Volume 2009, Article ID 691496, 9 pagesdoi:10.1155/2009/691496 Research Article Blow-Up Results for a Nonlinear Hyperbolic Equation with Lewis Function Faramarz Tahamtani Department of M
Trang 1Volume 2009, Article ID 691496, 9 pages
doi:10.1155/2009/691496
Research Article
Blow-Up Results for a Nonlinear Hyperbolic
Equation with Lewis Function
Faramarz Tahamtani
Department of Mathematics, Shiraz University, Shiraz 71454, Iran
Correspondence should be addressed to Faramarz Tahamtani,tahamtani@susc.ac.ir
Received 17 February 2009; Accepted 28 September 2009
Recommended by Gary Lieberman
The initial boundary value problem for a nonlinear hyperbolic equation with Lewis function in a bounded domain is considered In this work, the main result is that the solution blows up in finite time if the initial data possesses suitable positive energy Moreover, the estimates of the lifespan of solutions are also given
Copyrightq 2009 Faramarz Tahamtani 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
1 Introduction
Let Ω be a bounded domain in Rn with smooth boundary ∂Ω We consider the initial boundary value problem for a nonlinear hyperbolic equation with Lewis function αx which
depends on spacial variable:
α xu tt − ρΔu t− div|∇u| m−2 ∇u fu, x ∈ Ω, t ≥ 0, 1.1
u x, 0 u0x, u t x, 0 u1x, x ∈ Ω, 1.3
where αx ≥ 0, ρ > 0, m ≥ 2, and f is a continuous function.
The large time behavior of solutions for nonlinear evolution equations has been considered by many authorsfor the relevant references one may consult with 1 14.
In the early 1970s, Levine3 considered the nonlinear wave equation of the form
Trang 2in a Hilbert space where P are A are positive linear operators defined on some dense subspace
of the Hilbert space and h is a gradient operator He introduced the concavity method and
showed that solutions with negative initial energy blow up in finite time This method was later improved by Kalantarov and Ladyzheskaya4 to accommodate more general cases Very recently, Zhou 10 considered the initial boundary value problem for a quasilinear parabolic equation with a generalized Lewis function which depends on both spacial variable and time He obtained the blowup of solutions with positive initial energy
In the case with zero initial energy Zhou11 obtained a blow-up result for a nonlinear wave equation inRn A global nonexistence result for a semilinear Petrovsky equation was given
in14
In this work, we consider blow-up results in finite time for solutions of problem1.1
-1.3 if the initial datas possesses suitable positive energy and obtain a precise estimate for the lifespan of solutions The proof of our technique is similar to the one in10 Moreover,
we also show the blowup of solution in finite time with nonpositive initial energy
Throughout this paper · X denotes the usual norm of L XΩ
The source term fu in 1.1 with the primitive
F u
u 0
satisfies
f u ≤ c0|u| p−1
, c0> 0, p > m ≥ 2, 1.6
β1mF u β2m|∇u| m−1 ∇u t ≤ pFu < ufu, β1> 1, β2> 0. 1.7 LetB be the best constant of Sobolev embedding inequality
from W01,m Ω to L PΩ
We need the following lemma in4,Lemma 2.1
Lemma 1.1 Suppose that a positive, twice differentiable function Ψt satisfies for t ≥ 0 the inequality
If Ψ0 > 0, Ψ0 > 0, then
Ψ −→ ∞ as t −→ t1< t2 Ψ0
Trang 32 Blow-Up Results
We set
λ0 c0Bm−1/p−m , E0 p − m
pm c0Bp−m/p−m . 2.1 The corresponding energy to the problem1.1-1.3 is given by
E t 1
m
Ω|∇u| m
dx 1
2
Ωα xu2
t dx −
and one can find that Et ≤ E0 easily from
Et −ρ∇u2
whence
E t E0 − ρ
t 0
∇u τ2
We note that from1.6 and 1.7, we have
E t ≥ 1
m ∇u m
m− c0
p u p
and by Sobolev inequality1.8, Et ≤ Gu p , t ≥ 0, where
Note that Gλ has the maximum value E0at λ0which are given in2.1
Adapting the idea of Zhou10, we have the following lemma
ux, t p > λ0, ∇ux, t m > c0λ p01/m 2.7
for all t ≥ 0.
0 Ω and u1∈ L2Ω satisfy
μ x :
Ωα xu0u1dx > 0. 2.8
Trang 4If 0 < E0 ≤ E0, then the global solution of the problem1.1–1.3 blows up in finite time and the
lifespan
T < 2
∇u02
2−p − 2
μ x
p − 22
Proof To prove the theorem, it suffices to show that the function
A t
α xu
2 2
ρ
t 0
∇u2
2dτ ρ T0− t∇u02
2 γt t02 2.10
satisfies the hypotheses of theLemma 1.1, where T0 > t, t0 > 0 and γ > 0 to be determined
later To achieve this goal let us observe
2
t 0
Ω∇u∇u τ dxdτ
t 0
d
dτ ∇u2
2dτ
∇u2
2− ∇u02
2.
2.11
Hence,
∇u2
2 2
t 0
Ω∇u∇u τ dxdτ ∇u02
Let us compute the derivatives At and At Thus one has
At 2
Ωα xuu t dx ρ∇u22− ρ∇u02
2 2γt t0
2
Ωα xuu t dx 2ρ
t 0
Ω∇u∇u τ dxdτ 2γ t t0,
2.13
and
At 2
α xu t
2 2
− 2∇u m
m 2
Ωuf udx 2γ
≥ 2
α xu t
2 2
− 2∇u m
m 2p
ΩF udx 2γ
≥p 2 α xu t
2 2
2 p
m− 1
∇u m
m − 2pEt 2γ
≥p 2
α xu t
2 2
ρ
t 0
∇u τ2
2dτ 2 p
m− 1
∇u m
m − 2pE0 2γ
2.14
Trang 5for all t ≥ 0 In the above assumption 1.7, the definition of energy functionals 2.2 and 2.4 has been used Then, due to2.1 and 2.7 and taking γ 2E0− E0,
At ≥p 2
α xu t
2 2
ρ
t 0
∇u τ2
2dτ γ 2.15
Hence At ≥ 0 for all t ≥ 0 and by assumption 2.8 we have
A0 2μ x γt0
Therefore At ≥ 0 for all t ≥ 0 and by the construction of At, it is clearly that
A t ≥
α xu
2 2
ρ
t 0
∇u2
2dτ γt t02
whence, A0 > 0 Thus for all a, b ∈ R2, from2.13, 2.15, and 2.17 we obtain
a2A t abAt p 2−1
b2At ≥ a2
α xu
2 2
ρ
t 0
∇u2
2dτ γ t t02
2ab
Ωα xuu t dx ρ
t 0
Ω∇u∇u τ dxdτ γ t t0
b2
α xu t
2 2
ρ
t 0
∇u τ2
2dτ γ
α xau bu t
2 2
ρ
t 0
a∇u b∇u τ2
2dτ γa t t0 b2
≥ 0,
2.18 which implies
At2− 4
p 2 A tAt ≤ 0. 2.19 Then usingLemma 1.1, one obtain that At → ∞ as
t −→ 4A0
p − 2
A0
2 αxu02
2 T0∇u02
2 γt2 0
p − 2
μ x γt0
Trang 6Now, we are in a position to choose suitable t0and T0 Let t0be a number that depends on p,
E0− E0, ∇u0L2Ω, and μx as
t0> 2∇u02
2−p − 2
μ x
p − 2
To choose T0, we may fix t0as
T0 2
α xu022 2T0∇u02
2 2γt2 0
p − 2
μ x γt0
α xu022 γt2
0
p − 2
μ x γt0
− 2∇u02
2
.
2.22
Thus, for t ≥ t0the lifespan T is estimated by
T < 2α xu022 2γt2
p − 2
μ x γt− 2∇u02
2
< 2∇u02
2−p − 2
μ x
p − 22
E0− E0 ,
2.23
which completes the proof
u0∈ W 1,m
Then the corresponding solution to1.1–1.3 blows up in finite time.
Proof Let
B t
α xu
2 2
ρ
t 0
∇u2
Trang 7Bt 2
Ωα xuu t dx ρ ∇u2
Bt 2
α xu t
2 2 2
Ωα xuu tt dx 2ρ
Ω∇u∇u t dx
2
α xu t
2 2
− 2∇u m
m 2
Ωuf udx
> 2
α xu t
2 2
− 2∇u m
m 2β1m
ΩF udx 2β2m
Ω|∇u| m−1 ∇u t dx
> 2
β1 1 α xu t
2 2
2β1− 1∇u m
m 2β2 d
dt ∇u m
m − 2β1mE0
> 2
β1− 1∇u m
m 2β2d
dt ∇u m
m − 2β1mE 0, t > 0,
2.27
where the left-hand side of assumption1.7 and the energy functional 2.2 have been used Taking the inequality2.27 and integrating this, we obtain
Bt > 2β1− 1t
0
∇u m
m dτ 2β2∇u m
m − 2β1mE 0t B0, t > 0. 2.28
By using Poincare-Friedrich’s inequality
u2
2≤ λ1∇u2
and Holder’s inequality
∇u m
m ≥ λ1M−m/2|Ω|1−m/2
Ωα xu2dx
m/2
t 0
∇u m
m dτ ≥ t1−m/2
t
0
∇u2
2dτ
m/2
Trang 8where M maxΩ|αx| Using 2.30 and 2.31, we find from 2.28 that
Bt ≥ 2β2λ1M −m/2|Ω|1−m/2
Ωα xu2dx
m/2
2β1− 1t1−m/2
t 0
∇u2
2dτ
m/2
− 2β1mE 0t B0
≥ 2β2λ1M−m/2|Ω|1−m/2t1−m/2
Ωα xu2dx
m/2
2β1− 1t1−m/2
t 0
∇u2
2dτ
m/2
− 2β1mE 0t B0, t > 1.
2.32
Since−2β1mE0t B0 → ∞ as t → ∞ so, there must be a t1> 1 such that
By inequality
a1 a2r < 2 r−1
a r1 a r
2
, r > 1 2.34 and by virtue of2.33 and using 2.32, we get
where
C min
22−m/2
β1− 1, 22−m/2β2λ1M−m/2|Ω|1−m/2
Therefore, there exits a positive constant
T
⎧
⎨
⎩
C exp t1, m 2,
such that
This completes the proof
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