Volume 2010, Article ID 203248, 11 pagesdoi:10.1155/2010/203248 Research Article A Remark on the Blowup of Solutions to the Laplace Equations with Nonlinear Dynamical Boundary Conditions
Trang 1Volume 2010, Article ID 203248, 11 pages
doi:10.1155/2010/203248
Research Article
A Remark on the Blowup of Solutions to
the Laplace Equations with Nonlinear Dynamical Boundary Conditions
Hongwei Zhang and Qingying Hu
Department of Mathematics, Henan University of Technology, Zhengzhou 450001, China
Correspondence should be addressed to Hongwei Zhang,wei661@yahoo.com.cn
Received 24 April 2010; Revised 19 July 2010; Accepted 7 August 2010
Academic Editor: Zhitao Zhang
Copyrightq 2010 H Zhang and Q Hu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
We present some sufficient conditions of blowup of the solutions to Laplace equations with semilinear dynamical boundary conditions of hyperbolic type
1 Introduction
LetΩ be a bounded domain of R N , N ≥ 1, with a smooth boundary ∂Ω S S1∪S2, where S1
and S2are closed and disjoint and S1possesses positive measure We consider the following problem:
∂2u
∂t2 k ∂u
a ∂u
where a ≥ 0, b ≥ 0, a b 1, and k > 0 are constants, Δ is the Laplace operator with respect
to the space variables, and ∂/∂n is the outer unit normal derivative to boundary S u0, u1are
given initial functions For convenience, we take k 1 in this paper
Trang 2The problem1.1–1.4 can be used as models to describe the motion of a fluid in a container or to describe the displacement of a fluid in a medium without gravity; see 1 5 for more information In recent years, the problem has attracted a great deal of people Lions 6
following problem:
∂2u
∂t2 k ∂u
Hintermann 2
and uniqueness of the solution for problem 1.5–1.7 Cavalcanti et al 7 11
the existence and asymptotic behavior of solutions evolution problem on manifolds In this direction, the existence and asymptotic behavior of the related of evolution problem
hyperbolic problem with second-order boundary conditions
We will consider the blowup of the solution for problem1.1–1.4 with nonlinear
boundary source term gu Blowup of the solution for problem 1.1–1.4 was considered
by Kirane 3 1, by use of Jensen’s inequality and Glassey’s method 18
et al 19
type dynamical boundary inequality by the test function methods In this paper, we present some sufficient conditions of blowup of the solutions for the problem 1.1–1.4 when Ω is a
bounded domain and S2can be a nonempty set We use a different approach from those ones used in the prior literature 3,19
Another related problem to1.1–1.4 is the following problem:
∂u
∂t ∂u
of the solution of problem1.8–1.10 For more results concerning the related problem 1.8–
1.10, we refer the reader to 3,6,19–31
boundedness, asymptotic behavior, and nonexistence of global solutions for problem1.8–
1.10 were studied
In this paper, the definition of the usually space H1Ω, H s S, L p Ω, and L p S can
Trang 32 Blowup of the Solutions
In this paper, we always assume that the initial data u0∈ H s 1/2 S1, u1 ∈ H s S1, s > 1, and
g ∈ C and that the problem 1.1–1.4 possesses a unique local weak solution 2,3,6
u is in the class
u ∈ L∞
0, T; H s1Ω, u t ∈ L∞0, T, H s S1, u tt ∈ L∞
0, T; L2S1, 2.1 and the boundary conditions are satisfied in the trace sense 2
Lemma 2.1 see 33 t Ft, u, v t ≥ Ft, v, F ∈ C, t0 ≤ t < ∞, −∞ < u <
∞, and ut0 vt0 Then, vt ≥ ut, t ≥ t0.
Theorem 2.2 Suppose that ux, t is a weak solution of problem 1.1–1.4 and gs satisfies:
1 sgs ≥ KGs, where K > 2, Gs s
0g ρdρ, Gs ≥ β|s| p1, where β > 0, p > 1;
2 E0 u02
S1u12
S1b/au0S2−2S
1G σdσ ≤ −2/ K−2βC1p3−1 2/p−11−
e 1−p/44/p−1 < 0
where C1 mS1p1/p−1 Then, the solution of problem1.1–1.4 blows up in a finite time.
Proof Denote
E t u t2
S1 ∇u2
a u S2− 2
S1
then from1.1–1.4, we have
d
Hence
Let Ht ut2
S1t
0
τ
0us2
S1ds dτ Using condition1 ofTheorem 2.2, we have
˙
H t d
dt H t 2
S1
uu t dσ
t
0
us2
S1ds,
¨
H t d2
dt2H t 2
S1
u2t dσ 2
S1
uu tt dσ
S1
u2dσ
2
S1
u2t − u ∂u
∂n ugu 12u2
dσ
≥ 2
S
u2t − u ∂u
∂n KGu 12u2
dσ.
2.5
Trang 4Observing that
S1
u ∂u
∂n
Ω|∇u|2dxb
a
S2
K
S1
G udσ −E0 K − 2
S1
G udσ
S1
u2t dσ b
a
S2
u2dσ
Ω|∇u|2dx, 2.7
we know from2.5–2.7 that
¨
H t ≥ 4
S1
u2t dσ − 2E0
S1
u2dσ 2K − 2
S1
G udσ ≥ −2E0 2K − 2β
S1
|u| p1dσ.
2.8
It follows from2.8 that
˙
H t ≥ −2E0t 2K − 2β
t
0
S1
|u| p1dσ ds ˙H 0, 2.9
H t ≥ −E0t2 2K − 2β
t
0
τ
0
S1
|us| p1dσ ds dτ t ˙ H 0 H0, 2.10
where H0 u02
S1, ˙ H0 2S1u0u1dσ From2.8 and 2.10, we have
¨
H tHt≥2K−2β
S1
|u| p1dσ
t
0
τ
0
S1
|us| p1dσ ds dτ
t ˙ H 0 − E0t2 H0 − 2E0.
2.11
Using the inversion of the H ¨older inequality, we obtain
S1
|u| p1dσ≥
S1
|u|2dσ
p1/2
t
0
τ
0
S
|us| p1dσ ds dτ ≥ t
0
τ
0
S
|us|2dσ ds dτ
p1/2
1
2t
2mS1
p−1/2
. 2.13
Trang 5Substituting2.12 and 2.13 into 2.11, we have
¨
H t Ht
≥ 2K − 2βmS1p1/p−1
×
⎡
⎣
S1
|u|2dσ
p1/2
1
2t
0
τ
0
S1
|us| p1dσ ds dτ
2/p1⎤
⎦
t ˙ H 0 − E0t2 H0 − 2E0
≥ 2K − 2βmS1p1/p−1
⎡
⎣
S1
|u|2dσ
p1/2
0
τ
0
S1
|us| p1dσ ds dτ
p1/2⎤
⎦
t ˙ H 0 − E0t2 H0 − 2E0, t ≥ 1.
2.14 Noticing that
a b n≤ 2n−1a n b n , a > 0, b > 0, n > 1, 2.15
we have
¨
H t Ht ≥ 2 3−p/2 K − 2βmS1p1/p−1 H p1/2 t t ˙ H 0 − E0t2 H0 − 2E0.
2.16
We see from2.9 and 2.10 that ˙H t → ∞, Ht → ∞ as t → ∞ Therefore, there is a
t0≥ 1 such that
˙
Multiplying both sides of2.16 by 2 ˙H t and using 2.9, we get
d
dt
˙
H2t H2t≥ 1
p 325−p/2 K − 2βmS1p1/p−1 d
dt H
p3/2 t It, t ≥ t0,
2.18 where
I t −4E0t 2 ˙H0−E0t2 ˙H 0t H0 − 2E0
From2.18 we have
d dt
˙
H2t H2t − C2H p3/2 t≥ It, t ≥ t0, 2.20
Trang 6where C2 1/p 32 5−p/2 K − 2βmS1p1/p−1 Integrating2.20 over t, t0, we arrive at
˙
H2t H2t − C2H p3/2 t ≥
t
t0
I τdτ ˙ H2t0 H2t0 − C2H p3/2 t0, t ≥ t0.
2.21
Observe that when t → ∞, the right-hand side of 2.21 approaches to positive infinity
since It > 0 for sufficiently large t; hence, there is a t1 ≥ t0such that the right side of2.21
is larger than or equal to zero when t ≥ t1 We thus have
˙
H2t H2t ≥ C2H p3/2 t, t ≥ t1. 2.22 Extracting the square root of both sides of2.22 and noticing that ˙H tHt ≥ 0, we obtain
˙
H t Ht ≥ C3H p3/4 t ≥ C3t 1−p/2 H p3/4 t, t ≥ t1, 2.23
since 1− p < 0, t > t1> t0> 1, where C3C2.
Consider the following initial value problem of the Bernoulli equation:
˙
Z Z C3t 1−p/2 Z p3/4 , t ≥ t1, Z t1 Ht1. 2.24 Solving the problem2.24, we obtain the solution
Z t e −t−t1
H 1−p/4 t1 −p− 1
4
t
t1
C3τ 1−p/2 e 1−p/4τ−t1 dτ
4/1−p
e −t−t1 H t1J 4/1−p t, t ≥ t1,
2.25
where Jt 1 − p − 1/4H p−1/4 t1C3
t
t1τ 1−p/2 e 1−p/4τ−t1 dτ Obviously, J t1 1 > 0, and for t > t1 1
δ t p− 1
p−1/4 t1C3
t
t1
τ 1−p/2 e 1−p/4τ−t1 dτ
≥ p− 1
p−1/4 t1C3
t11
t1
τ 1−p/2 e 1−p/4τ−t1 dτ
≥ p− 1
p−1/4 t1C3t1 11−p/2
t11
t1
e 1−p/4τ−t1 dτ
H p−1/4 t1C3t1 11−p/2
1− e 1−p/4
.
2.26
Trang 7From2.10, we see that
H p−1/4 tt 1 1−p/2 ≥
−E0t2 ˙H 0t H0
t2 2t 1
p−1/4
−→ −E0p−1/4 2.27
as t → ∞ Take t1sufficiently large such that Hp1/4 t1t1 11−p/2 ≥ 1/2−E0p−1/4 It
follows from2.26 and the condition ofTheorem 2.2that
δ t ≥ 1
2−E0p−1/4 C3
1− e 1−p/4
Therefore,
By virtue of the continuity of Jt and the theorem of the intermediate values, there is a constant t1 < T ≤ t1 1 such that JT 0 Hence, Zt → ∞ as t → T− It follows from
Lemma 2.1that Ht ≥ Zt, t ≥ t1 Thus, H t → ∞ as t → T− The theorem is proved.
Theorem 2.3 Suppose that gs is a convex function, g0 0, gs ≥ ls p , where a is a real number
p > 1, and u x, t is a weak solution of problem 1.1–1.4
S1
u0σψ1σdσ α ≥ λ1
l
1/p−1
> 0,
S1
u1σψ1σdσ β > 0, 2.30
where ψ1 is the normalized eigenfunction (i.e., ψ1 ≥ 0,S1ψ1σdσ 1) corresponding the smallest
eigenvalue λ1> 0 of the following Steklov spectral problem [ 23 ]:
∂ψ
a ∂ψ
where Ω, S1, S2, k, a, b are defined as in Section 1 Then, the solution of problem1.1–1.4 blows up
in a finite time.
Proof Let
y t
S
Trang 8Then, y0 y0 α > 0, y t 0 y1 β > 0 It follows from 1.1–1.4 that yt satisfies
y tt −
S1
∂u
∂n ψ1dσ
S1
Using Green’s formula, we have
0
S
∂u
∂n ψ1dσ−
Ω∇u · ∇ψ1dx
S
∂u
∂n ψ1dσ−
S
u ∂ψ1
∂n dσ
Ωu Δψ1dx
S1
∂u
∂n ψ1dσ−
S1
u ∂ψ1
∂n dσ
S2
∂u
∂n ψ1dσ−
S2
u ∂ψ1
∂n dσ
Ωu Δψ1dx
B1 B2,
2.36
where we have used2.31 and the fact that ψ1is the eigenfunction of the problem1.1–1.4,
B1and B2are denoted as the expressions in the first and the second parenthesis, respectively From2.32, we have
B1
S1
∂u
∂n ψ1dσ − λ1
S1
If a 0, it is clear that B2 0 otherwise, by 1.3 and 2.33,
B2
S2
−b
a u ψ1dσ−
S2
u
−b
Therefore,2.36 implies that B1 0, that is,
S1
∂u
∂n ψ1dσ λ1
S1
Now,2.35 takes the form
y tt −λ1y
S1
From Jensen’s inequality and the condition gs ≥ ls p, we have
S
g uψ1dσ ≥ g
S
uψ1dσ
Trang 9
Substituting the above inequality into2.40, we get
y tt λ1y ≥ ly p , t > 0. 2.42
Since y0 α > 0, y t 0 β > 0, from the continuity of yt, it follows that there is a right
neighborhood0, δ of the point t 0, in which ˙yt > 0, and hence yt > y0 > 0 If there
exists a point t0 such that ˙yt > 0t ∈ 0, t0, but ˙yt0 0, then yt is monotonically
increasing on 0, t0 2.42 that on 0, t0
y tt ≥ yly p−1− λ1
≥ y0
ly0p−1− λ1
and thus y t t is monotonically increasing on 0, t0 0 0 Therefore,
˙yt > 0 and hence yt > y0as t > 0.
Multiplying both sides of2.42 by 2y tand integrating the product over
y2t ≥ 2l
p 1
y p1− y p1
0
− λ1
y2− y2 0
y2
1 By
Since By0 y2
1 > 0 and
B
y
2ly p − 2λ1y > 2y0
ly0p−1− λ1
then By > By0 > 0, ct > 0 Extracting the square root of both sides of 2.44, we have
y t≥
2l
p 1
y p1− y p1
0
− λ1
y2− y2 0
y2 1
−1/2
, t > 0. 2.46
Equation2.46
T ≤
∞
y0
2l
p 1
y p1− α p1
− λ1
y2− α2
β2
1/2
ds < ∞, 2.47
and yt → ∞ as t → T− The theorem is proved
Remark 2.4 The results of the above theorem hold when one considers1.1–1.4 with more general elliptic operator, like
Lu ≡ − divkx∇u cxu, 0 < k0≤ kx ≤ k1, c x ≥ 0, in Ω × 0, T, 2.48
Trang 10and the corresponding boundary conditions
∂2u
∂t2 kx ∂u
∂n gu, on S1× 0, T,
k x ∂u
∂n bu 0, bx ≥ 0, on S2× 0, T.
2.49
Acknowledgments
The authors are very grateful to the referee’s suggestions and comments The authors are supported by National Natural Science Foundation of China and Foundation of Henan University of Technology
References
Archive for Rational Mechanics and Analysis, vol 24, pp 352–362, 1967.
Society of Edinburgh Section A, vol 113, no 1-2, pp 43–60, 1989.
and hyperbolic type,” Hokkaido Mathematical Journal, vol 21, no 2, pp 221–229, 1992.
Mathematical Journal, vol 35, pp 260–275, 1932.
1969
equations on manifolds,” Di fferential and Integral Equations, vol 13, no 10–12, pp 1445–1458, 2000.
on manifolds with damping and source terms,” Journal of Mathematical Analysis and Applications, vol.
291, no 1, pp 109–127, 2004
wave equation on compact surfaces and locally distributed damping—a sharp result,” Transactions of
the American Mathematical Society, vol 361, no 9, pp 4561–4580, 2009.
wave equation on compact manifolds and locally distributed damping—a sharp result,” Journal of
Mathematical Analysis and Applications, vol 351, no 2, pp 661–674, 2009.
Ventcel problem with variable coefficients and dynamic boundary conditions,” Journal of Mathematical
Analysis and Applications, vol 328, no 2, pp 900–930, 2007.
stability for viscoelastic evolution problems on compact manifolds,” Journal of Computational Analysis
and Applications, vol 8, no 2, pp 173–193, 2006.
stability for viscoelastic evolution problems on compact manifolds II,” Journal of Computational
Analysis and Applications, vol 8, no 3, pp 287–301, 2006.
equation on the boundary,” Mathematical Methods in the Applied Sciences, vol 33, no 11, pp 1275–1283,
2010
de la Facult´e des Sciences de Toulouse, vol 11, no 1, pp 7–18, 2002.
condition,” Chinese Quarterly Journal of Mathematics, vol 24, no 3, pp 365–369, 2009.
Trang 11boundary damping,” Electronic Journal of Di fferential Equations, no 28, pp 1–10, 1998.
pp 183–203, 1973
with a dynamical boundary condition,” Boletim da Sociedade Paranaense de Matem´atica 3rd S´erie, vol.
22, no 2, pp 9–16, 2004
condition,” Acta Mathematica Universitatis Comenianae, vol 66, no 2, pp 321–328, 1997.
with semilinear dynamical boundary conditions of parabolic type,” Journal of Computational and
Applied Mathematics, vol 202, no 2, pp 414–434, 2007.
parabolic type,” Applied Mathematics and Computation, vol 191, no 1, pp 89–99, 2007.
parameters contained linearly in boundary conditions,” in Proceedings of the 8th International
Symposium on Algorithms and Computation (ISAAC ’97), Singapore, December 1997.
vol 210, no 3, pp 413–439, 1992
boundary condition,” Mathematical Methods in the Applied Sciences, vol 20, no 15, pp 1325–1333, 1997 nonlinear dynamical boundary condition,” in Topics in Nonlinear Analysis, vol 35 of Progr Nonlinear
Di fferential Equations Appl., pp 251–272, Birkh¨auser, Basel, Switzerland, 1999.
boundary conditions,” in Proceedings of the 4th International Conference on Large-Scale Scientific
Computing (LSSC ’03), vol 2907 of Lecture Notes in Computer Sciences, pp 105–123, Sozopol, Bulgaria,
June 2003
equation with semilinear dynamical boundary conditions,” Applied Mathematics and Computation, vol.
161, no 1, pp 69–91, 2005
manifolds,” Electronic Journal of Qualitative Theory of Di fferential Equations, no 13, pp 1–20, 2008.
of the London Mathematical Society, vol 93, no 2, pp 418–446, 2006.
Mathematik, vol 81, no 5, pp 567–574, 2003.
York, NY, USA, 1972
Scientiarum Naturalium Universitatis Jilinensis, vol 1, pp 257–293, 1960.
...condition,” Acta Mathematica Universitatis Comenianae, vol 66, no 2, pp 321–328, 1997.
with semilinear dynamical boundary conditions of parabolic type,” Journal of Computational...
boundary condition,” Mathematical Methods in the Applied Sciences, vol 20, no 15, pp 1325–1333, 1997 nonlinear dynamical boundary condition,” in Topics in Nonlinear Analysis, vol 35 of Progr...
The authors are very grateful to the referee’s suggestions and comments The authors are supported by National Natural Science Foundation of China and Foundation of Henan University of Technology