Volume 2009, Article ID 691496,9pages 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,[email protected] 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.
Copyrightq2009 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 functionαxwhich depends on spacial variable:
αxutt−ρΔut−div
|∇u|m−2∇u
fu, x∈Ω, t≥0, 1.1
u|∂Ω0, x∈∂Ω, t≥0, 1.2
ux,0 u0x, utx,0 u1x, x∈Ω, 1.3
whereαx≥0,ρ >0,m≥2, andfis 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 with1–14.
In the early 1970s, Levine3considered the nonlinear wave equation of the form
P uttAu hu 1.4
in a Hilbert space wherePareAare positive linear operators defined on some dense subspace of the Hilbert space andhis 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 Ladyzheskaya4to 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 Zhou11obtained 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- 1.3if 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 · Xdenotes the usual norm ofLXΩ.
The source termfuin1.1with the primitive
Fu u
0
fξdξ 1.5
satisfies
fu≤c0|u|p−1, c0>0, p > m≥2, 1.6 β1mFu β2m|∇u|m−1∇ut≤pFu< ufu, β1>1, β2>0. 1.7
LetBbe the best constant of Sobolev embedding inequality
up≤B∇um 1.8
fromW01,mΩtoLPΩ.
We need the following lemma in4,Lemma 2.1.
Lemma 1.1. Suppose that a positive, twice differentiable functionΨtsatisfies fort≥0 the inequality
ΨΨ−1 σΨ2 ≥0, σ >0. 1.9
IfΨ0>0,Ψ0>0, then
Ψ−→ ∞ as t−→t1< t2 Ψ0
σΨ0. 1.10
2. 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 problem1.1-1.3is given by
Et 1 m
Ω|∇u|mdx 1 2
Ωαxu2tdx−
ΩFudx, 2.2
and one can find thatEt≤E0easily from
Et −ρ∇u22≤0, 2.3
whence
Et E0−ρ t
0
∇uτ22dτ. 2.4
We note that from1.6and1.7, we have
Et≥ 1
m∇umm− c0
pupp, t≥0, 2.5
and by Sobolev inequality1.8,Et≤Gup,t≥0, where
Gλ mBm−1λm−c0p−1λp. 2.6 Note thatGλhas the maximum valueE0atλ0which are given in2.1.
Adapting the idea of Zhou10, we have the following lemma.
Lemma 2.1. Suppose thatux,0p> λ0andE0≤E0. Then
ux, tp> λ0, ∇ux, tm>c0λp01/m 2.7
for allt≥0.
Theorem 2.2. Forαx∈L∞Ω, suppose thatu0∈W01,mΩandu1∈L2Ωsatisfy
μx :
Ωαxu0u1dx >0. 2.8
If 0< E0≤E0, then the global solution of the problem1.1–1.3blows up in finite time and the lifespan
T < 2
∇u022− p−2
μx
p−22
E0−E0 . 2.9 Proof. To prove the theorem, it suffices to show that the function
At
αxu 2
2
ρ t
0
∇u22dτ ρT0−t∇u022 γt t02 2.10
satisfies the hypotheses of theLemma 1.1, whereT0 > t,t0 > 0 andγ > 0 to be determined later. To achieve this goal let us observe
2 t
0
Ω∇u∇uτdxdτ t
0
d
dτ∇u22dτ ∇u22− ∇u022.
2.11
Hence,
∇u22 2 t
0
Ω∇u∇uτdxdτ ∇u022. 2.12 Let us compute the derivativesAtandAt. Thus one has
At 2
Ωαxuutdx ρ∇u22−ρ∇u022 2γt t0 2
Ωαxuutdx 2ρ t
0
Ω∇u∇uτdxdτ 2γt t0,
2.13
and
At 2
αxut
2
2
−2∇umm 2
Ωufudx 2γ
≥2
αxut
2
2
−2∇umm 2p
ΩFudx 2γ
≥
p 2 αxut
2
2
2 p
m−1
∇umm−2pEt 2γ
≥
p 2
αxut
2
2
ρ t
0
∇uτ22dτ
2 p
m−1
∇umm−2pE0 2γ
2.14
for allt≥0. In the above assumption1.7, the definition of energy functionals2.2and2.4 has been used. Then, due to2.1and2.7and takingγ2E0−E0,
At≥
p 2
αxut
2
2
ρ t
0
∇uτ22dτ γ
. 2.15
HenceAt≥0 for allt≥0 and by assumption2.8we have A0 2
μx γt0
>0. 2.16
ThereforeAt≥0 for allt≥0 and by the construction ofAt, it is clearly that
At≥
αxu 2
2
ρ t
0
∇u22dτ γt t02, 2.17
whence,A0>0. Thus for alla, b∈R2, from2.13,2.15, and2.17we obtain
a2At abAt
p 2−1
b2At≥a2
αxu 2
2
ρ t
0
∇u22dτ γt t02
2ab
Ωαxuutdx ρ t
0
Ω∇u∇uτdxdτ γt t0
b2
αxut
2
2
ρ t
0
∇uτ22dτ γ
αxau but 2
2
ρ t
0
a∇u b∇uτ22dτ γat t0 b2
≥0,
2.18
which implies
At2− 4
p 2AtAt≤0. 2.19
Then usingLemma 1.1, one obtain thatAt → ∞as
t−→ 4A0 p−2
A0 2
αxu02
2 T0∇u022 γt20 p−2
μx γt0
. 2.20
Now, we are in a position to choose suitablet0andT0. Lett0be a number that depends onp, E0−E0,∇u0L2Ω, andμxas
t0> 2∇u022− p−2 μx
p−2
γ . 2.21
To chooseT0, we may fixt0as
T0 2
αxu022 2T0∇u022 2γt20 p−2
μx γt0
2
αxu022 γt20 p−2
μx γt0
−2∇u022.
2.22
Thus, fort≥t0the lifespanTis estimated by
T < 2
αxu022 2γt2 p−2
μx γt
−2∇u022
< 2∇u022− p−2 μx
p−22
E0−E0 ,
2.23
which completes the proof.
Theorem 2.3. Assume thatαx∈L∞Ωand the following conditions are valid:
u0∈W01,m, u1∈L2Ω, E0≤0. 2.24
Then the corresponding solution to1.1–1.3blows up in finite time.
Proof. Let
Bt
αxu 2
2
ρ t
0
∇u22dτ, 2.25
then
Bt 2
Ωαxuutdx ρ∇u22, 2.26
Bt 2
αxut
2
2
2
Ωαxuuttdx 2ρ
Ω∇u∇utdx 2
αxut
2
2
−2∇umm 2
Ωufudx
>2
αxut
2
2
−2∇umm 2β1m
ΩFudx 2β2m
Ω|∇u|m−1∇utdx
>2
β1 1 αxut
2
2
2 β1−1
∇umm 2β2 d
dt∇umm−2β1mE0
>2 β1−1
∇umm 2β2d
dt∇umm−2β1mE0, t >0,
2.27
where the left-hand side of assumption1.7and the energy functional2.2have been used.
Taking the inequality2.27and integrating this, we obtain
Bt>2
β1−1t
0
∇ummdτ 2β2∇umm−2β1mE0t B0, t >0. 2.28
By using Poincare-Friedrich’s inequality
u22≤λ1∇u22, 2.29
and Holder’s inequality
∇umm≥λ1M−m/2|Ω|1−m/2
Ωαxu2dx m/2
, 2.30
t
0
∇ummdτ≥t1−m/2 t
0
∇u22dτ m/2
, 2.31
whereMmaxΩ|αx|. Using2.30and2.31, we find from2.28that
Bt≥2β2λ1M−m/2|Ω|1−m/2
Ωαxu2dx m/2
2 β1−1
t1−m/2 t
0
∇u22dτ m/2
−2β1mE0t B0
≥2β2λ1M−m/2|Ω|1−m/2t1−m/2
Ωαxu2dx m/2
2 β1−1
t1−m/2 t
0
∇u22dτ m/2
−2β1mE0t B0, t >1.
2.32
Since−2β1mE0t B0 → ∞ast → ∞so, there must be at1>1 such that
−2β1mE0t B0≥0 ast > t1. 2.33
By inequality
a1 a2r <2r−1
ar1 ar2
, r >1 2.34
and by virtue of2.33and using2.32, we get
Bt≥Ct1−m/2Btm/2, 2.35
where
Cmin 22−m/2
β1−1
,22−m/2β2λ1M−m/2|Ω|1−m/2
. 2.36
Therefore, there exits a positive constant
T
⎧⎨
⎩
Cexpt1, m2,
Ct4−m/2−m1 , m >2, 2.37
such that
Bt−→ ∞ ast−→T−. 2.38
This completes the proof.
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