R E S E A R C H
Open Access
Study of solutions to an initial and boundary
value problem for certain systems with
variable exponents
Yunzhu Gao
1*and Wenjie Gao
2**Correspondence: [email protected]; [email protected]
1Department of Mathematics and Statistics, Beihua University, Jilin City, P.R. China
2Institute of Mathematics, Jilin University, Changchun, 130012, P.R. China
Abstract
In this paper, the existence and blow-up property of solutions to an initial and boundary value problem for a nonlinear parabolic system with variable exponents is studied. Meanwhile, the blow-up property of solutions for a nonlinear hyperbolic system is also obtained.
Keywords: existence; blow-up; parabolic system; hyperbolic system; variable exponent
1 Introduction
In this paper, we first consider the initial and boundary value problem to the following nonlinear parabolic system with variable exponents:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ut=u+f(u,v), (x,t)∈QT,
vt=v+f(u,v), (x,t)∈QT,
u(x,t) = , v(x,t) = , (x,t)∈ST,
u(x, ) =u(x), v(x, ) =v(x), x∈,
(.)
where⊂RN is a bounded domain with smooth boundary∂and <T <∞,Q
T=
×[,T),STdenotes the lateral boundary of the cylinderQT, and the source termsf,f
are in the form
f(u,v) =a(x)vp(x) and f(u,v) =a(x)up(x),
or
f(u,v) =a(x)
vp(y)(y,t)dy and f
(u,v) =a(x)
up(y)(y,t)dy,
respectively, wherep,p,a,aare functions satisfying conditions (.) below.
In the case when p, p are constants, system (.) provides a simple example of a
reaction-diffusion system. It can be used as a model to describe heat propagation in a two-component combustible mixture. There have been many results about the existence,
boundedness and blow-up properties of the solutions; we refer the readers to the bibliog-raphy given in [–].
The motivation of this work is due to [], where the following system of equations is studied.
⎧ ⎨ ⎩
ut–u=vp,
vt–v=uq,
(.)
wherex∈RN (N≥),t> , andp,qare positive numbers. The authors investigated the
boundedness and blow-up of solutions to problem (.). Furthermore, the authors also studied the uniqueness and global existence of solutions (see []).
Besides, this work is also motivated by [] in which the following problem is considered:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ut=u+f(x,u), (x,t)∈×[,T),
u(x, ) =u(x), x∈,
u(x,t) = , (x,t)∈∂×[,T),
(.)
where∈Rnis a bounded domain with smooth boundary∂, and the source term is of
the formf(x,u) =a(x)up(x) orf(x,u) =a(x)uq(y)(y,t)dy. The author studied the blow-up property of solutions for parabolic and hyperbolic problems. Parabolic problems with sources like the ones in (.) appear in several branches of applied mathematics, which
can be used to model chemical reactions, heat transfer or population dynamicsetc.We
also refer the interested reader to [–] and the references therein. We also study the following nonlinear hyperbolic system of equations:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
utt=u+f(u,v), (x,t)∈QT,
vtt=v+f(u,v), (x,t)∈QT,
u(x,t) = , v(x,t) = , (x,t)∈ST,
u(x, ) =u(x), v(x, ) =v(x), x∈,
ut(x, ) =u(x), vt(x, ) =v(x), x∈.
(.)
The aim of this paper is to extend the results in [, ] to the case of parabolic system (.) and hyperbolic system (.). As far as we know, this seems to be the first paper, where the blow-up phenomenon is studied with variable exponents for the initial and boundary value problem to some parabolic and hyperbolic systems. The main method of the proof is similar to that in [, ].
We conclude this introduction by describing the outline of this paper. Some preliminary results, including existence of solutions to problem (.), are gathered in Section . The blow-up property of solutions are stated and proved in Section . Finally, in Section , we prove the blow-up property of solutions for hyperbolic problem (.).
2 Existence of solutions
Throughout the paper, we assume that the exponentsp(x),p(x) :→(, +∞) and the
continuous functionsa(x),a(x) :→Rsatisfy the following conditions:
<p– =inf
x∈p(x)≤p(x)≤p + =sup
x∈
p(x) < +∞,
<p–=inf
x∈p(x)≤p(x)≤p + =sup
x∈
p(x) < +∞, (.)
<c≤a(x)≤C< +∞, <c≤a(x)≤C< +∞.
Definition . We say that the solution (u(x,t),v(x,t)) for problem (.) blows up in finite time if there exists an instantT∗<∞such that
(u,v) → ∞ ast→T∗,
where
(u,v) = sup
t∈[,T)
u(·,t)∞+v(·,t)∞.
Our first result here is the following.
Theorem . Let⊂RN be a bounded smooth domain,p
(x),p(x),a(x),a(x)satisfy the conditions in(.),and assume that u(x)and v(x)are nonnegative,continuous and bounded.Then there exists a T, <T≤ ∞,such that problem(.)has a nonnegative and bounded solution(u,v)in QT.
Proof We only prove the case whenf(u,v) =a(x)vp(x)andf(u,v) =a(x)up(x), and the
proofs to the casesf(u,v) =a(x)
vp(y)(y,t)dy andf(u,v) =a(x)
up(y)(y,t)dy are
similar.
Let us consider the equivalent systems of (.)
⎧ ⎨ ⎩
u(x,t) =g(x,y,t)u(y)dy+ t
g(x,y,t–s)a(y)vp(y)dy ds, v(x,t) =g(x,y,t)v(y)dy+
t
g(x,y,t–s)a(y)u
p(y)dy ds,
whereg(x,y,t) is the corresponding Green function. Then the existence and uniqueness of solutions for a given (u(x),v(x)) could be obtained by a fixed point argument.
We introduce the following iteration scheme:
u(x,t) = , v(x,t) = ,
un+(x,t) =
g(x,y,t)u(y)dy+ t
g(x,y,t–s)a(y)vpn(y)dy ds,
vn+(x,t) =
g(x,y,t)v(y)dy+ t
g(x,y,t–s)a(y)upn(y)dy ds,
and the convergence of the sequence{(un,vn)}follows by showing that
⎧ ⎨ ⎩
(v) = t
g(x,y,t–s)a(y)v
p(y)
n dy ds,
(u) = t
g(x,y,t–s)a(y)u
p(y)
n dy ds
Now, we define
(u,v) =(v),(u)
,
where
(v) = t
g(x,y,t–s)vp(y)
n dy ds,
(u) = t
g(x,y,t–s)up(y)
n dy ds.
We denote
(u,v) –(w,z) =(v) –(z),(u) –(w)
,
and for arbitraryT> , define the set
ET=
C,(T)∩C(T) (u,v) ≤M
,
whereT=×[,T],M>|(u(x),v(x))|is a fixed positive constant.
We claim thatis a contraction onET. In fact, for anyx∈fixed, we have
ξp(x)
–η
p(x)
=p(x)wp(x)–(ξ–η), wherew=s(x)ξ+
–s(x)
η,s(x)∈(, ),
ξp(x)
–η
p(x)
=p(x)wp(x)–(ξ–η),
wherew=s(x)ξ+
–s(x)
η,s(x)∈(, ),
and we always have
pi(x)wpii(x)–(ξi–ηi) ≤p+i(M)p
+
i–ξi–ηi∞, i= , . (.)
Now, we define
μ(t) = sup
x∈,≤τ<t
τ
g(x,y,t–s)dy ds.
It is obvious thatμ(t)→ whent→+. Then, by using inequality (.), we get
(v) –(z)∞+(u) –(w)∞
≤ t
a(y)g(x,y,t–s)
vp(y)
n –zpn(y)
dy ds
∞
+
t
a(y)g(x,y,t–s)
up(y)
n –wpn(y)
dy ds
∞
≤μ(t)(M)max{p+,p+}–(C
+C)
p∞+p∞
v–z∞+u–w∞
≤μ(t)(M)max{p+,p+}–(C
+C)
p∞+p∞
sup
t∈[,T)
v–z∞+u–w∞
Hence, for sufficiently smallt, we have
(u,v) –(w,z)
=(v) –(z),(u) –(w)
≤ sup
t∈[,T)
(v) –(z)∞+(u) –(w)∞
≤μ(t)(M)max{p+p+}–(C+C)p∞+p∞ (v–z,u–w)
<θ (v–z,u–w) ,
whereθ< is a constant. Thenis a strict contraction.
3 Blow-up of solutions
In this section, we study the blow-up property of the solutions to problem (.). We need the following lemma.
Lemma . Let y(t)be a solution of
y(t)≥–λy(t) +ayr(t) –C,
whereλ> ,a> ,r> and C> are given constants.Then,there exists a constant K>
such that if y()≥K,then y(t)cannot be globally defined;in fact,
y(t)≥
y()–r–a(r– )
t
–/(r–)
. (.)
Proof It is sufficient to takeK> such that
–λη+aηr–C≥a
η
r, ∀η≥K.
Hence, we have
y(t)≥a y(t)
r. (.)
By a direct integration to (.), then we get immediately (.), which gives an upper bound
for the blow-up timet∗=ya–(r–)r().
The next theorem gives the main result of this section.
Theorem . Let⊂RN be a bounded smooth domain,and let(u,v)be a positive
solu-tion of problem(.),with p(x),p(x),a(x),a(x)satisfying conditions in(.).Then any solutions of problem(.)will blow up at finite time T∗ if the initial datum(u(x),v(x)) satisfies
u(x) +v(x)
whereϕ> is the first eigenfunction of the homogeneous Dirichlet Laplacian onand C> is a constant depending only on the domainand the bounds C,Cgiven in condi-tion(.).
Proof Letλbe the first eigenvalue of
–ϕ=λϕ, x∈,
with the homogeneous Dirichlet boundary condition, and letϕbe a positive function sat-isfying
ϕdx= .
We introduce the functionη(t) =(u+v)ϕ. First of all, we consider the casef(u,v) = a(x)vp(x),f(u,v) =a(x)up(x). Then
η(t) =
(ut+vt)ϕ
=
(u+v)ϕ+
a(x)vp(x)+a(x)up(x)
ϕ
= –λη+
a(x)vp(x)+a(x)up(x)
ϕ.
We now deal with the term(a(x)vp(x)+a(x)up(x))ϕ. For eacht> , we divideinto
the following four sets:
=
x∈:v(x,t) < ,u(x,t) < , =
x∈:v(x,t) < ,u(x,t)≥,
=
x∈:v(x,t)≥,u(x,t) < , =
x∈:v(x,t)≥,u(x,t)≥.
Then we have
a(x)vp(x)+a(x)up(x) ϕ =
a(x)vp(x)+a(x)up(x)
ϕ+
a(x)vp(x)+a(x)up(x)
ϕ
+
a(x)
vp(x)+a
(x)up(x)
ϕ+
a(x)vp(x)+a(x)up(x)
ϕ
≥
a(x)upϕ+
a(x)vpϕ+
a(x)vp+a(x)up
ϕ
≥
a(x)upϕ+
a(x)vpϕ
+
a(x)vp+a(x)up
ϕ–
a(x)vp+a(x)up ϕ –
a(x)vp+a(x)up
ϕ–
a(x)vp+a(x)up
ϕ
≥γ
vp+upϕ–
vpϕ–
upϕ–
vp+upϕ ≥γ
vp+upϕ–
ϕ,
wherep=min{p–,p–}andγ =min{c,c},=max{C,C}.
From the convex property of the functionf(w) =wr,r> and Jensen’s inequality, we
obtain
a(x)vp(x)+a(x)up(x)
ϕ≥γ
vp+upϕ–
ϕ ≥γ v+u
p
ϕ–
ϕ
≥ γ p–η
p(t) – .
Then we get
η(t)≥–λη(t) +
γ
p–η
p(t) – .
Note that
η(t) =
(u+v)ϕ≤u(·,t)L∞()+v(·,t)L∞()
ϕ≤ (u,v) .
Hence, forη() big enough, the result follows from Lemma ..
Next, we state briefly the proof to the theorem in the casef(u,v) =a(x)
v
p(y)(y,t)dy
andf(u,v) =a(x)
u
p(y)(y,t)dy. We repeat the previous argument under definingη(t) =
(u+v)ϕ, and we obtain in much the same way
η(t)≥–λη(t) +
a(x)
vp(y)(y,t)dy+a
(x)
up(y)(y,t)dy
ϕ(x)dx.
In view of the property ofϕ, we get
a(x)
vp(y)(y,t)dy+a
(x)
up(y)(y,t)dy
ϕ(x)dx
=
vp(y)(y,t)dy
a(x)ϕ(x)dx
+
up(y)(y,t)dy
a(x)ϕ(x)dx
≥c
vp(y)(y,t) ϕ(y)
ϕ∞dy+c
up(y)(y,t) ϕ(y)
ϕ∞dy
≥γ
vp(y)(y,t) +up(y)(y,t) ϕ(y)
ϕ∞ dy.
According to the convex property of the functionf(w) =wr,r> , and by using Jensen’s inequality, by considering again,,,as before, we obtain
γ
vp(y)(y,t) +up(y)(y,t) ϕ(y)
ϕ∞dy≥γη
whereγdepends only onγ,pandϕ∞,||denotes the measure of. Hence,
η(t)≥–λη(t) +γηp(t) – ||.
By Lemma ., the proof is complete.
4 Blow-up of solutions for a hyperbolic system
Lemma .[] Let y(t)∈Csatisfying
y(t)≥hy(t),
y() =a> ,y() =b> ,and h(s)≥for all s≥a.Then y(t) > whenever y exists;and
t≤ y(t)
a
b+
s a
h(x)dx –/
ds. (.)
Now, let us study the following problem:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
utt=u+f(u,v), (x,t)∈QT,
vtt=v+f(u,v), (x,t)∈QT,
u(x,t) = , v(x,t) = , (x,t)∈ST,
u(x, ) =u(x), v(x, ) =v(x), x∈,
ut(x, ) =u(x), vt(x, ) =v(x), x∈,
(.)
whereu(x),v(x),u(x),v(x)≥ and they are not identically zero, andf(u,v),f(u,v) as
above respectively.
Theorem . Let(u,v)∈C×Cbe a solution of problem(.),and let the conditions in
(.)hold.Then there exist sufficiently large initial data u,v,u,vsuch that any solutions of problem(.)blew up at finite time T∗.
Proof Let (λ,ϕ) be the first eigenvalue and eigenfunction of Laplacian inwith
homo-geneous Dirichlet boundary conditions as before. We assume that f(u,v) =a(x)vp(x), f(u,v) =a(x)up(x), the other is similar. We also define the functionη(t) =
(u+v)ϕ,
so we have
η(t) =
(utt+vtt)ϕ
=
(u+v)ϕ+
a(x)vp(x)+a(x)up(x)
ϕ
= –λη+
a(x)vp(x)+a(x)up(x)
ϕ.
The term(a(x)vp(x)+a(x)up(x))ϕis dealt with as before, then we get
a(x)vp(x)+a(x)up(x)
ϕ≥γ
vp+upϕ–
By virtue of the convex property of the functionf(w) =wr,r> , and Jensen’s inequality,
we still obtain
γ
vp+upϕ–
ϕ≥γ
v+u
p
ϕ–
ϕ≥ γ
p–η
p(t) – .
Then we have
η(t)≥–λη(t) +
γ
p–η
p(t) – .
Now, we can apply Lemma (.) fora=η(),b=η() large enough such that –λη(t) + γ
p–ηp(t) – > , and note that
a=η() =
(u+v)ϕ,
b=η() =
(u+v)ϕ.
Moreover,
η(t) =
(u+v)ϕ≤u(·,t)L∞()+v(·,t)L∞()
ϕ≤ (u,v) .
Hence, (u,v) blows up before the maximal time of existence defined in inequality (.) is
reached.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YG performed the calculations and drafted the manuscript. WG supervised and participated in the design of the study and modified the draft versions. All authors read and approved the final manuscript.
Acknowledgements
Supported by NSFC (11271154) and by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education and by the 985 program of Jilin University.
Received: 15 February 2013 Accepted: 18 March 2013 Published: 5 April 2013 References
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doi:10.1186/1687-2770-2013-76
