Existence and non-existence of global solutions for nonlinear hyperbolic equations of higher order*
Chen Guowang, Wang Shubin
Abstract. The existence and uniqueness of classical global solution and blow up of non- global solution to the first boundary value problem and the second boundary value problem for the equation
utt−αuxx−βuxxtt=ϕ(ux)x
are proved. Finally, the results of the above problem are applied to the equation arising from nonlinear waves in elastic rods
utt−
a0+na1(ux)n−1
uxx−a2uxxtt= 0.
Keywords: nonlinear hyperbolic equation, initial boundary value problem, classical global solution, blow up of solutions
Classification: 35L35
1. Introduction
In the study of strain solitary waves in nonlinear elastic rods there exists a lon- gitudinal wave equation [1], [2]
(1.1) utt−h
a0+na1(ux)n−1i
uxx−a2uxxtt= 0,
where a0, a2 > 0 are constants, a1 is an arbitrary real number, n is a natural number. In [1], [2] the equation (1.1) is reduced approximately to KdV equation
(1.2) ut+uux+µuxxx= 0,
whereµis a constant. In [2], authors study the strain solitary waves of equation (1.2), but about the equation (1.1) there has not been any discussion. Obviously, the equation (1.1) is different from the equation (1.2). There are few results in dealing with the equation (1.1). The existence and uniqueness of the local classical solutions for the initial value problems and the first boundary value problems of the equation (1.1) have been proved by Galerkin’s method in [3].
* This project is supported by the National Natural Science Foundation of China and partially by the Natural Science Foundation of Henan Province.
In the present paper, we are going to consider the following initial boundary value problem
utt−αuxx−βuxxtt =ϕ(ux)x, x∈(0,1), t >0, (1.3)
u(0, t) =u(1, t) = 0, t≥0, (1.4)
u(x,0) =u0(x), ut(x,0) =u1(x), x∈[0,1], (1.5)
or the initial boundary value problem for the equation (1.3) ux(0, t) =ux(1, t) = 0, t≥0, (1.6)
u(x,0) =u0(x), ut(x,0) =u1(x), x∈[0,1], (1.7)
whereu(x, t) is an unknown function,α, β >0 are real constants,ϕ(s) is a given nonlinear function, u0(x) and u1(x) are given initial functions. Obviously, the equation (1.1) is a special case of the equation (1.3).
First of all, we reduce the problem (1.3)–(1.5) to an equivalent integral equa- tion by Green’s function of a boundary value problem for a second order ordinary differential equation, then making use of the contraction mapping principle we prove the existence and uniqueness of the local classical solutions for the integral equation in Section 2. In Section 3, under some conditions by use of a priori esti- mations of the solution we prove that the integral equation has a unique classical global solution, i.e. the problem (1.3)–(1.5) has a unique classical global solution.
In Section 4 the conditions of non-existence of global solutions are given. The ex- istence and non-existence theorems for the problem (1.3), (1.6), (1.7) are given in Section 5. In Section 6 the existence and non-existence theorems for the problem (1.1), (1.4), (1.5) and the problem (1.1), (1.6), (1.7) are given.
2. Existence and uniqueness of local solution for the problem (1.3)–(1.5)
Let K(x, ξ) be the Green’s function of the boundary value problem for the ordinary differential equation
y−βy′′= 0, y(0) =y(1) = 0, whereβ >0 is a real number, i.e.
(2.1) K(x, ξ) = 1
√βsinhh
√1β
i
sinhh
√1β(1−ξ)i sinhh
√1βxi
, 0≤x≤ξ, sinhh
√1βξi sinhh
√1β(1−x)i
, ξ≤x≤1.
Suppose thatu0(x) andu1(x) are appropriately smooth and satisfy the bound- ary condition (1.4),u(x, t) is the classical solution of the problem (1.3)–(1.5), then
the solution of the equation (1.3) satisfying the condition (1.4) satisfies the inte- gral equation
(2.2) utt(x, t) =α Z 1
0
K(x, ξ)uξξ(ξ, t)dξ+ Z 1
0
K(x, ξ)ϕ(uξ(ξ, t))ξdξ.
Hence the classical solution of the problem (1.3)–(1.5) should satisfy the integral equation
(2.3)
u(x, t) =u0(x) +u1(x)t+α Z t
0
Z 1
0
(t−τ)K(x, ξ)uξξ(ξ, τ)dξ dτ +
Z t
0
Z 1
0
(t−τ)K(x, ξ)ϕ(uξ(ξ, τ))ξdξ dτ.
Therefore any classical solution of the initial boundary value problem (1.3)–(1.5) is the solution of the integral equation (2.3). By use of the properties of Green’s functionK(x, ξ), it is easy to prove the following lemma.
Lemma 2.1. Suppose that u0(x), u1(x) ∈ C2[0,1], u0(0) = u0(1) = u1(0) = u1(1) = 0,ϕ(s)∈C1(R), and u(x, t)∈C([0, T];C2[0,1])is the solution of (2.3), thenu(x, t)must be the classical solution of the initial boundary value problem (1.3)–(1.5).
Now we are going to prove the existence of local classical solution for the integral equation (2.3) by the contraction mapping principle. For this purpose, we define the function space
X(T) =n
u(x, t)|u∈C([0, T];C2[0,1]), u(0, t) =u(1, t) = 0o , equipped with the norm defined by
kukX(T)= max
0≤t≤T
0max≤x≤1|ux(·, t)|+ max
0≤x≤1|uxx(·, t)|
=kukC([0,T];C2[0,1]), ∀u∈X(T).
It is easy to see thatX(t) is a Banach space.
LetU=ku0xkC1[0,1]+ku1xkC1[0,1]. We define the set P(U, T) =n
u|u∈X(T), kukX(T)≤2U+ 1o .
Obviously,P(U, T) is nonempty bounded closed convex set for eachU, T >0. We define the mapS as follows
(2.4)
Sw=u0(x) +u1(x)t+α Z t
0
Z 1
0
(t−τ)K(x, ξ)wξξ(ξ, τ)dξ dτ +
Z t
0
Z 1
0
(t−τ)K(x, ξ)ϕ(wξ(ξ, τ))ξdξ dτ, ∀w∈P(U, T).
Obviously,S mapsX(T) intoX(T). Our goal is to show thatS has a unique fixed point inP(U, T) for appropriately chosenT. For this purpose we employ the contraction mapping principle.
Lemma 2.2. Suppose thatu0, u1∈C2[0,1],u0(0) =u0(1) =u1(0) =u1(1) = 0 andϕ(s)∈C2(R), thenSmapsP(U, T)intoP(U, T)andS:P(U, T)→P(U, T) is strictly contractive if T is appropriately small relative toU.
Proof: Integrating by parts with respect toξin (2.4), we get
(2.5)
Sw=u0(x) +u1(x)t−α Z t
0
Z 1
0
(t−τ)Kξ(x, ξ)wξ(ξ, τ)dξ dτ
− Z t
0
Z 1
0
(t−τ)Kξ(x, ξ)ϕ(wξ(ξ, τ))dξ dτ.
Differentiating (2.5) with respect tox, we have
(2.6)
(Sw)x =u0x(x) +u1x(x)t−α β
Z t
0
(t−τ)wx(x, τ)dτ
−α Z t
0
Z 1
0
(t−τ)Kξx(x, ξ)wξ(ξ, τ)dξ dτ
−1 β
Z t
0
(t−τ)ϕ(wx(x, τ))dτ
− Z t
0
Z 1
0 (t−τ)Kξx(x, ξ)ϕ(wx(x, τ))dξ dτ.
Differentiating (2.6) also with respect tox, we obtain
(2.7)
(Sw)xx=u0xx(x) +u1xx(x)t−α β
Z t
0
(t−τ)wxx(x, τ)dτ
−α Z t
0
Z 1
0
(t−τ)Kξxx(x, ξ)wξ(ξ, τ)dξ dτ
−1 β
Z t
0
(t−τ)ϕ(wx(x, τ))xdτ
− Z t
0
Z 1
0 (t−τ)Kξxx(x, ξ)ϕ(wξ(ξ, τ))dξ dτ.
Let us defineφ: [0,∞)→[0,∞) by ϕ(η) = max
|s|≤η
|ϕ(s)|+|ϕ′(s)|+|ϕ′′(s)|
, ∀η≥0.
Observe thatφis continuous and nondecreasing on [0,∞). Using the boundedness of the Green’s functionK(x, ξ) and its derivatives which appear in (2.6) and (2.7), whenT ≤12, we get
kSwkX(T)≤U+U T+ α
β +C1α T2
2 (2U+ 1) +
1
β(2U+ 1) +C2 T2
2 φ(2U+ 1)
≤U+T[C3+C4φ(2U+ 1)](2U+ 1),
whereC1, C2, C3 andC4 are constants. IfT satisfies
(2.8) T ≤min
1
2, 1
2[C3+C4φ(2U+ 1)]
,
then kSwkX(T) ≤2U+ 1. Therefore, if (2.8) holds, then S maps P(U, T) into P(U, T).
Now we are going to prove thatS :P(U, T)→P(U, T) is strictly contractive.
LetT >0 andw1, w2∈P(U, T) be given. We have
(2.9)
(Sw1−Sw2)x=
=−α β
Z t
0
(t−τ)[w1x(x, τ)−w2x(x, τ)]dτ
−α Z t
0
Z 1
0
(t−τ)Kξx(x, ξ)[w1ξ(ξ, τ)−w2ξ(ξ, τ)]dξ dτ
−1 β
Z t
0
(t−τ)[ϕ(w1x(x, τ))−ϕ(w2x(x, τ))]dτ
− Z t
0
Z 1
0
(t−τ)Kξx(x, τ)[ϕ(w1ξ(ξ, τ))−ϕ(w2ξ(ξ, τ))]dξ dτ and
(2.10)
(Sw1−Sw2)xx =
=−α β
Z t
0
(t−τ)[w1xx(x, τ)−w2xx(x, τ)]dτ
−α Z t
0
Z 1
0 (t−τ)Kξxx(x, ξ)[w1ξ(ξ, τ)−w2ξ(ξ, τ)]dξ dτ
− 1 β
Z t
0
(t−τ)[ϕ′(w1x(x, τ))w1xx(x, τ)−ϕ′(w2x(x, τ))w2xx(x, τ)]dτ
− Z t
0
Z 1
0
(t−τ)Kξxx(ξ, τ)[ϕ(w1ξ(ξ, τ))−ϕ(w2ξ(ξ, τ))]dξ dτ.
From (2.9) and (2.10) it follows that
kSw1−Sw2kX(T)≤ {C5+C6φ(2U+ 1)}T2
2 kw1−w2kX(T), whereC5 andC6 are constants.
IfT satisfies
(2.11) T ≤min 1
2, 1
2[C3+C4φ(2U+ 1)], 1
pC5+C6φ(2U+ 1)
! ,
then
kSw1−Sw2kX(T)≤ 1
2kw1−w2kX(T).
The lemma is proved.
Theorem 2.1. Let the assumptions of Lemma 2.2 hold. Then the integral equation (2.3) has a unique solution u(x, t) ∈ C([0, T0);C2[0,1]), where [0, T0) is a maximal time interval. Moreover, if
(2.12) sup
t∈[0,T0) kuxkC1[0,1]+kuxtkC1[0,1]
<∞,
thenT0=∞.
Proof: It follows from Lemma 2.2 and the contraction mapping principle that, for appropriately chosen T > 0, S has a unique fixed point u(x, t) ∈ P(U, T) which is obviously a solution of the integral equation (2.3). It is easy to prove that for eachT′ >0, the equation (2.3) has at most one solution which belongs toX(T′).
Let [0, T0) be the maximal time interval of existence for u∈ X(T0). It only remains to show that if (2.12) is satisfied, thenT0=∞. This can be done in the usual way: If (2.12) holds andT0<∞, we can reapply the contraction mapping principle extending the solution to an interval [0, T0+δ],δ >0, which contradicts the assumption that [0, T0) is maximal.
Suppose that (2.12) holds andT0<∞. For anyT′ ∈[0, T0), we consider the integral equation
(2.13)
v(x, t) =u(x, T′) +ut(x, T′)t+α Z t
0
Z 1
0
(t−τ)K(x, ξ)vξξ(ξ, τ)dξ dτ +
Z t
0
Z 1
0
(t−τ)K(x, ξ)ϕ(vξ(x, τ))ξdξ dτ.
By virtue of (2.12),kux(·, T′)kC1[0,1]+kuxt(·, T′)kC1[0,1]is uniformly bounded in T′ ∈ [0, T0), which allows us to choose T∗ ∈ (0, T0), such that for each T′ ∈[0, T0), the integral equation (2.13) has a unique solution v(x, t) ∈X(T∗).
The existence of such aT∗ follows from Lemma 2.2 and the contraction mapping principle. In particular (2.8) and (2.11) reveal thatT∗ can be selected indepen- dently ofT′∈[0, T0). SetT′=T0−T2∗, letv denote the corresponding solution of (2.13), and defineu(x, t) : [0,b 1]×[0, T0+T2∗]→Rby
(2.14) bu(x, t) =
u(x, t), t∈[0, T′],
v(x, t−T′), t∈
T′, T0+T2∗ .
By construction, bu(x, t) is a solution of (2.3) on [0, T0 + T2∗], and by local uniqueness,buextends u. This violates the maximality of [0, T0). Hence if (2.12) holds,T0=∞.
This completes the proof of the theorem.
3. The classical global solution of the problem (1.3)–(1.5)
Lemma 3.1. Suppose that u0(x), u1(x) ∈ H01[0,1], ϕ(s) ∈ C1(R), then the classical solution of the problem(1.3)–(1.5)satisfies the following identity
(3.1)
E(t)≡ kutk2L2[0,1]+αkuxk2L2[0,1]+βkuxtk2L2[0,1]
+ 2 Z 1
0
Z ux
0
ϕ(s)ds dx
=ku1k2L2[0,1]+αku0xk2L2[0,1]+βkuxtk2L2[0,1]
+ 2 Z 1
0
Z u0x
0 ϕ(s)ds dx
≡E(0), ∀t∈[0, T].
Proof: Multiplying both sides of the equation (1.3) byut, integrating the prod- uct with respect toxover [0,1] and integrating by parts we get
(3.2)
d dt
kutk2L2[0,1]+αkuxk2L2[0,1]+βkuxtk2L2[0,1]
+ 2 Z 1
0
Z ux
0
ϕ(s)ds dx
= 0.
Integrating (3.2) with respect tot, we obtain (3.1). The lemma is proved.
Theorem 3.1. Suppose that the condition of Theorem 2.1 and the following condition
(3.3) |ϕ(s)| ≤A
Z s
0
ϕ(y)dy+B
hold, where A and B are positive constants. Then the problem (1.3)–(1.5) has a unique classical global solutionu(x, t).
Remark 3.1. The functionϕ(s) satisfying (3.3) exists. For example,ϕ(s) =es satisfies the inequality (3.3). ϕ(s) =rsn is the second example, where r > 0 is a real number andnis a natural number. Whennis an odd number,ϕ(s) =rsn satisfies the inequality (3.3), i.e.
(3.4) |rsn| ≤n
Z s
0
ryndy+ r n+ 1.
In fact, takingp=n+1n ,p′=n+ 1 and using Young’s inequality we have
|rsn|=r|sn| ≤r |s|np
p + 1 p′
=r n
n+ 1sn+1+ r n+ 1
=n Z s
0
ryndy+ r n+ 1.
Proof of Theorem 3.1: By virtue of Theorem 2.1, we are only required to prove that (2.12) holds. Integrating by parts in (2.3), we obtain
(3.5)
u(x, t) =u0(x) +u1(x)t−α Z t
0
Z 1
0
(t−τ)Kξ(x, ξ)uξ(ξ, τ)dξ dτ
− Z t
0
Z 1
0 (t−τ)Kξ(x, ξ)ϕ(uξ(ξ, τ))dξ dτ.
It follows from (3.5) that
(3.6)
ux(x, t) =u0x(x) +u1x(x)t−α β
Z t
0
(t−τ)ux(x, τ)dτ
−α Z t
0
Z 1
0 (t−τ)Kξx(x, ξ)uξ(ξ, τ)dξ dτ
−1 β
Z t
0
(t−τ)ϕ(ux(x, τ))dτ
− Z t
0
Z 1
0
(t−τ)Kξx(x, ξ)ϕ(uξ(ξ, τ))dξ dτ,
(3.7)
uxtt(x, t) =−α
βux(x, t)−α Z 1
0
Kξx(x, ξ)uξ(ξ, t)dξ
− 1
βϕ(ux(x, t))− Z 1
0
Kξx(x, ξ)ϕ(uξ(ξ, t))dξ.
Multiplying both sides of (3.7) byuxtwe get
(3.8) d dt
h
u2xt+α βu2x+2
β Z ux
0
ϕ(s)dsi
= 2h
−α Z 1
0
Kξx(x, ξ)uξ(ξ, t)dξ
− Z 1
0
Kξx(x, ξ)ϕ(uξ(ξ, t))dξi uxt.
Let us denoteu21x(x) +αβu20x(x) +β2|Ru0x(x)
0 ϕ(s)ds|byE1(x). Integrating both sides of (3.8) with respect tot and making use of the conditions (3.3) and (3.1), we can obtain
u2xt+α βu2x+ 2
β Z ux
0
ϕ(s)ds
≤E1(x) + 2 Z t
0
h−α Z 1
0
Kξx(x, ξ)uξ(ξ, τ)dξ (3.9)
− Z 1
0
Kξx(x, ξ)ϕ(uξ(ξ, τ))dξ i
uxtdτ
≤E1(x) + Z t
0
nhZ 1
0 4αKξx2 (x, ξ)dξi12hZ 1
0
αu2x(x, τ)dxi12 +C7
Z 1
0 |ϕ(ux(x, τ))|dxo
|uxt|dτ
≤E1(x) + Z t
0
n C8+α
Z 1
0
u2x(x, τ)dx +C7
Z 1
0
h A
Z ux(x,τ)
0 ϕ(s)ds+Bi dxo
|uxt|dτ
≤E1(x) + Z t
0
C9+C10E(0) |uxt|dτ
≤E1(x) +1
4[C9+C10E(0)]2T+ Z t
0
u2xtdτ.
Multiplying both sides of (3.9) byA, adding the product to 2Bβ and using (3.3), we get
(3.10) Au2xt+Aα β u2x+ 2
β|ϕ(ux)| ≤M1(T) +A Z t
0
u2xtdτ, whereM1(T) is a constant dependent onT.
It follows from (3.10) by Gronwall’s inequality that Au2xt+Aα
β u2x+2
β|ϕ(ux)| ≤M1(T)eAT. Therefore
(3.11) sup
0≤t≤TkuxkC[0,1]+ sup
0≤t≤TkuxtkC[0,1]+ sup
0≤t≤Tkϕ(ux)kC[0,1]≤M2(T).
Differentiating (3.6) with respect tox, we obtain
(3.12)
uxx(x, t) =u0xx(x) +u1xx(x)t−α β
Z t
0 (t−τ)uxx(x, τ)dτ
−α Z t
0
Z 1
0
(t−τ)Kξxx(x, ξ)uξ(ξ, τ)dξ dτ
−1 β
Z t
0
(t−τ)ϕ′(ux(x, τ))uxx(x, τ)dτ
− Z t
0
Z 1 0
(t−τ)Kξxx(x, ξ)ϕ(uξ(ξ, τ))dξ dτ.
It follows from (3.12) that
|uxx(x, t)| ≤ max
0≤x≤1|u0xx(x)|+ max
0≤x≤1|u1xx|T+C11T2+C12T Z t
0 |uxx(x, τ)|dτ.
Making use of Gronwall’s inequality, we have
(3.13) sup
0≤t≤Tkuxx(·, t)kC[0,1]≤M3(T).
Differentiating (3.12) with respect tot, we obtain
(3.14)
uxxt(x, t) =u1xx(x)−α β
Z t
0
uxx(x, τ)dτ
−α Z t
0
Z 1
0
Kξxx(x, ξ)uξ(ξ, τ)dξ dτ
−1 β
Z t
0
ϕ′(ux(x, τ))uxx(x, τ)dτ
− Z t
0
Z 1
0
Kξxx(x, ξ)ϕ(uξ(ξ, τ))dξ dτ.
It follows from (3.14) that
(3.15) sup
0≤t≤Tkuxxt(·, t)kC[0,1]≤M4(T).
From (3.11), (3.13) and (3.15) it follows that sup
0≤t≤T kuxkC1[0,1]+kuxtkC1[0,1]
<∞.
By virtue of Theorem 2.1 and Lemma 2.1 we know that the problem (1.3)–(1.5) has a unique classical global solutionu(x, t). Theorem 3.1 is proved.
4. Blow-up of solutions of the problem (1.3)–(1.5)
In this section we are going to consider blow-up of solutions of the problem (1.3)-(1.5).
Theorem 4.1. Suppose that the following conditions hold:
(1) R1
0(u0u1+βu0xu1x)dx >0, (2) E(0)≤0,
(3) ϕ∈C1(R),ϕ(s)s≤2(2δ+ 1)Rs
0 ϕ(y)dy+ 2δαs2,
whereδ >0is a constant. Then the classical solutions of the problem(1.3)–(1.5) must blow up in finite time.
Proof: The proof is made by use of so called “concavity” arguments. Assume that u(x, t) is the classical solution of the problem (1.3)–(1.5) on [0,1]×[0, T].
Let
F(t) = Z 1
0
(u2+βu2x)dx.
We have
F′(t) = 2 Z 1
0 (uut+βuxuxt)dx,
F′′(t) = 2 Z 1
0
(u2t+βu2xt)dx+ 2 Z 1
0
(uutt+βuxuxtt)dx
= 2 Z 1
0
(u2t+βu2xt)dx+ 2 Z 1
0
u(utt−βuxxtt)dx.
Using Cauchy’s inequality, we see that (4.1) [F′(t)]2≤4hZ 1
0 (u2+βu2x)dxi hZ 1
0 (u2t+βu2xt)dxi .
Therefore, using (1.3) and (4.1), we find that
(4.2)
F F′′−(1 +δ)(F′)2 ≥Fnh 2
Z 1
0 (u2t+βu2xt)dx + 2
Z 1
0 u(αuxx+ϕ(ux)x)dxi
−4(1 +δ)h Z 1
0 (u2t +βu2xt)dxio
= 2Fh
− Z 1
0
(αu2x+ϕ(ux)ux)dx−(2δ+ 1) Z 1
0
(u2t+βu2xt)dxi .
Thus from (4.2), (3.1) and the conditions (2), (3) it follows that F F′′−(1 +δ)(F′)2 ≥2Fh
−2(2δ+ 1) Z 1
0
Z ux
0
φ(s)ds dx
−(2δ+ 1)α Z 1
0
u2xdx−(2δ+ 1) Z 1
0
(u2t +βu2xt)dxi
=−2F(2δ+ 1)E(0)≥0, t∈[0, T].
We see thatF(t)>0 for allt∈[0, T] and that from the condition (1),F′(0)>0.
From “concavity” arguments (see [4], [5]) we know that there exists a constantt0 such that
lim
t→t−0
kuk2L2[0,1]+βkuxk2L2[0,1]
= +∞ and
T < t0= ku0k2L2[0,1]+βku0xk2L2[0,1]
2δR1
0(u0u1+βu0xu1x)dx .
Theorem 4.1 is proved.
5. Initial boundary value problem (1.3), (1.6), (1.7)
This section is concerned with the problem (1.3), (1.6), (1.7). We consider problem (1.3), (1.6), (1.7) by the method used in Sections 1–4. Observe that
K(x, ξ) = 1
√βsinh√1β
cosh√x
βcosh1√−ξ
β , x≤ξ, cosh1√−x
β cosh√ξ
β, x > ξ,
is the Green’s function of the boundary value problem for the ordinary differential equation
y−βy′′= 0, y′(0) =y′(1) = 0, whereβ >0 is a real number.
The following theorems can be proved analogously as Theorems 3.1 and 4.1 above.
Theorem 5.1. Assume that the following conditions hold:
(1) u0(x), u1(x)∈C2[0,1]andu0x(0) =u0x(1) =u1x(0) =u1x(1) = 0, (2) ϕ(s)∈C2(R),ϕ(0) = 0and|ϕ(s)| ≤ARs
0 ϕ(y)dy+B,
whereA, B >0 are constants. Then the problem(1.3),(1.6),(1.7)has a unique classical global solutionu(x, t).
Theorem 5.2. Assume that the following conditions hold:
(1) R1
0(u0u1+βu0xu1x)dx >0, (2) E(0)≤0,
(3) ϕ∈C1(R),ϕ(0) = 0andϕ(s)s≤2(2δ+ 1)Rs
0 ϕ(s)ds+ 2δαs2, whereδ >0 is a constant.
Then the classical solutions of the problem (1.3), (1.6), (1.7) must blow up in finite time.
6. On the problems (1.1), (1.4), (1.5) and (1.1), (1.6), (1.7)
Here we apply the results of the problem (1.3), (1.4), (1.5) to the problem (1.1), (1.4), (1.5) and the results of the problem (1.3), (1.6), (1.7) to the problem (1.1), (1.6), (1.7).
Theorem 6.1. Suppose that
u0(x), u1(x)∈C2[0,1], u0(0) =u0(1) =u1(0) =u1(1) = 0, a0, a2>0.
(1) If n is an odd number, a1 > 0, then the problem (1.1), (1.4), (1.5) has a unique classical global solutionu(x, t).
(2)If n(n6= 1)is an odd number,a1<0,
ku1k2L2[0,1]+a0ku0xk2L2[0,1]+a2ku1xk2L2[0,1]
+ 2a1 n+ 1
Z 1
0
(u0x)n+1dx≡E(0)b ≤0
and the condition Z 1
0
(u0u1+a2u0xu1x)dx >0
holds, then the classical solutions of the problem(1.1),(1.4),(1.5)must blow up in finite time.
(3)If nis an even number,a16= 0,E(0)b ≤0 and the condition Z 1
0
(u0u1+a2u0xu1x)dx >0
holds, then the classical solutions of the problem(1.1),(1.4),(1.5)must blow up in finite time.
Theorem 6.2. Suppose thatu0(x), u1(x)∈C2[0,1],u0x(0) =u0x(1) =u1x(0) = u1x(1) = 0,a0, a2>0.
(1) If n is an odd number, a1 > 0, then the problem (1.1), (1.6), (1.7) has a unique classical global solutionu(x, t).
(2)If n(n6= 1)is an odd number,a1<0,E(0)b ≤0 and the condition Z 1
0
(u0u1+a2u0xu1x)dx >0
holds, then the classical solutions of the problem(1.1),(1.6),(1.7)must blow up in finite time.
(3)If nis an even number,a16= 0,E(0)b ≤0 and the condition Z 1
0
(u0u1+a2u0xu1x)dx >0
holds, then the classical solutions of the problem(1.1),(1.6),(1.7)must blow up in finite time.
References
[1] Zhuang Wei, Yang Guitong,Propagation of solitary waves in the nonlinear rods, Applied Mathematics and Mechanics7(1986), 571–581.
[2] Zhang Shangyuan, Zhuang Wei, Strain solitary waves in the nonlinear elastic rods (in Chinese), Acta Mechanica Sinica20(1988), 58–66.
[3] Chen Guowang, Yang Zhijian, Zhao Zhancai, Initial value problems and first boundary problems for a class of quasilinear wave equations, Acta Mathematicae Applicate Sinica9 (1993), 289–301.
[4] Levine H.A.,Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal.5(1974), 138–146.
[5] ,Instability&nonexistence of global solutions to nonlinear wave equations of the formP utt=−Au+F(u), Trans. of AMS192(1974), 1–21.
Department of Mathematics, Zhengzhou University, Zhengzhou 450052, China Department of Mathematics and Mechanics, Zhengzhou Institute of Technology, Zhengzhou 450002, China
(Received February 21, 1995)
