# Global Nonexistence of Positive Initial-Energy Solutions for Coupled Nonlinear Wave Equations with Damping and Source Terms

Authors: Liang Fei; Gao Hongjun
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

This work is concerned with a system of nonlinear wave equations with nonlinear damping and source terms acting on both equations. We prove a global nonexistence theorem for certain solutions with positive initial energy.


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## Body

## 1. Introduction

In this paper we study the initial-boundary-value problem(1.1)utt-div⁡(g(|∇u|2)∇u)+|ut|m-1ut=f1(u,v),(x,t)∈Ω×(0,T),vtt-div⁡(g(|∇v|2)∇v)+|vt|r-1vt=f2(u,v),(x,t)∈Ω×(0,T),u(x,t)=v(x,t)=0,x∈∂Ω×(0,T),u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,v(x,0)=v0(x),vt(x,0)=v1(x),x∈Ω,
where Ω is a bounded domain in ℝn with a smooth boundary ∂Ω, m,r≥1, and fi(·,·):ℝ2→ℝ (i=1,2) are given functions to be specified later. We assume that g is a function which satisfies (1.2)g∈C1,g(s)>0,g(s)+2sg′(s)>0
for s>0.To motivate our work, let us recall some results regardingg≡1. The single-wave equation of the form (1.3)utt-Δu+h(ut)=f(u),x∈Ω,t>0
in Ω×(0,∞) with initial and boundary conditions has been extensively studied, and many results concerning global existence, blow-up, energy decay have been obtained. In the absence of the source term, that is, (f=0), it is well known that the damping term h(ut) assures global existence and decay of the solution energy for arbitrary initial data (see [1]). In the absence of the damping term, the source term causes finite time blow-up of solutions with a large initial data (negative initial energy) (see [2, 3]). The interaction between the damping term and the source term makes the problem more interesting. This situation was first considered by Levine [4, 5] in the linear damping case h(ut)=aut and a polynomial source term of the form f(u)=b|u|p-2u. He showed that solutions with negative initial energy blow up in finite time. The main tool used in [4, 5] is the “concavity method.” Georgiev and Todorova in [6] extended Levine's result to the nonlinear damping case h(ut)=a|ut|m-2ut. In their work, the authors considered problem (1.3) with f(u)=b|u|p-2u and introduced a method different from the one known as the concavity method and showed that solutions with negative energy continue to exist globally in time if m≥p≥2 and blow up in finite time if p>m≥2 and the initial energy is sufficiently negative. This latter result has been pushed by Messaoudi [7] to the situation where the initial energy E(0)<0 and has been improved by the same author in [8] to accommodate certain solutions with positive initial energy.In the case ofg being a given nonlinear function, the following equation: (1.4)utt-g(ux)x-uxxt+δ|ut|m-1ut=μ|u|p-1u,x∈(0,1),t>0,
with initial and boundary conditions has been extensively studied. Equation of type of (1.4) is a class of nonlinear evolution governing the motion of a viscoelastic solid composed of the material of the rate type, see [9–12]. It can also be seen as field equation governing the longitudinal motion of a viscoelastic bar obeying the nonlinear Voigt model, see [13]. In two- and three-dimensional cases, they describe antiplane shear motions of viscoelastic solids. We refer to [14–16] for physical origins and derivation of mathematical models of motions of viscoelastic media and only recall here that, in applications, the unknown u naturally represents the displacement of the body relative to a fixed reference configuration. When δ=μ=0, there have been many impressive works on the global existence and other properties of solutions of (1.4), see [9, 10, 17, 18]. Especially, in [19] the authors have proved the global existence and uniqueness of the generalized and classical solution for the initial boundary value problem (1.4) when we replace δ|ut|m-1ut and μ|u|p-1u by g(ut) and f(u), respectively. But about the blow-up of the solution for problem, in this paper there has not been any discussion. Chen et al. [20] considered problem (1.4) and first established an ordinary differential inequality, next given the sufficient conditions of blow-up of the solution of (1.4) by the inequality. In [21], Hao et al. considered the single-wave equation of the form (1.5)utt-div⁡(g(|∇u|2)∇u)+h(ut)=f(u),x∈Ω,t>0
with initial and Dirichlet boundary condition, where g satisfies condition (1.2) and (1.6)g(s)≥b1+b2sq,q≥0.
The damping term has the form (1.7)h(ut)=d1ut+d2|ut|r-1ut,r>1.
The source term is (1.8)f(u)=a1u+a2|u|p-1u
with p≥1 for n=1,2 and 1≤n≤2n/(n-2) for n≥3, a1,a2,b1,b2,d1,d2 are nonnegative constants, and b1+b2>0. By using the energy compensation method [7, 8, 22], they proved that under some conditions on the initial value and the growth orders of the nonlinear strain term, the damping term, and the source term, the solution to problem (1.5) exists globally and blows up in finite time with negative initial energy, respectively.Some special cases of system (1.1) arise in quantum field theory which describe the motion of charged mesons in an electromagnetic field, see [23, 24]. Recently, some of the ideas in [6, 22] have been extended to study certain systems of wave equations. Agre and Rammaha [25] studied the system of (1.1) with g≡1 and proved several results concerning local and global existence of a weak solution and showed that any weak solution with negative initial energy blows up in finite time, using the same techniques as in [6]. This latter blow-up result has been improved by Said-Houari [26] by considering a larger class of initial data for which the initial energy can take positive values. Recently, Wu et al. [27] considered problem (1.1) with the nonlinear functions f1(u,v) and f2(u,v) satisfying appropriate conditions. They proved under some restrictions on the parameters and the initial data several results on global existence of a weak solution. They also showed that any weak solution with initial energy E(0)<0 blows up in finite time.In this paper, we also consider problem (1.1) and improve the global nonexistence result obtained in [27], for a large class of initial data in which our initial energy can take positive values. The main tool of the proof is a technique introduced by Payne and Sattinger [28] and some estimates used firstly by Vitillaro [29], in order to study a class of a single-wave equation.


## 2. Preliminaries and Main Result

First, let us introduce some notation used throughout this paper. We denote by||·||q the Lq(Ω) norm for 1≤q≤∞ and by ||∇·||2 the Dirichlet norm in H01(Ω) which is equivalent to the H1(Ω) norm. Moreover, we set (2.1)(φ,ψ)=∫Ωφ(x)ψ(x)dx
as the usual L2(Ω) inner product.Concerning the functionsf1(u,v) and f2(u,v), we take (2.2)f1(u,v)=[a|u+v|2(p+1)(u+v)+b|u|pu|v|(p+2)],f2(u,v)=[a|u+v|2(p+1)(u+v)+b|u|(p+2)|v|pv],
where a,b>0 are constants and p satisfies (2.3){p>-1,ifn=1,2,-1<p≤4-nn-2,ifn≥3.
One can easily verify that (2.4)uf1(u,v)+vf2(u,v)=2(p+2)F(u,v),∀(u,v)∈R2,
where (2.5)F(u,v)=12(p+2)[a|u+v|2(p+2)+2b|uv|p+2].We have the following result.Lemma 2.1 (see [30, Lemma  2.1]).
There exist two positive constantsc0 and c1 such that
(2.6)c02(p+2)(|u|2(p+2)+|v|2(p+2))≤F(u,v)≤c12(p+2)(|u|2(p+2)+|v|2(p+2)).Throughout this paper, we defineg by (2.7)g(s)=b1+b2sq,q≥0,b1+b2>0,
where b1,b2 are nonnegative constants. Obviously, g satisfies conditions (1.2) and (1.6). Set (2.8)G(s)=∫0sg(s)ds,s≥0.In order to state and prove our result, we introduce the following function space:(2.9)Z={(u,v)∣u,v∈L∞([0,T);W01,2(q+1)(Ω)∩L2(p+2)(Ω)),ut∈L∞([0,T);L2(Ω))∩Lm+1(Ω×(0,T)),vt∈L∞([0,T);L2(Ω))∩Lr+1(Ω×(0,T)),utt,vtt∈L∞([0,T),L2(Ω))W01,2(q+1)}.Define the energy functionalE(t) associated with our system
(2.10)E(t)=12(‖ut(t)‖22+‖vt(t)‖22)+12∫Ω(G(|∇u|2)+G(|∇v|2))dx-∫ΩF(u,v)dx.
A simple computation gives (2.11)dE(t)dt=-‖u‖m+1m+1-‖v‖r+1r+1≤0.Our main result reads as follows.Theorem 2.2.
Assume that (2.3) holds. Assume further that  2(p+2)>max{2q+2,m+1,r+1}. Then any solution of (1.1) with initial data satisfying
(2.12)(∫Ω(G(|∇u0|2)+G(|∇v0|2))dx)1/2>α1,E(0)<E2,
cannot exist for all time, where the constant α1 and E2 are defined in (3.7).


## 3. Proof of Theorem2.2

In this section, we deal with the blow-up of solutions of the system (1.1). Before we prove our main result, we need the following lemmas.Lemma 3.1.
LetΘ(t) be a solution of the ordinary differential inequality
(3.1)dΘ(t)dt≥CΘ1+ε(t),t>0,
where ε>0. If Θ(0)>0, then the solution ceases to exist for t≥Θ-ε(0)C-1ε-1.Lemma 3.2.
Assume that (2.3) holds. Then there exists η>0 such that for any (u,v)∈Z, one has
(3.2)‖u+v‖2(p+2)2(p+2)+2‖uv‖p+2p+2≤η(∫Ω(G(|∇u|2)+G(|∇v|2))dx)p+2.Proof.
By using Minkowski's inequality, we get(3.3)‖u+v‖2(p+2)2≤2(‖u‖2(p+2)2+‖v‖2(p+2)2).
Also, Hölder's and Young's inequalities give us
(3.4)‖uv‖p+2≤‖u‖2(p+2)‖v‖2(p+2)≤12(‖u‖2(p+2)2+‖v‖2(p+2)2).
If b1>0, then we have
(3.5)∫Ω(G(|∇u|2)+G(|∇v|2))dx≥c(‖∇u‖22+‖∇v‖22).
If b1=0, from b1+b2>0, we have b2>0. Since W01,2(q+1)(Ω)↪H01(Ω), we have
(3.6)‖∇u‖22+‖∇v‖22≤c1(‖∇u‖2(q+1)2+‖∇v‖2(q+1)2),
which implies that (3.5) still holds for b1=0. Combining (3.3), (3.4) with (3.5) and the embedding H01(Ω)↪L2(p+2)(Ω), we have (3.2).In order to prove our result and for the sake of simplicity, we takea=b=1 and introduce the following: (3.7)B=η1/(2(p+2)),α1=B-(p+2/(p+1)),E1=(12-12(p+2))α12,E2=(12(q+1)-12(p+2))α12,
where η is the optimal constant in (3.2). The following lemma will play an essential role in the proof of our main result, and it is similar to a lemma used first by Vitillaro [29].Lemma 3.3.
Assume that (2.3) holds. Let (u,v)∈Z be the solution of the system (1.1). Assume further that E(0)<E1 and
(3.8)(∫Ω(G(|∇u0|2)+G(|∇v0|2))dx)1/2>α1.
Then there exists a constant α2>α1 such that
(3.9)(∫Ω(G(|∇u|2)+G(|∇v|2))dx)1/2≥α2,fort>0,(3.10)(‖u+v‖2(p+2)2(p+2)+2‖uv‖p+2p+2)1/(2(p+2))≥Bα2,fort>0.Proof.
We first note that, by (2.10), (3.2), and the definition of B, we have
(3.11)E(t)≥12∫Ω(G(|∇u|2)+G(|∇v|2))dx-12(p+2)(‖u+v‖2(p+2)2(p+2)+2‖uv‖p+2p+2)≥12∫Ω(G(|∇u|2)+G(|∇v|2))dx-B2(p+2)2(p+2)(∫Ω(G(|∇u|2)+G(|∇v|2))dx)p+2=12α2-B2(p+2)2(p+2)α2(p+2),
where α=(∫Ω(G(|∇u|2)+G(|∇v|2))dx)1/2. It is not hard to verify that g is increasing for 0<α<α1, decreasing for α>α1, g(α)→-∞ as α→+∞, and
(3.12)g(α1)=12α12-B2(p+2)2(p+2)α12(p+2)=E1,
where α1 is given in (3.7). Since E(0)<E1, there exists α2>α1 such that g(α2)=E(0).
Setα0=(∫Ω(G(|∇u0|2)+G(|∇v0|2))dx)1/2. Then by (3.11) we get g(α0)≤E(0)=g(α2), which implies that α0≥α2. Now, to establish (3.9), we suppose by contradiction that
(3.13)(∫Ω(G(|∇u(t0)|2)+G(|∇v(t0)|2))dx)1/2<α2,
for some t0>0. By the continuity of ∫Ω(G(|∇u|2)+G(|∇v|2))dx, we can choose t0 such that
(3.14)(∫Ω(G(|∇u(t0)|2)+G(|∇v(t0)|2))dx)1/2>α1.
Again, the use of (3.11) leads to
(3.15)E(t0)≥g((∫Ω(G(|∇u(t0)|2)+G(|∇v(t0)|2))dx)1/2)>g(α2)=E(0).
This is impossible since E(t)≤E(0) for all t∈[0,T). Hence (3.9) is established.
To prove (3.10), we make use of (2.10) to get
(3.16)12∫Ω(G(|∇u|2)+G(|∇v|2))dx≤E(0)+12(p+2)(‖u+v‖2(p+2)2(p+2)+2‖uv‖p+2p+2).
Consequently, (3.9) yields
(3.17)12(p+2)(‖u+v‖2(p+2)2(p+2)+2‖uv‖p+2p+2)≥12∫Ω(G(|∇u|2)+G(|∇v|2))dx-E(0)≥12α22-E(0)≥12α22-g(α2)=B2(p+2)2(p+2)α22(p+2).
Therefore, (3.17) and (3.7) yield the desired result.Proof of Theorem2.2.
We suppose that the solution exists for all time and we reach to a contradiction. Set(3.18)H(t)=E2-E(t).
By using (2.10) and (3.18), we have
(3.19)0<H(0)≤H(t)=E2-12(‖ut(t)‖22+‖vt(t)‖22)-12∫Ω(G(|∇u|2)+G(|∇v|2))dx+12(p+2)(‖u+v‖2(p+2)2(p+2)+2‖uv‖p+2p+2).
From (3.9), we have
(3.20)E2-12(‖ut(t)‖22+‖vt(t)‖22)-12∫Ω(G(|∇u|2)+G(|∇v|2))dx≤E2-12α12≤E1-12α12=-12(p+2)α12<0,∀t≥0.
Hence, by the above inequality and (2.6), we have
(3.21)0<H(0)≤H(t)≤12(p+2)(‖u+v‖2(p+2)2(p+2)+2‖uv‖p+2p+2),(3.22)≤c12(p+2)(‖u‖2(p+2)2(p+2)+‖v‖2(p+2)2(p+2)).
We then define
(3.23)Θ(t)=H1-δ(t)+ϵ∫Ω(uut+vvt)dx,
where ϵ small enough is to be chosen later and
(3.24)0<δ≤min⁡{p+12(p+2),2(p+2)-(m+1)2m(p+2),2(p+2)-(r+1)2r(p+2)}.
Our goal is to show that Θ(t) satisfies the differential inequality (3.1) which leads to a blow-up in finite time. By taking a derivative of (3.23), we get
(3.25)Θ′(t)=(1-δ)H-δ(t)H′(t)+ϵ(‖ut‖22+‖vt‖22)-ϵ∫Ω(g(|∇u|2)|∇u|2+g(|∇v|2)|∇v|2)dx-ϵ∫Ω(|ut|m-1utu+|vt|r-1vtv)dx+ϵ∫Ω(uf1(u,v)+vf2(u,v))dx=(1-δ)H-δ(t)H′(t)+ϵ(‖ut‖22+‖vt‖22)-b1ϵ(‖∇u‖22+‖∇v‖22)-ϵb2‖∇u‖2(q+2)2(q+2)-ϵb2‖∇v‖2(q+2)2(q+2)-ϵ∫Ω(|ut|m-1utu+|vt|r-1vtv)dx+ϵ(‖u+v‖2(p+2)2(p+2)+2‖uv‖p+2p+2).
From the definition of H(t), it follows that
(3.26)-b2‖∇u‖2(q+2)2(q+2)-b2‖∇v‖2(q+2)2(q+2)=2(q+1)H(t)-2(q+1)E2+(q+1)(‖ut‖22+‖vt‖22)+(q+1)b1(‖∇u‖22+‖∇v‖22)-2(q+1)∫ΩF(u,v)dx,
which together with (3.25) gives
(3.27)Θ′(t)=(1-δ)H-δ(t)H′(t)+ϵ(q+2)(‖ut‖22+‖vt‖22)+b1qϵ(‖∇u‖22+‖∇v‖22)-ϵ∫Ω(|ut|m-1utu+|vt|r-1vtv)dx+ϵ(1-q+1p+2)(‖u+v‖2(p+2)2(p+2)+2‖uv‖p+2p+2)+2(q+1)H(t)-2(q+1)E2.
Then, using (3.10), we obtain
(3.28)Θ′(t)≥(1-δ)H-δ(t)H′(t)+ϵ(q+2)(‖ut‖22+‖vt‖22)+b1qϵ(‖∇u‖22+‖∇v‖22)+2(q+1)H(t)+ϵc¯(‖u+v‖2(p+2)2(p+2)+2‖uv‖p+2p+2)-ϵ∫Ω(|ut|m-1utu+|vt|r-1vtv)dx,
where c¯=1-(q+1)/(p+2)-2(q+1)E2(Bα2)-2(p+2). It is clear that c¯>0, since α2>B-(p+2)/(p+1). We now exploit Young's inequality to estimate the last two terms on the right side of (3.28)
(3.29)|∫Ω|ut|m-1utudx|≤η1m+1m+1‖u‖m+1m+1+mη1-((m+1)/m)m+1‖ut‖m+1m+1,|∫Ω|vt|r-1vtvdx|≤η2r+1r+1‖v‖r+1r+1+rη2-((r+1)/r)r+1‖vt‖r+1r+1,
where η1,η2 are parameters depending on the time t and specified later. Inserting the last two estimates into (3.28), we have
(3.30)Θ′(t)≥(1-δ)H-δ(t)H′(t)+ϵ(q+2)(‖ut‖22+‖vt‖22)+b1qϵ(‖∇u‖22+‖∇v‖22)+2(q+1)H(t)+ϵc¯(‖u+v‖2(p+2)2(p+2)+2‖uv‖p+2p+2)-ϵη1m+1m+1‖u‖m+1m+1-ϵmη1-((m+1)/m)m+1‖ut‖m+1m+1-ϵη2r+1r+1‖v‖r+1r+1-ϵrη2-((r+1)/r)r+1‖vt‖r+1r+1.
By choosing η1 and η2 such that
(3.31)η1-(m+1)/m=M1H-δ(t),η2-(r+1)/r=M2H-δ(t),
where M1 and M2 are constants to be fixed later. Thus, by using (2.6) and (3.31), inequality (3.31) then takes the form
(3.32)Θ′(t)≥((1-δ)-Mϵ)H-δ(t)H′(t)+ϵ(q+2)(‖ut‖22+‖vt‖22)+b1qϵ(‖∇u‖22+‖∇v‖22)+2(q+1)H(t)+ϵc2(‖u‖2(p+2)2(p+2)+2‖v‖2(p+2)2(p+2))-ϵM1-mHδm(t)‖u‖m+1m+1-ϵM2-rHδr(t)‖v‖r+1r+1,
where M=m/(m+1)M1+r/(r+1)M2 and c2 is a positive constant.
Since2(p+2)>max{m+1,r+1}, taking into account (2.6) and (3.21), then we have
(3.33)Hδm(t)‖u‖m+1m+1≤c3(‖u‖2(p+2)2δm(p+2)+(m+1)+‖v‖2(p+2)2δm(p+2)‖u‖m+1m+1),Hδr(t)‖v‖r+1r+1≤c4(‖v‖2(p+2)2δr(p+2)+(r+1)+‖u‖2(p+2)2δr(p+2)‖v‖r+1r+1),
for some positive constants c3 and c4. By using (3.24) and the algebraic inequality
(3.34)zν≤z+1≤(1+1a)(z+a),∀z≥0,0<ν≤1,a≥0,
we have
(3.35)‖u‖2(p+2)2δm(p+2)+(m+1)≤d(‖u‖2(p+2)2(p+2)+H(0))≤d(‖u‖2(p+2)2(p+2)+H(t)),∀t≥0,
where d=1+1/H(0). Similarly,
(3.36)‖v‖2(p+2)2δr(p+2)+(r+1)≤d(‖v‖2(p+2)2(p+2)+H(t)),∀t≥0.
Also, since
(3.37)(X+Y)s≤C(Xs+Ys),X,Y≥0,s>0,
by using (3.24) and (3.34), we conclude that
(3.38)‖v‖2(p+2)2δm(p+2)‖u‖m+1m+1≤C(‖v‖2(p+2)2(p+2)+‖u‖(m+1)2(p+2))≤C(‖v‖2(p+2)2(p+2)+‖u‖2(p+2)2(p+2)),‖u‖2(p+2)2δr(p+2)‖v‖r+1r+1≤C(‖u‖2(p+2)2(p+2)+‖v‖(r+1)2(p+2))≤C(‖u‖2(p+2)2(p+2)+‖v‖2(p+2)2(p+2)),
where C is a generic positive constant. Taking into account (3.33)–(3.38), estimate (3.32) takes the form
(3.39)Θ′(t)≥((1-δ)-Mϵ)H-δ(t)H′(t)+ϵ(q+2)(‖ut‖22+‖vt‖22)+ϵ(2(q+1)-C1M1-m-C1M2-r)H(t)+ϵ(c2-C2M1-m-C2M2-r)(‖u‖2(p+2)2(p+2)+‖v‖2(p+2)2(p+2)),
where C1=max{c3d+C,c4d+C}, C2=max{c3d,c4d}. At this point, and for large values of M1 and M2, we can find positive constants κ1 and κ2 such that (3.39) becomes
(3.40)Θ′(t)≥((1-δ)-Mϵ)H-δ(t)H′(t)+ϵ(q+2)(‖ut‖22+‖vt‖22)+ϵκ1H(t)+ϵκ2(‖u‖2(p+2)2(p+2)+‖v‖2(p+2)2(p+2)).
Once M1 and M2 are fixed, we pick ϵ small enough so that (1-δ)-Mϵ≥0 and
(3.41)Θ(0)=H1-δ(0)+ϵ∫Ω(u0u1+v0v1)dx>0.
Since H′(t)≥0, there exists Λ>0 such that (3.40) becomes
(3.42)Θ′(t)≥ϵΛ(H(t)+‖ut‖22+‖vt‖22+‖u‖2(p+2)2(p+2)+‖v‖2(p+2)2(p+2)).
Then, we have
(3.43)Θ(t)≥Θ(0),∀t≥0.
Next, we have by Hölder's and Young's inequalities(3.44)(∫Ωuutdx+∫Ωvvtdx)1/(1-δ)≤C(‖u‖2(p+2)τ/(1-δ)+‖ut‖2s/(1-δ)+‖v‖2(p+2)τ/(1-δ)+‖vt‖2s/(1-δ)),
for 1/τ+1/s=1. We take s=2(1-δ), to get τ/(1-δ)=2/(1-2δ). Here and in the sequel, C denotes a positive constant which may change from line to line. By using (3.24) and (3.34), we have
(3.45)‖u‖2(p+2)2/(1-2δ)≤d(‖u‖2(p+2)2(p+2)+H(t)),‖v‖2(p+2)2/(1-2δ)≤d(‖v‖2(p+2)2(p+2)+H(t)),∀t≥0.
Therefore, (3.44) becomes
(3.46)(∫Ωuutdx+∫Ωvvtdx)1/(1-δ)≤C(‖u‖2(p+2)2(p+2)+‖v‖2(p+2)2(p+2)+‖ut‖22+‖vt‖22).
Note that
(3.47)Θ1/(1-δ)(t)=(H1-δ(t)+ϵ∫Ω(uut+vvt)dx)1/(1-δ)≤C(H(t)+|∫Ωuutdx+∫Ωvvtdx|1/(1-δ))≤C(H(t)+‖u‖2(p+2)2(p+2)+‖v‖2(p+2)2(p+2)+‖ut‖22+‖vt‖22).
Combining (3.42) with (3.47), we have
(3.48)Θ(t)≥CΘ1/(1-δ)(t),∀t≥0.
A simple application of Lemma 3.1 gives the desired result.



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*Source: 760209-2011-08-11.xml*