#### Study of solutions to an initial and boundary value problem for certain systems with variable exponents

Boundary Value Problems
Study of solutions to an initial and boundary value problem for certain systems with
Yunzhu Gao 1 2 3
Wenjie Gao 0 1 2
0 Institute of Mathematics , Jilin
1 City , P.R. China
2 Statistics, Beihua University , Jilin
3 Department of Mathematics
system is also obtained. 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
exponent; existence; blow-up; parabolic system; hyperbolic system; variable
1 Introduction
In this paper, we first consider the initial and boundary value problem to the following
nonlinear parabolic system with variable exponents:
u + f(u, v), (x, t) ∈ QT ,
v + f(u, v), (x, t) ∈ QT ,
v(x, t) = , (x, t) ∈ ST ,
v(x, ) = v(x), x ∈ ,
and < T < ∞, QT =
× [, T ), ST denotes the lateral boundary of the cylinder QT , and the source terms f, f
are in the form
f(u, v) = a(x)vp(x)
and f(u, v) = a(x)up(x),
f(u, v) = a(x)
vp(y)(y, t) dy and f(u, v) = a(x)
up(y)(y, t) dy,
respectively, where p, p, a, a are 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,
u = vp,
v = uq,
u + f (x, u), (x, t) ∈
× [, T ),
× [, T ),
where x ∈ RN (N ≥ ), t > , and p, q are 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:
where ∈ Rn is a bounded domain with smooth boundary ∂ , and the source term is of
the form f (x, u) = a(x)up(x) or f (x, u) = a(x) uq(y)(y, t) dy. The author studied the
blowup 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 dynamics etc. We
also refer the interested reader to [–] and the references therein.
We also study the following nonlinear hyperbolic system of equations:
v(x, t) = , (x, t) ∈ ST ,
⎪⎪⎪⎪⎩⎪⎪⎪⎪ uut((xx,,))==uu((xx)),,
v(x, ) = v(x), 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 exponents p(x), p(x) : → (, +∞) and the
continuous functions a(x), a(x) : → R satisfy the following conditions:
< p– = inf p(x) ≤ p(x) ≤ p+ = sup p(x) < +∞,
x∈ x∈
< p– = inf p(x) ≤ p(x) ≤ p+ = sup p(x) < +∞,
x∈ 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 instant T ∗ < ∞ such that
as 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 .
g(x, y, t – s)a(y)vp(y) dy ds,
g(x, y, t – s)a(y)up(y) dy ds,
where g(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 +
vn+(x, t) =
g(x, y, t)v(y) dy +
g(x, y, t – s)a(y)vpn(y) dy ds,
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
(u) = t
g(x, y, t – s)a(y)vpn(y) dy ds,
g(x, y, t – s)a(y)upn(y) dy ds
is a contraction in the set ET to be defined below.
Now, we define
(v) =
(u) =
g(x, y, t – s)vpn(y) dy ds,
g(x, y, t – s)upn(y) dy ds.
(u, v) – (w, z) =
(v) – (z), (u) – (w) ,
and for arbitrary T > , define the set
ET = C,( T ) ∩ C( T )
(u, v) ≤ M ,
where T = × [, T ], M > |(u(x), v(x)) | is a fixed positive constant.
We claim that is a contraction on ET . In fact, for any x ∈ fixed, we have
and we always have
Now, we define
It is obvious that μ(t) → when t → +.
Then, by using inequality (.), we get
g(x, y, t – s) dy ds.
pi(x)wipi(x)–(ξi – ηi) ≤ pi+(M)pi+– ξi – ηi ∞, i = , .
(u) – (w) ∞
a(y)g(x, y, t – s) vpn(y) – znp(y) dy ds
a(y)g(x, y, t – s) upn(y) – wpn(y) dy ds
≤ μ(t) (M)max{p+,p+}–(C + C) p ∞ + p ∞
≤ μ(t) (M)max{p+,p+}–(C + C) p ∞ + p ∞
= μ(t) (M)max{p+,p+}–(C + C) p ∞ + p ∞
(v – z, u – w) .
v – z ∞ + u – w ∞
Hence, for sufficiently small t, we have
(u, v) – (w, z)
(v) – (z), (u) – (w)
(v) – (z) ∞ +
(u) – (w) ∞
≤ μ(t) (M)max{p+p+}–(C + C) p ∞ + p ∞
(v – z, u – w)
is 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
Proof It is sufficient to take K > such that
Hence, we have
y (t) ≥ a y(t)r.
where λ > , 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,
By a direct integration to (.), then we get immediately (.), which gives an upper bound
for the blow-up time t∗ = ay(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
solution 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) ϕ > C,
where ϕ > is the first eigenfunction of the homogeneous Dirichlet Laplacian on and
C > is a constant depending only on the domain and the bounds C, C given in
condition (.).
x ∈
ϕ dx = .
We introduce the function η(t) =
a(x)vp(x), f(u, v) = a(x)up(x). Then
(ut + vt)ϕ
with the homogeneous Dirichlet boundary condition, and let ϕ be a positive function
satisfying
a(x)vp(x) + a(x)up(x) ϕ
a(x)vp(x) + a(x)up(x) ϕ.
We now deal with the term
the following four sets:
(a(x)vp(x) + a(x)up(x))ϕ. For each t > , we divide
= x ∈
= x ∈
: v(x, t) < , u(x, t) < ,
= x ∈
: v(x, t) < , u(x, t) ≥ ,
: 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)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
a(x)vp(x) + a(x)up(x) ϕ ≥ γ
Then we get
η (t) ≥ –λη(t) + γp– ηp(t) – .
(u + v)ϕ ≤
u(·, t) L∞( ) + v(·, t) L∞( )
ϕ ≤ (u, v) .
Hence, for η() big enough, the result follows from Lemma ..
Next, we state briefly the proof to the theorem in the case f(u, v) = a(x) vp(y)(y, t) dy
and f(u, v) = a(x) up(y)(y, t) dy. We repeat the previous argument under defining η(t) =
(u + v)ϕ, and we obtain in much the same way
vp(y)(y, t) dy + a(x)
up(y)(y, t) dy ϕ(x) dx.
In view of the property of ϕ, we get
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) + up(y)(y, t)
According to the convex property of the function f (w) = wr, r > , and by using Jensen’s
inequality, by considering again , , , as before, we obtain
vp(y)(y, t) + up(y)(y, t)
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) ∈ C satisfying
y (t) ≥ h y(t) ,
t ≤
Now, let us study the following problem:
⎨⎪⎪⎪⎪⎪⎪⎪⎧⎪ vuutt(ttx==,t) =vu++,ff((uu,,vvv))(,,x, t()(xx=,,tt)),∈∈QQ(xTT,,,t) ∈ ST ,
⎪⎪⎪⎪⎩⎪⎪⎪⎪ uut((xx,,))==uu((xx)),, vv(tx(,x,))==vv(x(x),), xx∈∈ , ,
where u(x), v(x), u(x), v(x) ≥ and they are not identically zero, and f(u, v), f(u, v) as
above respectively.
Theorem . Let (u, v) ∈ C × C be a solution of problem (.), and let the conditions in
(.) hold. Then there exist sufficiently large initial data u, v, u, v such that any solutions
of problem (.) blew up at finite time T ∗.
Proof Let (λ, ϕ) be the first eigenvalue and eigenfunction of Laplacian in with
homogeneous 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 function η(t) = (u + v)ϕ,
so we have
(utt + vtt)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 is dealt with as before, then we get
a(x)vp(x) + a(x)up(x) ϕ ≥ γ
ϕ ≥
we still obtain
Then we have
> , and note that
(u + v)ϕ,
(u + v)ϕ.
(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
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.
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