0,u0,x=u0x,ut0,x=u1x,v0,x=v0x,vt0,x=v1x,on R×RN, where αi>0,βi>0,i=1,2,3,mj>0,j=1,2. For the physics of the problem we refer to [13, 14].Applying(-Δ)-1 to the system above, we get (101)GB∗-Δ-1+α2Idutt+-α3Δ+m1-Δ-1+α1Idu=Fu,v,-Δ-1+β2Idvtt+-β3Δ+m2-Δ-1+β1Idv=Gu,v,u0,x=u0x,ut0,x=u1x,v0,x=v0x,vt0,x=v1x,on R×RN. Then, (102)Pu˙=-Δ-1+α2Idu˙,Au=-α3Δ+m1-Δ-1+α1Idu,Qv˙=-Δ-1+β2Idv˙,Bu=-β3Δ+m2-Δ-1+β1Idv,and H=L2(RN), (103)WP=WQ=u∈L2RN:-Δ-1/2u∈L2RN,VA=VB=u∈H1RN:-Δ-1/2u∈L2RN.Moreover, if (104)u∗2=u,u∗≡-Δ-1/2u,-Δ-1/2u2,then (105)uWP2=u∗2+α2u22,uVA2=α3∇u22+m1u∗2+α1u22,vWQ2=v∗2+β2v22,vVB2=β3∇v22+m2v∗2+β1v22.Consequently, hypothesis (H0) holds with c=minm1,α1/α2,m2,β1/β2. We assume that the source terms (F,G) and the corresponding potential operator K do not have any particular form but they satisfy (H1). We do not know any reference proving an existence and uniqueness result for this system. We assume that Theorem 1 is true then nonexistence of global solutions is due to blow-up. By Theorem 3 and Corollary 4 for every positive initial energy E0, where (106)E0=12u1∗2+α2u122+α3∇u022+m1u0∗2+α1u022+12v1∗2+β2v122+β3∇v022+m2v0∗2+β1v022-Ku0,v0,there exists initial data such that (107)u0∗2+α2u022+v0∗2+β2v022>0,u0,u1∗+α2u0,u12+v0,v1∗+β2v0,v12>0imply the nonexistence of global solutions in the norm of H.



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*Source: 7405725-2019-06-04.xml*