Electronic Journal of Differential Equations, Vol. 2004(2004), No. 35, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
EXISTENCE OF SOLUTIONS TO THE ROSENAU AND
BENJAMIN-BONA-MAHONY EQUATION IN DOMAINS WITH
MOVING BOUNDARY
RIOCO K. BARRETO, CRUZ S. Q. DE CALDAS,
PEDRO GAMBOA, & JUAN LIMACO
Abstract. In this article, we prove the existence of solutions for a hyperbolic
equation known as the the Rosenau and Benjamin-Bona-Mahony equations.
We study increasing, decreasing, and mixed non-cylindrical domains. Our
main tools are the Galerkin method, multiplier techniques, and energy estimates.
1. Introduction
To investigate the dynamics of certain discrete systems, Philip Rosenau obtained
the equation ut + (u + u2 )x + uxxxxt = 0. The study of this equation in cylindrical
domains was done by Mi Ai Park [13], who proved the existence and uniqueness
of local and global solutions. The Rosenau equation could be seen as a variant of
Benjamin-Bona-Mahony (BBM) equation, ut + (u + u2 )x − uxxt = 0, which models
long waves in a non linear dispersive system. In [3], Benjamin-Bona-Mahony proved
the existence and uniqueness of global solutions for the BBM equation in cylindrical
domains. In this work, we study the existence of solutions for the Rosenau and BBM
equations for increasing, decreasing, and mixed noncylindrical domains.
We introduce the following notation: Let α, β, γ = β − α, be C 2 -functions of a
real variable, such that α(t) < β(t), for all t ≥ 0. We represent the noncylindrical
domain by
b = {(x, t) ∈ R2 : α(t) < x < β(t), ∀t ≥ 0},
Q
b=S
and its lateral boundary by Σ
0≤t≤T {α(t), β(t)} × {t}.
In the present work we investigate the following two equations:
ut + (u + u2 )x + uxxxxt = 0
b
in Q
u(x, t) = 0
b
for (x, t) ∈ Σ
ux (x, t) = 0
b
for (x, t) ∈ Σ
u(x, 0) = u0 (x)
for α(0) < x < β(0)
1991 Mathematics Subject Classification. 35M10, 35B30.
Key words and phrases. Benjamin-Bona-Mahony equation, Rosenau equation,
noncylindrical domains.
c
2004
Texas State University - San Marcos.
Submitted April 28, 2003. Published March 11, 2004.
1
(1.1)
2
R. K. BARRETO, C. DE CALDAS, P. GAMBOA, & J. LIMACO
and
ut + (u + u2 )x − uxxt = 0
u(x, t) = 0
EJDE-2004/35
b
in Q
b
for (x, t) ∈ Σ
(1.2)
0
u(x, 0) = u (x) in Ω0 .
This paper is organized as follows: The next section is devoted to the existence and
uniqueness of solution for (1.1) and (1.2), satisfying the hypothesis
(H1) α0 (t) ≥ 0 and β 0 (t) ≤ 0 for t ∈ [0, T ].
b decreases in the sense that if t2 > t1 , then the
Note that this hypothesis implies Q
projection of [α(t2 ), β(t2 )] in the subspace t = 0 is contained in the projection of
[α(t1 ), β(t1 )] in the same subspace.
In the third section of this article, we study the existence of solutions for (1.1)
and (1.2) satisfying the hypothesis
(H2) α0 (t) ≤ 0 and β 0 (t) ≥ 0 for t ∈ [0, T ].
b increases
Analogously hypothesis (H2) implies that Q
In the last section of this article, we study the (1.1) and (1.2), satisfying the
hypothesis:
b=Q
c1 ∪ Q
c2 where Q
c1 is increasing and Q
c2 is decreasing.
(H3) Q
In the following, by Ω we represent the interval ]0, 1[, Ωt and Ω0 denote the
intervals ]α(t), β(t)[ and ]α(0), β(0)[ respectively. We denote, as usual, by (., .), k · k
respectively the scalar product and norm in L2 (Ω). In the sequel, wm,x denotes
∂wm
∂ 2 wm
∂ 2 wm
∂x , analogously wm,xx = ∂x2 , wm,xt = ∂t ∂x , etc.
2. Solutions on decreasing domains
In this section we study the existence and uniqueness for (1.1) and (1.2) satisfying
the hypothesis (H1). Let γ(t) = β(t) − α(t) > 0, for all t ≥ 0. Then 0 < x−α(t)
γ(t) < 1,
for all t ∈ [0, T ]. With the change of variable u(x, t) = v(y, t) where y =
all t ∈ [0, T ], problem (1.1) is transformed into
vt +
1
(α0 + γ 0 y)
4γ 0
1
(v + v 2 )y + 4 vyyyyt −
vy − 5 vyyyy
γ
γ
γ
γ
(α0 + γ 0 y)
−
vyyyyy = 0 in Ω×]0, T [
γ5
v(0, t) = v(1, t) = 0 in ]0, T [
vy (0, t) = vy (1, t) = 0
x−α(t)
γ(t) ,
for
(2.1)
in ]0, T [
0
v(y, 0) = v (y) in Ω .
Also problem (1.2) is transformed into
vt +
1
1
(α0 + γ 0 y)
2γ 0
vy + 3 vyy
(v + v 2 )y −
vyyt −
γ
γ2
γ
γ
(α0 + γ 0 y)
+
vyyy = 0 in Ω×]0, T [
γ3
v(0, t) = v(1, t) = 0 in ]0, T [
v(y, 0) = v 0 (y) in Ω .
Under these conditions, we establish the following existence results.
(2.2)
EJDE-2004/35
ROSENAU AND BENJAMIN-BONA-MAHONY EQUATIONS
3
Theorem 2.1. For each u0 ∈ H02 (Ω0 ) ∩ H 4 (Ω0 ), there exists a unique function
b → R, satisfying u ∈ C 1 ([0, T ]; H 2 (Ωt )) ∩ C(0, T ; H 3 (Ωt ) ∩ H 2 (Ωt )) and
u:Q
0
0
Z
Z
Z
ut φ dx dt + (u + u2 )x φ dx dt +
uxxt φxx dx dt = 0,
b
Q
2
for all φ ∈ L
(0, T ; H02 (Ωt )),
b
Q
b
Q
0
u(x, 0) = u (x), for all x ∈ Ω0 .
Theorem 2.2. For each u0 ∈ H01 (Ω0 ) ∩ H 2 (Ω0 ), there exists a unique function
b → R, satisfying u ∈ L∞ (0, T ; H 1 (Ωt )), ut ∈ L∞ (0, T ; H 1 (Ωt )) and
u:Q
0
0
Z
Z
Z
ut φ dx dt + (u + u2 )x φ dx dt +
uxt φx dx dt = 0,
b
Q
2
for all φ ∈ L
(0, T ; H01 (Ωt )),
b
Q
b
Q
0
u(x, 0) = u (x), for all x ∈ Ω0 .
To prove Theorem 2.1, we need the following lemmas.
Lemma 2.3. For each v 0 ∈ H02 (Ω) ∩ H 4 (Ω), there exists a unique function v :
Ω×]0, T [→ R, satisfying v ∈ L∞ (0, T ; H02 (Ω) ∩ H 4 (Ω)), vt ∈ L∞ (0, T ; H02 (Ω)), and
Z
1
1
[vt ψ + (v + v 2 )y ψ + 4 vyyt ψyy
γ
γ
Ω×]0,T [
−
(α0 + γ 0 y)
4γ 0
(α0 + γ 0 y)
vy ψ − 5 vyy ψyy + (
ψ)y vyyyy ]dy dt = 0,
γ
γ
γ5
for all ψ ∈ L2 (0, T ; H02 (Ω)), v(y, 0) = v 0 (y), for all y ∈ Ω.
Lemma 2.4. For each f ∈ C([0, T ]; H −2 (Ω)), there exists a unique function z :
Ω×]0, T [→ R, satisfying z ∈ C([0, T ]; H02 (Ω)) and z + γ14 ∆2 z = f
Lemma 2.5. For each v 0 ∈ H02 (Ω) ∩ H 4 (Ω), there exists a unique function v :
Ω×]0, T [→ R, satisfying v ∈ C 1 ([0, T ]; H02 (Ω)) ∩ C([0, T ]; H 3 (Ω) ∩ H02 (Ω)) and
Z
1
1
[vt ψ + (v + v 2 )y ψ + 4 vyyt ψyy
γ
γ
Ω×]0,T [
−
(α0 + γ 0 y)
4γ 0
(α0 + γ 0 y)ψ
vy ψ − 5 vyy ψyy + (
)y vyyyy ] dy dt = 0 ,
γ
γ
γ5
for all ψ ∈ L2 (0, T ; H02 (Ω)); v(y, 0) = v 0 (y), for all y ∈ Ω.
Lemma 2.6. For each v 0 ∈ H01 (Ω) ∩ H 2 (Ω), there exists a unique function v :
Ω×]0, T [→ R, satisfying v ∈ L∞ (0, T ; H01 (Ω) ∩ H 2 (Ω)), vt ∈ L∞ (0, T ; H01 (Ω))
and
Z
1
1
[vt ψ + (v + v 2 )y ψ − 2 vyyt ψ
γ
γ
Ω×]0,T [
−
(α0 + γ 0 y)
2γ 0
(α0 + γ 0 y)ψ
vy ψ + 3 vyy ψ − (
)y vyy ]dy dt = 0,
γ
γ
γ3
for all ψ ∈ L2 (0, T ; H01 (Ω)); v(y, 0) = v 0 (y), for all y ∈ Ω.
In this article, we prove Theorem 2.1 and Lemmas 2.3, 2.4, 2.5 which correspond
to Rosenau Equation. However, we omit the proofs of Theorem 2.2 and Lemma
2.6 which correspond to Benjamin Bona-Mahony Equation; because the proofs are
made in a similar way.
4
R. K. BARRETO, C. DE CALDAS, P. GAMBOA, & J. LIMACO
EJDE-2004/35
Proof of Lemma 2.3. Let (wi )i∈N be the special basis of H02 (Ω), such that
wi,yyyy = λi wi ,
in Ω
wi (0) = wi (1) = wi,y (0) = wi,y (1) = 0,
i ∈ N.
We denote by Vm the subspace generated by w1 , . . . , wm . Our starting point is to
construct the Galerkin approximation of the solution vm ∈ Vm , which is given by
the solution of the approximate equation
1
4γ 0
1
2
)y , w) + 4 (vm,yyyyt , w) − 5 (vm,yyyy , w)
(vm,t , w) + ( (vm + vm
γ
γ
γ
0
0
0
0
(α + γ y)
(α + γ y)
+(−
vm,yyyyy , w) + (−
vm,y , w) = 0 for all w ∈ Vm
γ5
γ
0
vm (0) = vm
→ v 0 in H 4 (Ω)
(2.3)
First Estimate. Taking w = vm (t) in (V)1, we have:
d
1
γ0
(kvm (t)k2 + 4 kvm,yy (t)k2 ) + 5 kvm,yy (t)k2
dt
γ
γ
0
0
γ
α 2
β0 2
(0) − 5 vm,yy
(1) = 0
+ kvm (t)k2 + 5 vm,yy
γ
γ
γ
(2.4)
Integrating this equation over [0, t] and using hypothesis (H1), we obtain
1
kvm,yy (t)k2
γ4
Z
Z
γ2 t 1
γ2 t
1
0 2
2
k +
kv
(s)k
ds
+
kvm (s)k2 ds
≤ kv 0 k2 + 4 kvyy
m,yy
γ (0)
γ0 0 γ 4
γ05 0
kvm (t)k2 +
(2.5)
where γ0 , γ1 , γ2 are positive constants, such that γ0 ≤ γ(t) ≤ γ1 , and
γ2 = max |γ 0 (t)|,
γ1 = max |α0 (t)|.
0≤t≤T
0≤t≤T
This implies
kvm (t)k2 +
1
kvm,yy (t)k2 ≤ C0 + C1
γ4
Z
0
t
[kvm (s)k2 +
1
kvm,yy (s)k2 ] ds
γ4
(2.6)
where C0 , C1 , . . . denote positive constants. Applying Gronwall inequality, we have
the first estimate
1
kvm (t)k2 + 4 kvm,yy (t)k2 ≤ C2
(2.7)
γ
Second Estimate Taking w = vm,yyyy (t) in the first equation of (2.3), we obtain
1 d
1
[kvm,yy (t)k2 + 4 kvm,yyyy (t)k2 ]
2 dt
γ
3γ2 1
1
≤
kvm,yyyy (t)k2 + kvm,y (t)kkvm,yyyy (t)k
4
2γ0 γ
γ
(α1 + γ2 )
+
kvm,y (t)k2 kvm,yyyy (t)k.
γ0
(2.8)
EJDE-2004/35
ROSENAU AND BENJAMIN-BONA-MAHONY EQUATIONS
5
From (2.7) and using Schwartz’s inequality and Poincare’s inequality, we obtain
1 d
1
[kvm,yy (t)k2 + 4 kvm,yyyy (t)k2 ]
2 dt
γ
1
γ12
3γ2 1
2
2
≤
kv
(t)k
+
kv
(t)k
+
C3 kvm,yy (t)k2
m,yyyy
m,yyyy
2γ0 γ 4
2γ 4
2
+ C3 kvm,yy (t)k2 kvm,yyyy (t)k
1
(1 +
2
1
≤ (1 +
2
≤
Let c4 =
3γ2 1
) kvm,yyyy (t)k2 +
γ0 γ 4
3γ2 1
) kvm,yyyy (t)k2 +
γ0 γ 4
(α1 +γ2 )
C3 C2
γ0
and let c5 =
γ12
(α1 + γ2 )
C3 C2 +
C3 C2 kvm,yyyy (t)k
2
γ0
γ12
γ2
1
C3 C2 + 1 C42 + 4 kvm,yyyy (t)k2 .
2
2
2γ
γ12
2 C3 C2
+
γ12 2
2 C4 .
Then
d
1
3γ2 1
[kvm,yy (t)k2 + 4 kvm,yyyy (t)k2 ] ≤ C5 + (2 +
) kvm,yyyy (t)k2 .
dt
γ
γ0 γ 4
(2.9)
Integrating this inequality over [0, t] and applying Gronwall inequality, we obtain
kvm,yy (t)k2 +
1
kvm,yyyy (t)k2 ≤ C6
γ4
(2.10)
Third Estimate Taking w = vm,t (t) in (V)1 , we have
kvm,t (t)k2 +
1
kvm,yyt (t)k2
γ4
1
4γ 0
2
= − ((vm (t) + vm
(t))y , vm,t (t)) + 5 (vm,yyyy (t), vm,t (t))
γ
γ
(α0 + γ 0 y)
(α0 + γ 0 y)
−(
v
(t),
v
(t))
+
(
vm,y (t), vm,t (t)).
m,yyyy
m,yt
γ5
γ5
(2.11)
¿From (2.7), (2.10), and (2.11), we obtain
kvm,t (t)k2 +
1
kvm,yyt (t)k2 ≤ C7 .
γ4
(2.12)
These three estimates permit to pass to the limit in the approximate equation and
we obtain a weak solution v in the sense of Lemma 2.3. The uniqueness of solution
and the verification of initial data are showed by the standard arguments.
Proof of Lemma 2.4. To prove the existence we consider two stages: First stage
f ∈ C([0, T ]; H02 (Ω)). Let (wi )i∈N be the special basis of H02 (Ω)
Pnused in the proof
of Lemma 2.3. Consider the sequence (fn ), such that fn (t) = i=1 (f (t), wi )wi . It
is clear that fn → f strongly in C([0, T ]; H02 (Ω)).
Pm
The approximated solution zm (t) to z + γ14 ∆2 z = f is zm (t) = i=1 gim (t)wi ,
where gim are solutions of the approximated system
(zm (t), wi ) +
1
(∆zm (t), ∆wi ) = (fm (t), wi ) ,
γ4
i = 1, . . . m
(2.13)
A priori estimate. Let us prove that (zm ) is a Cauchy sequence in C([0, T ]; H02 (Ω)).
In fact, let m and n be positive integer such that m > n and gin (t) = 0 for
6
R. K. BARRETO, C. DE CALDAS, P. GAMBOA, & J. LIMACO
EJDE-2004/35
n ≤ i ≤ m. Then zm and zn are solutions of (2.13) in Vm . Consider the Cauchy
difference zm − zn . We have from (2.13) that, for i = 1, . . . , m,
1
(zm (t) − zn (t), wi ) + 4 (∆(zm (t) − zn (t)), ∆wi ) = (fm (t) − fn (t), wi ) , (2.14)
γ
Taking wi = zm (t) − zn (t) in (2.14), using the Cauchy-Schwarz inequality and the
equivalent norms, we obtain
|zm − zn |C([0,T ];H02 (Ω)) ≤ c|fm − fn |C([0,T ];H02 (Ω))
Then zn → z strongly in C([0, T ]; H02 (Ω)). Therefore, taking limit in (2.13), we
obtain z + γ14 ∆2 z = f in C([0, T ]; H −2 (Ω)).
Second stage f ∈ C([0, T ]; H −2 (Ω)). By density, there exists a sequence (fn ),
fn ∈ C([0, T ]; H02 (Ω)), such that fn → f strongly in C([0, T ]; H −2 (Ω)). Using the
first stage we have that there exist a sequence (zn ), zn ∈ C([0, T ]; H02 (Ω)) such that
1 2
∆ zn = fn in C([0, T ]; H −2 (Ω)).
γ4
Consider the Cauchy difference zm − zn , m > n. We obtain
1
zm− zn + 4 ∆2 (zm − zn ) = fm − fn in C([0, T ]; H −2 (Ω)).
γ
zn +
(2.15)
(2.16)
Composing (2.16) with zm − zn ∈ C([0, T ]; H02 (Ω)). and integrating in Ω, we have
|zm − zn |C([0,T ];H02 (Ω)) ≤ c|fm − fn |C([0,T ];H −2 (Ω)) ;
therefore, zn → z strongly in C([0, T ]; H02 (Ω)) and taking limit in (2.15) we have
z + γ14 ∆2 z = f in C([0, T ]; H −2 (Ω)). The uniqueness of the solutions is showed by
the standard arguments.
Proof of Lemma 2.5. From Lemma 2.4 we can define the operator B(t) = (I +
1
2 −1
from C([0, T ]; H −2 (Ω)) to C([0, T ]; H02 (Ω)) by B(t)f = z with f ∈
γ4 ∆ )
C([0, T ]; H −2 (Ω)) where z is a solution of z + γ14 ∆2 z = f . Note that B(t) is
linear and continuous.
By Lemma 2.3, v ∈ L2 (0, T ; H 4 (Ω) ∩ H02 (Ω)) and vt ∈ L2 (0, T ; H02 (Ω)). From
Lions-Magenes, Theorems 3.1 and 9.6, chapter I [10], we conclude that
v ∈ C([0, T ]; H02 (Ω) ∩ H 3 (Ω))
On the other hand, from the transformed problem (2.1), we obtain
1
(I + 4 ∆2 )vt = f ,
γ
where,
1
(α0 + γ 0 y)
4γ 0
(α0 + γ 0 y)
f = − (v + v 2 )y +
vy + 5 vyyyy + (
)vyyyyy
γ
γ
γ
γ5
(2.17)
(2.18)
From (2.17), we conclude that f ∈ C([0, T ]; H −2 (Ω)). Then from (2.18) we have
that vt = B(t)f , where vt ∈ C([0, T ]; H02 (Ω)) and we get the required result.
The proof of Theorem 2.1 follows immediately from Lemma 2.5 and the Change
of Variable Theorem. Therefore, we omit it.
Observe that Theorem 2.1 in a cylindrical domain, has the regularity u ∈
C 1 ([0, T ]; H02 (Ω)) ∩ C([0, T ]; H 4 (Ω) ∩ H02 (Ω)). In fact, as we consider an additional estimate with wi = um,txxxx , in the Galerkin approximation, that allows us
EJDE-2004/35
ROSENAU AND BENJAMIN-BONA-MAHONY EQUATIONS
7
to obtain the regularity ut ∈ L2 (0, T ; H 4 (Ω) ∩ H02 (Ω)). However, in our noncylindrical domain, this is not possible since the transformed problem (III), contains a
term vyyyyy that does not allow us to use the estimate with wi = vm,tyyyy in the
Galerkin approximation.
3. Solutions on increasing domains
In this section we study the existence of solution for the systems (1.1) and (1.2)
satisfying the hypothesis (H2). We use the Penalization Method given by Lions
b ⊂ Q. We define the function
[10]. Let Q =]a, b[×]0, T [ be the cylinder such that Q
M : Q → R, by
(
b
1 in Q \ Q
M (x, t) =
b
0 in Q
To show the existence result we will use the following Lemma.
Lemma 3.1. If u, ut ∈ L2 (0, T ; L2 (a, b)), then
Z t
1
1
(M u(s), ut (s))ds ≥ kM (t)u(t)k2L2 (a,b) − kM (0)u(0)k2L2 (a,b) .
2
2
0
Proof. We have
Z
t
0
Z Z
1 t b
(M u(s), ut (s))ds =
M (u2 (s))t dξ ds
2 0 a
Z
1
=
M (u2 (s))t dξ ds.
2 [0,t]×[a,b]
From Fubini’s Theorem and recalling the definition of M , it follows that
Z t
(M u(s), ut (s))ds
0
Z
Z −1
1 α(0) α (x) 2
[u (s)]t ds dξ +
[u (s)]t ds dξ
2 α(t) 0
a
0
Z
Z −1
Z
Z t
1 β(t) β (x) 2
1 b
+
[u (s)]t ds dξ +
[u2 (s)]t ds dξ
2 β(0) 0
2 β(t) 0
Z
Z
1 α(t) 2
1 α(0) 2 −1
2
=
[u (t, ξ) − u (0, ξ)] dξ +
[u (α (x), 0) − u2 (0, ξ)] dξ
2 a
2 α(t)
Z
Z
1 β(t) 2 −1
1 b
+
[u (β (x), 0) − u2 (0, ξ)] dξ +
[u2 (t, ξ) − u2 (0, ξ)] dξ
2 β(0)
2 β(t)
Z
Z b
Z α(t)
1 α(t) 2
≥ [
u (t, ξ) dξ +
u2 (t, ξ) dξ − (
u2 (0, ξ) dξ
2 a
β(t)
a
Z α(0)
Z β(t)
Z b
+
u2 (0, ξ) dξ +
u2 (0, ξ) dξ +
u2 (0, ξ) dξ)]
1
=
2
Z
α(t)
α(t)
Z
b
Z
t
2
β(0)
β(t)
Z
b
1
1
M (t, ξ) u2 (t, ξ) dξ −
M (0, ξ)u2 (0, ξ) dξ
2 a
2 a
1
1
= kM (t)u(t)k2L2 (a,b) − kM (0)u(0)k2L2 (a,b)
2
2
=
8
R. K. BARRETO, C. DE CALDAS, P. GAMBOA, & J. LIMACO
EJDE-2004/35
which completes the proof.
The existence of solution for (1.1) and (1.2), satisfying the hypothesis (H2), is
established in the next theorems.
b → R, satisfying
Theorem 3.2. For each u0 ∈ H02 (Ω0 ), there exists a function u : Q
∞
2
∞
2
u ∈ L (0, T ; H0 (Ωt )), ut ∈ L (0, T ; H0 (Ωt )) and
Z
Z
Z
ut φ dx dt + (u + u2 )x φ dx dt +
uxxt φxx dx dt = 0
(3.1)
b
Q
b
Q
b
Q
for all φ ∈ L2 (0, T ; H02 (Ωt )); u(x, 0) = u0 (x)
b → R, satisfying
Theorem 3.3. For each u0 ∈ H01 (Ω0 ), there exists a function u : Q
∞
1
∞
1
u ∈ L (0, T ; H0 (Ωt )), ut ∈ L (0, T ; H0 (Ωt )) and
Z
Z
Z
2
ut φ dx dt + (u + u )x φ dx dt +
uxt φx dx dt = 0,
b
Q
b
Q
b
Q
for all φ ∈ L2 (0, T ; H01 (Ωt )); u(x, 0) = u0 (x)
Proof of Theorem 3.2. To prove this result we use the penalization method. For
each > 0 we consider the problem
1
1
u,t + (u + u2 )x + u,xxxxt + M u,t − (M u,xt )x = 0 in Q
u (a, t) = u (b, t) = u,x (a, t) = u,x (b, t) = 0 in ]0, T [
u (x, 0) = u
e0 (x)
(3.2)
in ]a, b[
Let {wi }i∈N be a basis of H02 (a, b), such that w1 = u
e0 . We denote by Vm =
[w1 , . . . , wm ] the subspace of H02 (a, b), generated by u
e0 , w2 , . . . , wm . We seek um (t)
in Vm solution to the approximate problem
(um,t , w) + ((um + u2m )x , w) + (um,xxxxt , w)
1
1
+ (M um,t , w) − ((M um,xt )x , w) = 0 for all w ∈ Vm
um (0) = u0 (x) → u
e0 in H02 (a, b)
(3.3)
First Estimate. Taking w = um in (3.3) and applying Lemma 3.1, we obtain
1
1
k|um (t)k|2 + k|um,xx (t)k|2 + k|M (t)um (t)k|2 + k|M (t)um,x (t)k|2 ≤ c8 , (3.4)
2
where k| · k| denotes the norm in L (a, b).
Second Estimate. Taking w = um,t (t) in (3.3) and using (3.4) we have
1
k|um,t (t)k|2 + k|um,xxt (t)k|2 + (M (t)um,t (t), um,t (t))
1
(3.5)
(M (t)um,xt (t), um,xt (t))
1
≤ c9 + k|um,t (t)k|2
2
From where we obtain
1
1
k|um,t (s)k|2 + k|um,xxt (s)k|2 + k|M (t)um,t (t)k|2 + k|M (t)um,xt (t)k|2 ≤ c9
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ROSENAU AND BENJAMIN-BONA-MAHONY EQUATIONS
9
From the estimates above, we pass to the limit in the approximate equation, and
we obtain that u is solution of the penalized problem
Z TZ b
Z TZ b
Z TZ b
u,t v dx dt +
(u + u2 )x v dx dt +
u,xxt vxx dx dt
0
a
0
a
0
a
(3.6)
Z Z
Z Z
1 T b
1 T b
+
M u,t v dx dt +
M u,xt vx dx dt = 0
0 a
0 a
for all v ∈ L2 (0, T ; H02 (a, b)). From (3.4), (3.5) and the Banach-Steinhauss Theorem, we pass to the limit as → 0 in (3.6) and we obtain (3.1).
Regularity. From the first estimate, we have
Z
1 t
(M um,t (s), um (s)) ds ≤ c
0
On the other hand, from Lemma 3.1 we obtain
Z
1 t
1
(M um,t (s), um (s)) ds ≥ k|M (t)um (t)k|2 .
0
2
RT Rb
2
Then k|M (t)um (t)k| ≤ 2c. Thus 0 a M (t) u2m (t) dx dt ≤ 2cT or
Z TZ b
Z TZ b
2
|M (t)u (t)| dx dt ≤ lim inf
|M (t)u (t)|2 dx dt ≤ 2cT
0
a
0
2
a
2
Then M u → 0 in L (0, T ; L (a, b)).
On the other hand, M u → M u in L2 (0, T ; L2 (a, b)) and M um → M u in
2
b
L (0, T ; L2 (a, b)). So, we conclude that M u = 0 a.e. in Q or u = 0 in Q \ Q.
Analogously, applying the Lemma 3.1 to um,xt instead of um,t , we obtain:
b Since u ∈ L∞ (0, T ; H 1 (a, b)) and ut ∈ L∞ (0, T ; H 1 (a, b)), then u ∈
ux = 0 in Q\ Q.
0
0
C([0, T ]; H01 (a, b)). Therefore, u(t) ∈ H01 (a, b) for all t and u = 0 in ]a, b[\]α(t), β(t)[.
From where u(t) ∈ H01 (α(t), β(t)), for all t. Thus u ∈ L∞ (0, T ; H01 (Ωt )). Analogously, ux ∈ L∞ (0, T ; H01 (Ωt )). From these two statements, we have that u ∈
L∞ (0, T ; H02 (Ωt )). From the second estimate,
Z TZ b
Z TZ b
2
|M (t)um,t (t)| dx dt +
|M (t)um,xt (t)|2 dx dt ≤ 2cT.
0
a
0
a
By similar arguments, we obtain that ut ∈ L∞ (0, T ; H02 (Ωt )), which prove the
regularity of the solution.
The proof of Theorem 3.3 is similar to the proof of Theorem 3.2 and is ommitted.
Remark 3.4. Theorems 2.1, 2.2, 3.2 and 3.3 are invariable by translation. In fact,
the particular problem
ut + (u + u2 )x + uxxxxt = 0
b ⊂ Ω×]T0 , T1 [
in Q
u(x, t) = 0
b
in Σ
ux (x, t) = 0
b
in Σ
u(x, T0 ) = u0 (x) in ΩT0 ,
with the change of variable u(x, t) = u(x, t−T0 ), can be transformed into a problem
of type (1.1).
10
R. K. BARRETO, C. DE CALDAS, P. GAMBOA, & J. LIMACO
EJDE-2004/35
4. Solutions on mixed domains
b is a mixed domain; i.e., Q
b = B1 ∪ B2 where
Here we analyze the case when Q
b
b
B1 = {(x, t) ∈ Q : 0 < t ≤ T1 } and B2 = {(x, t) ∈ Q : T1 < t < T }, where B1 is
ci by
decreasing satisfying (H1), and B2 is increasing satisfying (H2). We define Q
ci = int(Bi ), i = 1, 2. i.e.,
Q
c1 = {(x, t) ∈ Q
b : 0 < t < T1 } and Q
c2 = {(x, t) ∈ Q;
b
Q
T1 < t < T }.
b=Q
c1 ∪ Q
c2 , we consider the following two cases:
To find a solution to (1.1) in Q
0
2
c
(1) Solution on Q1 : For each u ∈ H0 (Ω0 ) ∩ H 4 (Ω0 ), by Theorem 2.1, there exist
u1 solution of
b1
u1,t + (u1 + u21 )x + u1,xxxxt = 0 in Q
u1 (x, t) = 0
b1
inΣ
u1,x (x, t) = 0
u1 (x, 0) = uo (x)
∞
(4.1)
b1
inΣ
in Ω0
∞
(0, T1 ; H02 (Ωt )),
satisfying u1 ∈ L
u1t ∈ L (0, T1 ; H02 (Ωt )); therefore u1 is in
2
C([0, T1 ]; H0 (Ωt )).
c2 : For each u0 = u1 (T1 ) ∈ H 2 (ΩT ), by Theorem 3.2, there exist
(2) Solution on Q
0
1
u2 solution of
b2
u2,t + (u2 + u22 )x + u2,xxxxt = 0 in Q
u2 (x, t) = 0
b2
inΣ
u2,x (x, t) = 0
u2 (x, T1 ) = u0 (x)
∞
(T1 , T ; H02 (Ωt )),
(4.2)
b2
inΣ
in ΩT1
∞
satisfying u2 in L
u2t in L (T1 , T ; H02 (Ωt )), and u2 in
2
C([T1 , T ]; H0 (Ωt )).
b We define u : Q
b → R by
(3) Solution on Q:
(
c1
u1 (x, t), (x, t) ∈ Q
u(x, t) =
c2 ,
u2 (x, t), (x, t) ∈ Q
where u1 and u2 are solutions of (4.1) and (4.2), respectively. Given that u0 ∈
H02 (Ω0 ), by Remark 3.4, we deduce that the function u defined above is solution of
b ⊂ Ω×]0, T [
ut + (u + u2 )x + uxxxxt = 0 in Q
u(x, t) = 0
b
in Σ
ux (x, t) = 0
b
in Σ
u(x, 0) = u0 (x)
(4.3)
in Ω0
satisfying u ∈ L∞ (0, T ; H02 (Ωt )), ut ∈ L∞ (0, T ; H02 (Ωt )) and
Z
Z
Z
2
ut φ dx dt + (u + u )x φ dx dt +
uxxt φxx dx dt = 0,
b
Q
b
Q
b
Q
for all φ ∈ L2 (0, T ; H02 (Ωt )); u(x, 0) = u0 (x). This result is summarized in the
following theorem.
EJDE-2004/35
ROSENAU AND BENJAMIN-BONA-MAHONY EQUATIONS
11
b a mixed domain defined
Theorem 4.1. For each u0 ∈ H02 (Ω0 ) ∩ H 4 (Ω0 ), and Q
b → R satisfying u ∈ L∞ (0, T ; H 2 (Ωt )),
as above, there exists a function u : Q
0
∞
2
ut ∈ L (0, T ; H0 (Ωt )) and
Z
Z
Z
2
ut φ dx dt + (u + u )x φ dx dt +
uxxt φxx dx dt = 0,
b
Q
2
for all φ ∈ L
(0, T ; H02 (Ωt ));
b
Q
b
Q
0
u(x, 0) = u (x) for all x ∈ Ω0 .
In analogous way we have the following result
b a mixed domain defined as
Theorem 4.2. For each u0 ∈ H01 (Ω0 ) ∩ H 2 (Ω0 ), and Q
b
above, there exists a unique function u : Q → R, satisfying u ∈ L∞ (0, T ; H01 (Ωt )),
ut ∈ L∞ (0, T ; H01 (Ωt )) and
Z
Z
Z
ut φ dx dt + (u + u2 )x φ dx dt +
uxt φx dx dt = 0,
b
Q
b
Q
b
Q
for all φ ∈ L2 (0, T ; H01 (Ωt )); u(x, 0) = u0 (x), for all x ∈ Ω0 .
Acknowledgement. We wish to acknowledge the anonymous referee at the Electronic Journal of Differential Equations, for his constructive remarks and corrections
on the original manuscript.
References
[1] Adams, R. A.; Sobolev Spaces, Academic, New York, 1975.
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arbitrary dimensions, Nonlinear Analysis TMA, 9 (1985), 861-865.
[3] Bona, J. L. and Dougalis, V.A.; An initial boundary-value problem for a model equation for
propagation of long waves, J. Math. Analysis Appl. 75 (1980), 503-522.
[4] Brill, H.; A semilinear Sobolev equation in a Banach space, J. Diff. Eqns., (1977) 412-425.
[5] Caldas, C. S. Q., Limaco, J. and Barreto, K.; Linear thermoelastic system in noncylindrical
domains, Funckcialaj Ekvacioj, 1 (1999), 115-127.
[6] Friedman, A.; Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[7] Goldstein, J. A. Semigrups of Linear Operators and Applications, Oxford University, New
York, 1985.
[8] Goldstein, J. A.; Mixed problems for the generalized Benjamin-Bona-Mahony equation, Nonlinear Analysis TMA, 4 (1980), 665-675.
[9] Lions, J. L.; Une remarque sur les problèmes d’ èvolution linéaires dans des domaines non
cylindriques, Revue Roumaine de Math. Pures et Appliquèes, V. 9, (1964) 11-18.
[10] Lions, J. L. and Magenes E.; Problèmes aux Limites non Homogènes et Applications V. 1,
Dunod, Paris 1968.
[11] Medeiros, L. A., Limaco J. and Bezerra S.; Vibrations of Elastic Strings, J. of Computational
Analysis and Application 4(2002), No. 2, 91-127, No. 3, 211-263.
[12] Park, M. A.; On the Rosenau Equation, Mat. Aplic. Comp. V. 9, (1990), 145-152.
[13] Rosenau, P. H.; Dynamics of dense discrete systems, Prog. Theoretical Phys., 79 (1988),
1028-1042.
Rioco K. Barreto
Instituto de Matemática - UFF, Rua Mário Santos Braga s/no , CEP: 24020-140, Niterói,
RJ, Brasil
E-mail address: rikaba@vm.uff.br
Cruz S. Q. de Caldas,
Instituto de Matemática - UFF, Rua Mário Santos Braga s/no , CEP: 24020-140, Niterói,
RJ, Brasil
E-mail address: gmacruz@vm.uff.br
12
R. K. BARRETO, C. DE CALDAS, P. GAMBOA, & J. LIMACO
EJDE-2004/35
Pedro Gamboa
Instituto de Matemática - UFRJ, Caixa Postal 68530, CEP 21945-970, Rio de Janeiro,
RJ, Brasil
E-mail address: pgamboa@dmm.im.ufrj.br
Juan Limaco
Instituto de Matemática - UFF, Rua Mário Santos Braga s/no , CEP: 24020-140, Niterói,
RJ, Brasil
E-mail address: juanbrj@hotmail.com