R E S E A R C H
Open Access
Blow-up of solution for an
integro-differential equation with arbitrary
positive initial energy
Liu Jie and Liang Fei
**Correspondence: fl[email protected]
Department of Mathematics, Xi’an University of Science and Technology, Xi’an, 710054, P.R. China
Abstract
In this paper, we consider the integro-differential equationutt–M(∇u22)u+
t
0g(t–
τ
)u(τ)dτ
+ut=f(u), (x,t)∈×(0,T), with initial and Dirichlet boundary conditions. Under suitable assumptions on the functionsgand the initial data, a blow-up result with arbitrary positive initial energy is established.
MSC: 35L05; 35L55; 35L70
Keywords: blow-up; arbitrary positive initial energy; integro-differential equation
1 Introduction
In this paper we study the following nonlinear integro-differential equations:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
utt–M(∇u)u+ t
g(t–τ)u(τ)dτ+ut=f(u), (x,t)∈×(,T),
u(x,t) = , x∈∂×(,T),
u(x, ) =u(x), ut(x, ) =u(x), x∈,
(.)
whereis a bounded domain inRnwith a smooth boundary∂,Mis a positiveC -func-tion likeM(s) =m+bsγ,m> ,b≥,γ ≥, ands≥,grepresents the kernel of the memory term andf is a nonlinear function likef(u) =|u|p–u,p> , they will be specified later.
Before going further, (.) without the viscoelastic term, that is,g≡, for the case that
M≡, (.) becomes a nonlinear wave equation which has been extensively studied and several results concerning existence and nonexistence have been established [–]. When
Mis not a constant function, a special case of (.) is Kirchhoff equation which has been introduced in order to describe the nonlinear vibrations of an elastic string. Kirchhoff [] was the first one to study the oscillations of stretched strings and plates. In this case the existence and nonexistence of solutions have been discussed by many authors; see [–] and the references cited therein.
For (.) withg= , in the case thatM≡, (.) becomes a semilinear viscoelastic equa-tion which has been extensively studied and many results concerning global existence and blow-up in finite time have been proved. See in this regard [–]. For instance, Messaoudi [] studied (.) with damping terma|ut|m–utandf(u) =b|u|p–uand proved
a blow-up result for solutions with negative initial energy ifp>m≥ and a global result for ≤p≤m. This result has later been improved by the same author in [] to accommo-date certain solutions with positive initial energy. In [], Song and Zhong considered (.) with strong damping –utandf(u) =|u|p–uand proved a blow-up result for solutions with positive initial energy by using the ideas of the ‘potential well’ theory introduced by Payne and Sattinger [].
Forg= andMis not a constant function, (.) is a model to describe the motion of deformable solids as hereditary effect is incorporated. It may also be used to describe the dynamics of an extensible string with fading memory. This equation states that the dy-namic equilibrium of a body depends not only on the present state of deformation, but also on the previous history of the deformation []. Also, (.) is applied to the theory of the heat conduction with memory; see [, ]. Therefore, the dynamics of (.) is of great importance and interest as they have wide applications in natural sciences.
This type of problem have been considered by many authors and several results concern-ing existence, nonexistence, and asymptotic behavior have been established. Equation (.) was first studied by Torrejón and Young [], who proved the existence of weakly asymp-totic stable solution for a large analytical datum. Later, Munoz Rivera [] showed the existence of global solutions for small datum and the total energy decays to zero expo-nentially under some restrictions. In [], Wu and Tsai studied (.) for a strong damping –utand proved the global existence, decay result, and blow-up properties. Recently, they [] discussed the local existence and blow-up of solutions with positive initial energy for nonlinear damping under some conditions.
In this paper, we consider problem (.) and will establish a blow-up result for (.) with arbitrary positive initial energy under suitable assumptions on the functions g and the initial data. This result extends earlier ones [, ], in which only some a positive initial energy is considered. The main tool in proving blow-up result is the ‘concavity method’ where the basic idea of the method is to construct a positive defined functionalF(t) of the solution by the energy inequality and show thatF–α(t) is a concave function oft.
The rest of this paper is organized as follows. In Section , we give some preliminaries and state the main result. In Section , we prove the blow-up result by a concavity method. Section is devoted to a simple discussion of the main result.
2 Preliminaries
First, let us introduce some notation used throughout this paper. We denote by · qthe
Lq() norm for ≤q≤ ∞and by∇ ·
the Dirichlet norm inH(), which is equivalent to theH() norm. Moreover, we set
(ϕ,ψ) =
ϕ(x)ψ(x)dx
as the usualL() inner product.
We next state some assumptions onf,M, andg:
(A) f() = and there are two positive constantscandδsuch that
and
sf(s)≥( + δ)F(s)
fors,s∈R, <p≤(n–)n– ifn> and <p<∞ifn≤, whereF(s) =sf(τ)dτ. (A) Mis a positiveC-function likeM(s) =m
+bsγ,m> ,b≥,γ ≥, ands≥, and it satisfies
(δ+ )M(s) – M(s) + δm
s≥, ∀s≥,
whereM(s) =sM(τ)dτ andδis the constant appeared in (A). (A) Assumeg(t) :R+→R+belongs toC(R+)and satisfy
g(t)≥, g(t)≤ fort≥ and
l= ∞
g(s)ds<δm + δ.
(A) The functionetg(t)is of positive type in the following sense:
t
υ(s)
s
es–τg(s–τ)dτds≥,
∀υ∈C([,∞))and∀t> .
Remark . It is clear thatf(u) =|u|p–u,p≥γ + , andm(s) =m
+bsγ, wherem> ,
b≥,γ ≥ satisfy the assumptions (A) and (A) withα/≤δ≤(p– )/. It is also obvious thatg(t) =e–t with <<m
satisfies the assumptions (A) and (A).
Now we are ready to state the local existence of problem (.), whose proof can be found in [].
Theorem . Assume that(A)-(A)hold,and that u∈H()∩H(),u∈L(),then
there exists a unique solution u of(.)satisfying
u∈C [,T);H()∩H() and ut∈C [,T);L()
∩L [,T);H().
Moreover,at least one of the following statements holds true: (i) T=∞,
(ii) ut(t)+u(t)→ ∞ast→T–.
We introduce the energy functionalE(t) associated to our equation,
E(t) = ut(t)
+
M ∇u(t)
–
t
g(τ)dτ∇u(t)
+
(g◦ ∇u)(t) –
where
(g◦ ∇w)(t) = t
g(t–τ)∇w(t) –∇w(τ)dτ.
As in [], we see that
d
dtE(t) = –
ut(t)
dx+ g
◦ ∇u(t) –
g(t)∇u(t) ≤, which implies
E(t)≤E() – t
ut(τ)
dx dτ. (.)
Let
I(u) =M ∇u(t)∇u(t)–
f(u)u dx
foru∈H(). We finally state our main blow-up result for problem (.).
Theorem . Assume(A)-(A)hold.If u∈H()∩H(),u∈L(),satisfy the
fol-lowing conditions:
E() > ,
u(x)u(x)dx> (.)
and
I(u) < , u>
(δ+ )E() (δm– (δ+ )l)min{λ, }
, (.)
whereδis the constant appeared in(A)andλis the constant of the Poincaré inequality on,then the corresponding solution u(t)of problem(.)blows up in a finite time T∗> .
3 Proof of Theorem 2.2
In this section, we deal with the blow-up solutions of equation (.). Before we prove our blow-up result, we need the following lemmas.
Lemma .(see [], Lemma .) Assume that g(t)satisfies the assumptions (A)and
(A),and(t)is a function that is twice continuously differentiable,satisfying
⎧ ⎨ ⎩
(t) +(t) >tg(t–τ)∇u(τ,x)∇u(t,x)dx dτ, () > , () >
for every t∈[,T),where u(t)is the corresponding solution of(.)with uand u.Then
the function(t)is strictly increasing on[,T).
Lemma . Assume u∈H()∩H(),u∈L(),satisfy
If the local solution u(t)of(.)satisfies
I(u) < ,
thenu(t)is strictly increasing on[,T).
Proof Sinceu(t) is the local solution of (.), by a simple computation we have
d
dt
u(t,x)
dx=
ut(t)
+uutt
dx
=
ut(t,x)dx–
uutdx–M ∇u(t)∇u(t)
+
f(u)u dx+ t
g(t–τ)
∇u(t)· ∇u(τ)dx dτ
> –
uutdx+ t
g(t–τ)
∇u(t)· ∇u(τ)dx dτ,
where the last inequality usesI(u) < , which implies
d
dt
u(t,x)dx+ d
dt
u(t,x)dx> t
g(t–τ)
∇u(t)· ∇u(τ)dx dτ.
Therefore, this lemma comes from Lemma ..
Proof of Theorem. We next prove Theorem . in two steps. First, by a contradiction argument we claim that
I u(t)< (.)
and
u(t)> (δ+ )E() (δm– (δ+ )l)min{λ, }
(.)
for everyt∈[,T). If this was not the case, then there would exist a timetsuch that
t=min
t∈(,T) :I u(t)= > . (.)
By the continuity of the solutionu(t) as a function oft, we see thatI(u(t)) < whent∈
(,t) andI(u(t)) = . Thus by Lemma . we have
u(t)>u>
(δ+ )E() (δm– (δ+ )l)min{λ, } for everyt∈[,t). In addition, it is obvious thatu(t)
is continuous on [,t]. Thus the following inequality is obtained:
u(t)> (δ+ )E() (δm– (δ+ )l)min{λ, }
On the other hand, it follows from the definition ofE(t) and (.) that
M ∇u(t)
–
t
g(τ)dτ∇u(t)
+
(g◦ ∇u)(t) –
F u(t)
dx≤E(). (.)
Using the assumptions (A) and (A), we have
M ∇u(t)
∇u(t)
+ δm∇u(t)
– (δ+ )l∇u(t)
–
f u(t)u(t)dx≤(δ+ )E().
Noting the fact thatI(u(t)) = , we then have
δm– (δ+ )l∇u(t)≤(δ+ )E().
Thus, by the Poincaré inequality, we have
u(t) ≤
(δ+ )E() (δm– (δ+ )l)min{λ, }
. (.)
Obviously, there is a contradiction between (.) and (.). Thus, we have proved that (.) is true for everyt∈[,T). Furthermore, by Lemma . we see that (.) is also valid on
t∈[,T).
Secondly, we prove that the solution of problem (.) blows up in a finite time. Assume by contradiction that the solutionuis global. Then, for sufficiently largeT> , we consider
H(t) : [,T]→R+defined by
H(t) =u(t)+ t
u(τ)dτ+ (T–t)u+α(t+t),
wheretandα are positive constants, which will be determined in the sequel. A direct computation yields
H(t) =
u(t)ut(t)dx+u(t)–u+ α(t+t)
=
u(t)ut(t)dx+ t
u(τ),ut(τ)dτ+ α(t+t)
and
H(t) = utt,u(t)
+ ut(t)
+ u(t),ut(t)
+ α
= ut(t)
– M ∇u
∇u +
t
g(t–τ)
∇u(τ)∇u(t)dx dτ
+
Therefore, we have
H(t)H(t) – (δ+ )H(t)
= H(t)ut(t)–M ∇u∇u+ t
g(t–τ)
∇u(τ)∇u(t)dx dτ
+
f(u)u dx+α
– (δ+ )
u(t)ut(t)dx+ t
u(τ),ut(τ)
dτ+α(t+t)
= H(t)ut(t)–M ∇u∇u+ t
g(t–τ)
∇u(τ)∇u(t)dx dτ
+
f(u)u dx+α
+ (δ+ ) G(t) – H(t) – (T–t)u
(t), (.)
where(t),G(t) : [,T]→R+are the functions defined by
(t) =ut(t) +
t
ut(τ)dτ+α
and
G(t) =(t)u(t)+ t
u(τ)dτ+α(t+t)
–
u(t)ut(t)dx+ t
(u,ut)dτ+α(t+t)
.
Using the Schwarz inequality, we have
uutdx
≤u(t)ut(t), t
(u,ut)dτ
≤
t
u(τ)dτ t
ut(τ) dτ
and
u(t)ut(t)dx
t
u(τ),ut(τ)
dτ
≤u(t) t
ut(τ) dτ
ut(t) t
u(τ)dτ
≤
u(t)
t
ut(τ)dτ+ ut(t)
t
u(τ)dτ
and
α(t+t)
u(t)ut(t)dx
≤√α√α(t+s)u(t)ut(t)
≤
αu(t) +
α(t+t) u
Similarly, we have
α(t+t) t
u(τ),ut(τ)dτ≤ α
t
u(τ)dτ+
α(t+t)
t
ut(τ)dτ.
The previous inequalities entailG(t)≥ for every [,T]. Using (.), we get
H(t)H(t) – (δ+ )H(t)≥H(t)L(t), (.) where
L(t) = –(δ+ )ut(t)– M ∇u
∇u
+ t
g(t–τ)
∇u(τ)∇u(t)dx dτ
+
f(u)u dx– (δ+ ) t
ut(τ)dτ– (δ+ )α
= –(δ+ )ut(t) –
M ∇u– t
g(τ)dτ
∇u
+
f(u)u dx– (δ+ )α– (δ+ ) t
ut(τ)dτ
+ t
g(t–τ)
∇u(t) ∇u(τ) –∇u(t)dx dτ. (.)
Using Young’s inequality, we have
t
g(t–τ)
∇u(t)∇ u(τ) –u(t)dx dτ
≥–(δ+ )(g◦ ∇u)(t) – (δ+ )
t
g(τ)dτ∇u(t). (.)
Inserting (.) into (.), we have
L(t)≥–(δ+ )ut(t)
– (δ+ )(g◦ ∇u)(t) –
M ∇u– t
g(τ)dτ
∇u
+
f(u)u dx– (δ+ ) t
ut(τ) dτ
– δ+
t
g(τ)dτ∇u– (δ+ )α
≥–(δ+ )E(t) + (δ+ )M ∇u–M∇u∇u
– (δ+ ) t
ut(τ)dτ–
(δ+ )F(u) –f(u)udx
–
δ+ δ+
t
g(τ)dτ∇u– (δ+ )α
+ δ t
ut(τ)dτ–
(δ+ )F(u) –f(u)udx
–
δ+ δ+
t
g(τ)dτ∇u– (δ+ )α.
Using the assumptions (A) and (A), we have
L(t)≥–(δ+ )E() +
δm–
δ+ δ+
t
g(τ)dτ
∇u– (δ+ )α
≥–(δ+ )E() + δm– (δ+ )l
λu(t)– (δ+ )α
≥–(δ+ )E() + δm– (δ+ )l
λu– (δ+ )α,
where the last inequality follows from Lemma . and the Poincaré inequality. From (.), we have
(δ+ )E() < δm– (δ+ )l
λu< δm– (δ+ )l
λu. Thus, we can letαsatisfy
(δ+ )α< δm– (δ+ )l
λu– (δ+ )E(),
which implies that there existsθ> (independent ofT) such that
L(t)≥θ fort∈[,T]. (.)
By (.) and (.), it follows that
H(t)H(t) – (δ+ )H(t)> .
Moreover, we lettsatisfy
αt+
uudx> ,
which meansH() > . Thus byH(t) > we see thatH(t) andH(t) are strictly increasing on [,T].
Settingy(t) =H(t)–δ, then we have
y(t) = –δH(t)–(δ+)H(t) <
and
y(t) = –δH–δ– H(t)H(t) – (δ+ )H(t)<
for allt∈[,T], which implies thaty(t) reaches in finite time, say ast→T∗. SinceT∗is independent of the initial choice ofT, we may assume thatT∗<T. This tells us that
lim
4 Conclusions
In this paper, we consider the integro-differential equation
utt–M ∇u
u+ t
g(t–τ)u(τ)dτ+ut=f(u), (x,t)∈×(,T),
with initial and Dirichlet boundary conditions which arises in the dynamics of an exten-sible string with fading memory. Under suitable assumptions on the relax functiongand the initial data, we establish a blow-up result with arbitrary positive initial energy. The main tool in proving the blow-up result is the ‘concavity method’ where the basic idea of the method is to construct a positive defined functionalF(t) of the solution by the energy inequality and show thatF–α(t) is a concave function oft.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The article is a joint work of the two authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
Acknowledgements
The authors are indebted to the referee for giving some important suggestions which improved the presentation of this paper. This work is supported in part by the scientific research program funded by Shanxi Provincial Education Department No. 14JK1474, the Shanxi Province Postdoctoral Science Foundation, the China Postdoctoral Science Foundation Grant No. 2013M540767, the China NSF Grant No. 11402194 and the doctor scientific research start fund project of Xi’an University of Science and Technology Grant No. 2014QDJ042.
Received: 29 January 2015 Accepted: 28 May 2015 References
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