# Growth of Solutions with Positive Initial Energy to Systems of Nonlinear Wave Equations with Damping and Source Terms

Authors: Erhan Pişkin
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

We consider initial-boundary conditions for coupled nonlinear wave
equations with damping and source terms. We prove that the solutions of the problem are
unbounded when the initial data are large enough in some sense.


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## Body

## 1. Introduction

In this work, we consider the following initial-boundary value problem:(1)utt-Δut+utp-1ut=divρ∇u2∇u+f1u,v,hhhhhhhhhhx,t∈Ω×0,T,vtt-Δvt+vtq-1vt=divρ∇v2∇v+f2u,v,hhhhhhhhhhx,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 Ω is a bounded domain with smooth boundary ∂Ω in Rn, n=1,2,3; p,q≥1; fi:R2→R are given functions to be specified later.Throughout this paper, we defineρ by
(2)ρs=b1+b2sm,s≥0,
where b1, b2 are nonnegative constants and b1+b2>0.This type of problems not only is important from the theoretical point of view, but also arises in material science and physics that deal with system of nonlinear wave equations.Ye [1] obtained the local existence and the blowup of the solution of problem (1), for ρs=sm. In the absence of the strong damping ΔutandΔvt terms, problem (1) becomes
(3)utt+utp-1ut=divρ∇u2∇u+f1u,v,vtt+vtq-1vt=divρ∇v2∇v+f2u,v.
Wu et al. [2] obtained the global existence and blowup of the solution of problem (3) under some suitable conditions. Fei and Hongjun [3] considered problem (3) and improved the blowup result obtained in [2], for a large class of initial data in positive initial energy, using the same techniques as in Payne and Sattinger [4] and some estimates used firstly by Vitillaro [5]. Recently, Pişkin and Polat [6, 7] studied the local and global existence, energy decay, and blowup of the solution of problem (3). Also, for more information about (1) and (3), see [2, 3, 7].The many problems associated with (1) are studied from various aspects in many papers [8–13].In this work, we will consider the blowup property in infinity time, that is, exponential growth.This work is organized as follows. In Section2, we state the local existence result. In Section 3, we establish that the energy will grow up as an exponential as time goes to infinity, provided that the initial data are large enough or E0<E1, where E0 and E1 are defined in (9) and (15).


## 2. Preliminaries

In this section, we introduce some notations and lemmas and local existence theorem needed in the proof of our main results. Let· and ·p denote the usual L2Ω norm and LpΩ norm, respectively.Concerning the functionsf1u,v and f2u,v, we take
(4)f1u,v=au+v2r+1u+v+buruvr+2,f2u,v=au+v2r+1u+v+bvrvur+2,
where a,b>0 are constants and r satisfies
(5)-1<rifn≤2,-1<r≤1ifn=3.
According to the above equalities they can easily verify that
(6)uf1u,v+vf2u,v=2r+2Fu,v,∀u,v∈R2,
where
(7)Fu,v=12r+2au+v2r+2+2buvr+2.We have the following result.Lemma 1 (see [14]).
There exist two positive constantsc0 and c1 such that
(8)c0u2r+2+v2r+2≤2r+2Fu,v≤c1u2r+2+v2r+2
is satisfied.We define the energy function as follows:(9)Et=12ut2+vt2+12∫Ω‍P∇u2+P∇v2dx-∫Ω‍Fu,vdx,
where Ps=∫0sρξdξ, s≥0.Lemma 2 (see [7]).
E
t is a nonincreasing function for t≥0 and
(10)E′t=-∇ut2+∇vt2+utp+1p+1+vtq+1q+1≤0.Lemma 3 (Sobolev-Poincare inequality [15]).
Letp be a number with 2≤p<∞n=1,2 or 2≤p≤2n/(n-2)(n≥3), then there is a constant C*=C*Ω,p such that
(11)up≤C*∇uforu∈H01Ω;Next, we state the local existence theorem [1, 7].Theorem 4 (local existence).
Suppose that (5) holds. Then there exist p, q satisfying
(12)1≤p,qifn≤2,1≤p,q≤5ifn=3,
and further u0,v0∈(W01,2m+1Ω∩L2r+2Ω)×(W01,2m+1Ω∩L2r+2Ω), (u1,v1)∈L2Ω×L2Ω. Then problem (1) has a unique local solution
(13)u,v∈C0,T;W01,2m+1Ω∩L2r+2Ω,ut∈C0,T;L2Ω∩Lp+1Ω×0,T,vt∈C0,T;L2Ω∩Lq+1Ω×0,T.


## 3. Exponential Growth

In this section, we will prove that the energy is unbounded when the initial data are large enough in some sense. Our techniques of proof follow very carefully the techniques used in [16].Lemma 5 (see [3]).
Suppose that (5) holds. Then there exists η>0 such that for any u,v∈(W01,2m+1Ω∩L2r+2Ω)×(W01,2m+1Ω∩L2r+2Ω) the inequality
(14)u+v2r+22r+2+2uvr+2r+2≤η∫Ω‍P∇u2+P∇v2dxr+2
holds.For the sake of simplicity and to prove our result, we takea=b=1 and introduce
(15)B=η1/2r+2,α1=B-r+2/r+1,E1=12-12r+2α12,
where η is the optimal constant in (14). Next, we will state a lemma which is similar to the one introduced firstly by Vitillaro in [5] to study a class of a single wave equation.Lemma 6 (see [3]).
Suppose that (5) holds. Let u,v be the solution of problem (1). Assume further that E0<E1 and
(16)∫Ω‍P∇u02+P∇v02dx1/2>α1.
Then there exists a constant α2>α1 such that
(17)∫Ω‍P∇u2+P∇v2dx1/2≥α2,(18)u+v2r+22r+2+2uvr+2r+21/2r+2≥Bα2,
for all t∈0,T.Theorem 7.
Suppose that (5) and 2r+2>max⁡p+1,q+1 hold. Then any solution of problem (1) with initial data satisfying
(19)∫Ω‍P∇u02+P∇v02dx1/2>α1,E0<E1,
grows exponentially.Proof.
We set(20)Ht=E1-Et,t≥0.
From (10) and (20) we get
(21)H′t=-E′t=utp+1p+1+vtq+1q+1≥0;
hence we have Ht≥H0=E1-E0>0.
We consider the following functional:(22)Lt=Ht+ɛ∫Ω‍uut+vvtdx+ɛ2∇u2+∇v2
for small ɛ to be specified later.
Our goal is to show thatLt satisfies a differential inequality of the form
(23)L′t≥CLt∀t≥0.
This, of course, will lead to exponential growth.
By taking a derivative of (22) and using (1), it follows that
(24)L′t=H′t+ɛut2+vt2+ɛ∫Ω‍uutt+vvttdx+ɛ∫Ω‍∇u∇ut+∇v∇vtdx=utp+1p+1+vtq+1q+1+ɛut2+vt2-ɛb1∇u2+∇v2-ɛb2∇u2m+12m+1+∇v2m+12m+1-ɛ∫Ω‍uf1u,v+vf2u,vdx-ɛ∫Ω‍uutp-1ut+vvtq-1vtdx.
From (9) and (20), it follows that
(25)-b2∇u2m+12m+1+∇v2m+12m+1=2m+1Ht-2m+1E1+m+1ut2+vt2+b1m+1∇u2+∇v2-2m+1∫Ω‍Fu,vdx.
Inserting (25) into (24), we get
(26)L′t=utp+1p+1+vtq+1q+1+ɛm+2ut2+vt2+ɛb1m∇u2+∇v2+ɛ1-m+1r+2u+v2r+22r+2+2uvr+2r+2+2ɛm+1Ht-2ɛm+1E1-ɛ∫Ω‍uutp-1ut+vvtq-1vtdx.
Then using (18), we obtain
(27)L′t≥utp+1p+1+vtq+1q+1+ɛm+2ut2+vt2+ɛb1m∇u2+∇v2+2ɛm+1Ht+ɛc′u+v2r+22r+2+2uvr+2r+2-ɛ∫Ω‍uutp-1ut+vvtq-1vtdx,
where c′=1-m+1/r+2-2m+1E1Bα2-2r+2. It is clear that c′>0, since α2>B-r+2/r+1. In order to estimate the last two terms in (27), we use the following Young inequality:
(28)XY≤δkXkk+δ-lYll,
where X,Y≥0, δ>0, k,l∈R+ such that 1/k+1/l=1. Consequently, applying the above inequality we have
(29)∫Ω‍uututp-1dx≤δ1p+1p+1up+1p+1+pδ1-p+1/pp+1utp+1p+1,∫Ω‍vvtvtq-1dx≤δ2q+1q+1vq+1q+1+qδ2-q+1/qq+1vtq+1q+1.
Inserting estimates (29) into (27), we have
(30)L′t≥ɛm+2ut2+vt2+ɛb1m∇u2+∇v2+2ɛm+1Ht+ɛc2u2r+22r+2+v2r+22r+2-ɛδ1p+1p+1up+1p+1+1-ɛpδ1-p+1/pp+1utp+1p+1-ɛδ2q+1q+1vq+1q+1+1-ɛqδ2-q+1/qq+1vtq+1q+1,
where c2=c′c0.
Since2r+2>max⁡p+1,q+1, from the embedding L2r+2Ω↪Lp+1Ω and embedding L2r+2Ω↪Lp+1Ω, we have
(31)up+1p+1≤c3u2r+2p+1,vq+1q+1≤c4v2r+2q+1
for some positive constants c3 and c4. Using the algebraic inequality
(32)zυ≤z+1≤1+1az+a,∀z≥0,0<υ≤1,a≥0,
and since Ht≥H0, we get
(33)u2r+2p+1≤du2r+22r+2+H0≤du2r+22r+2+Ht,
where d=1+1/H0. Similarly
(34)v2r+2q+1≤dv2r+22r+2+Ht.
Inserting (33) and (34) into (30), we have
(35)L′t≥ɛm+2ut2+vt2+ɛb1m∇u2+∇v2+ɛ2m+1-dδ1p+1p+1-dδ2q+1q+1Ht+ɛc2-dδ1p+1p+1-dδ2q+1q+1u2r+22r+2+v2r+22r+2+1-ɛpδ1-p+1/pp+1utp+1p+1+1-ɛqδ2-q+1/qq+1vtq+1q+1.
Now, once δ1 and δ2 are fixed, we can choose ɛ small enough such that
(36)1-ɛpδ1-p+1/pp+1>0,1-ɛqδ2-q+1/qq+1>0,L0=H0+ɛ∫Ω‍u0u1+v0v1dx>0.
Consequently (35) takes the form
(37)L′t≥ɛm+2ut2+vt2+ɛb1m∇u2+∇v2+ɛ2m+1-dδ1p+1p+1-dδ2q+1q+1Ht+ɛc2-dδ1p+1p+1-dδ2q+1q+1u2r+22r+2+v2r+22r+2≥θut2+vt2+Ht+∇u2v2r+22r+2ssssss+∇v2+u2r+22r+2+v2r+22r+2,
where θ=min⁡⁡{ɛm+2,ɛb1m,(2m+1-d(δ1p+1/p+1)-d(δ2q+1/q+1)),ɛ(c2-d(δ1p+1/p+1) -d(δ2q+1/q+1))}.
Then we have(38)Lt≥L0>0,∀t≥0.
On the other hand, applying Hölder inequality, we obtain
(39)∫Ω‍uutdx+∫Ω‍vvtdx≤uut+vvt≤Cu2r+2ut+v2r+2vt.
Young inequality gives
(40)∫Ω‍uutdx+∫Ω‍vvtdx≤C2u2r+22+ut2+v2r+22+vt2≤C2u2r+22r+21/r+2+ut2aaaasa+v2r+22r+21/r+2+vt2.
Since r>-1, algebraic inequality (32) yields
(41)∫Ω‍uutdx+∫Ω‍vvtdx≤C21+1H0u2r+22r+2+Ht+ut2+1+1H0v2r+22r+2+Ht+vt2.
Note that
(42)Lt≤Ht+ɛ∫Ω‍uut+vvtdx≤Ht+∫Ω‍uutdx+∫Ω‍vvtdx≤C′ut2+vt2+Ht+∇u2+∇v2v2r+22r+2sssssss+u2r+22r+2+v2r+22r+2.
Combining with (37) and (42), we arrive at
(43)L′t≥CLt∀t≥0.
Integrating differential inequality (43) between 0 and t gives the following estimate for Lt:
(44)Lt≥L0eCt.
The proof of Theorem 7 is completed.Remark 8.
WhenE0<0, by setting Ht=-Et, the similar result is obtained by applying the same arguments in the proof of Theorem 7.



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*Source: 523254-2015-02-22.xml*