Stability of a quasi-variational system

Nonlinear Analysis 34 (1998) 701 – 717

Stability of a quasi-variational system Xiaodong Zhu Department of Mathematics, University of Nevada, Reno, NV 89557, USA Received 23 December 1996; accepted 17 March 1997

Keywords: Stability; Asymptotic stability; Variational identity

1. Introduction In this paper we consider the following quasi-variational system (∇G(u; u0 ))0 − ∇u G(u; u0 ) + f(t; u) = Q(t; u; u0 );

(1.1)

for t ∈ J = [R; ∞). Our purpose is to study stability of the rest state without assuming a damping condition on Q. First, we establish a stability theorem for (1.1). Second, we carry out a similar method to study the following inhomogeneous system, (∇G(u; u0 ))0 − ∇u G(u; u0 ) + f(t; u) = Q(t; u; u0 ) + e(t)

(1.2)

Finally, we study the stability of the rest state solution of a special case of system (1.1), in which Q is nonautonomous, nonlinear, and in particular not a damping. Canonical models for (1.1) are harmonic oscillators with damping or amplifying, namely, u00 + u = a(t)u0 ;

(1.3)

and quasilinear second order dierential systems of the form u00 + f(u) = Q(t; u; u0 ):

(1.4)

The system (1.1) also arises from the calculus of variations. Suppose that L is a Lagrangian function with L(t; u; p) = T (u; p) − V (t; u); 0362-546X/98/$19.00 ? 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 7 ) 0 0 5 8 9 - 0

(1.5)

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X. Zhu / Nonlinear Analysis 34 (1998) 701 – 717

where T and V are the corresponding kinetic energy and potential energy. According to the famous Euler–Lagrange equation we have Lu (t; u; u0 ) −

dLp (t; u; u0 ) = 0: dt

(1.6)

Considering in uence of damping and amplifying, we also have the following quasi Euler–Lagrange system Lu (t; u; u0 ) −

dLp (t; u; u0 ) = Q(t; u; u0 ): dt

(1.7)

If we take G(u; u0 ) = 1=2|u0 |2 in (1.3), and G(u; u0 ) = L(t; u; u0 ) in (1.6), (1.7), then (1.4), (1.6) and (1.7) become special cases of (1.1). Suppose that u = (u1 ; : : : ; uN ) is a vector in RN . A vector u is said to be a solution of (1.1) if u = (u1 ; : : : ; uN ) ∈ C 1 (J ; RN );

∇G(u(t); u0 (t)) ∈ C 1 (J ; RN );

and (1.1) holds. According to assumptions in Section 2, u ≡ 0 is a solution of (1.1), and it is called a rest state. Stability and asymptotic stability of the rest state of (1.1) have been carefully studied by Pucci and Serrin in a series of papers, see [1–3]. They considered in particular, an important special case of (1.1), u00 + A(t; u; u0 )u0 + f(u) = 0:

(1.8)

In (1.8) A is a continuous N × N damping matrix for which there exist nonnegative measurable functions ; : J → R such that (A(t; u; p)p; p) ≥ const.|A(t; u; p)p||p|

(1.9)

and (t)|p| ≤ |A(t; u; p)p| ≤ (t)|p|

(1.10)

for all t ∈ J , u; p ∈ R. The function f is a restoring force which is derivable from a potential, i.e. ∇F(u) = f(u);

(f(u); u) ≥ 0 for u 6= 0:

According to [2] the following conclusions hold for bounded solutions of (1.8). Theorem 1.1. (i) Suppose  is bounded on J . If  is continuous and of bounded variation, or if log  is uniformly Lipschitz continuous, then a sucient condition for the rest state of (1.8) to be a global attractor is that  ∈= L1 (J ); while a necessary condition is that  ∈= L1 (J ):

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(ii) Suppose 1=() is bounded on J . If 1= is continuous and of bounded variation, p or if 1= is absolutely continuous and |1=0 | ≤ const. =, then a sucient condition for the rest state of (1.8) to be a global attractor is that 1= ∈= L1 (J ); while a necessary condition is that 1= ∈= L1 (J ): It is noticed that the importance of the damping condition in [1–3], namely, they assumed (Q(t; u; p); p) ≤ 0:

(1.11)

Recently, Kosecki proved a stability theorem of the rest state for a quasi harmonic oscillator without assuming a damping condition on Q, see [4]. Kosecki’s result indicates that (1.11) may not be essential to stability when the initial data is very closed to the rest state. In the present paper, we extend Pucci and Serrin’s results to a system without assuming a damping condition on Q, and also generalize Kosecki’s result to a nonautonomous harmonic oscillator. In fact, instead of (1.10), we assume |(Q(t; u; p); p)| ≤ (t)(H (u; p) + F(t; u))

with  ∈ L1 (J );

(1.12)

where H (u; ·) is the Legendre transform of G(u; ·). Theorem 1.2. Suppose that u is a solution of system (1.1), and (H1 ) to (H3 ) (see Section 2) hold. If we assume additionally (I) (t)F(t; u) ≤ Ft (t; u) ≤ (t)(H (u; u0 ) + F(t; u)); for all t; u, and † ’− ∈ L1 (J );

+

∈ L1 (J ):

Then H (u; u0 ) + F(t; u) → l¿0

as

t → 0:

Comparing Theorem 1.2 with Theorem 1.1, we give up the damping assumption on Q, the bounded assumption on solutions, and also generalize the assumption on Ft . Since it is only assumed that  ∈ L1 (J ), we can show the rest state is stable but not necessarily asymptotically stable. The condition:  ∈ L1 (J ), is essential in the proof of the above theorem. However, the case,  6∈ L1 (J ), seems more delicate. In [4], the case that  ≡ const, had been considered for a special case of (1.1) (see 1.13). Kosecki’s result implies: when  6∈ L1 (J ), asymptotically stability of the rest state essentially depends on algebraic structure of Q, namely, the rst few Taylor’s coecients of Q. †

’− = min{0; ’} and

+

= max{0; }.

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X. Zhu / Nonlinear Analysis 34 (1998) 701 – 717

We consider the following special case of (1.1) ( 00 u + u = (t)G(u; u0 ) u(0) = u0

(1.13)

u0 (0) = u1

where  ∈ C 1 (J )

and

G(x; y) =

∞ X X

cij xi yj :

n=2 i+j=n

For small , the damping condition on Q is not a main factor in stability and asymptotic stability studies of (1.13). But  and algebraic structure of G(u; u0 ) are of the most important ones, especially, interaction between  and G(u; u0 ). Here we show in uence of both  and algebraic structure of G(u; u0 ) on stability and asymptotic stability of the rest state of (1.13). Using Theorem 1.2, we show the stability of the rest state, and also an oscillatory behavior of solutions around the rest state. Theorem 1.3. If  ∈ L1 (J ), then the rest state of (1.13) is stable but not asymptotically stable, provided that  is small enough; if 0¡m ≤  ≤ M; (0 )− ∈ L1 (R+ ) and sup A1 = A¡0; t≥0

where

  2 2 0 (c11 − 2c02 ) ; M1 = sup c11 c02 + 4 t≥0

 M2 = sup t≥0

0 c02 c20 2



A1 = c21 + 3c03 + (c11 c20 − 2c02 c11 ) + 3M1 + 3M2 ; then the rest state of (1.13) is asymptotically stable provided that  is small enough; if 0¡m ≤  ≤ M; (0 )+ ∈ L1 (R+ ) and sup A1 = A ¿ 0; t≥0

where



M1 = inf

t≥0

2 2 0 (c11 − 2c02 ) c11 c02 + 4



 ;

M2 = inf

t≥0

0 c02 c20 2



A1 = c21 + 3c03 + (c11 c20 − 2c02 c11 ) + 3M1 + 3M2 ; then the rest state of (1.13) is not asymptotically stable provided that  is small enough. We arrange the paper in the following order: Sections 2 and 3 are devoted to general stability theorems for homogeneous and inhomogeneous quasi-variational systems, in Section 4, with the help of Theorem 1.2, Kosecki’s result is generalized to nonautonomous equations, and conditions for stability, and asymptotic stability of the rest

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state are also given. In the course of the paper we need to construct a Lyapunov function for the system (1.1), and a modi ed version of the general variational identity which has been proved in [3]. 2. Preliminaries In this section, rst we impose some assumptions on G; f; Q. Second, we introduce notations used in this paper. At last we prove a lemma which is important for the course of the paper. The most important of assumptions which will be imposed on (1.1) are as follows. (H1 ) Suppose that G(u; ·) is a strictly convex function in RN ; G(u; 0) = 0, ∇G(u; 0) = 0 and the function ∇G(·; ·) is continuously dierentiable with respect to its variables. (H2 ) Suppose that F(t; u) satisfying ∇Fu (t; u) = f(t; u), and (∇u F(t; u); u) ≥ k¿0 when t ∈ J; |u| ≥ u0 ¿0, and F(t; 0) = 0 for all t ∈ J . (H3 ) Q(t; u; p) is continuous in all its variables, Q(t; u; 0) = 0, and
|(Q(t; u; p); p | ≤ (t)(H (u; p) + F(t; u)) with  ∈ L1 (J ); for all t, u, p in consideration. Here (·; ·) denotes the inner product in RN , and     @ @ @ @ ; ∇u = : ;:::; ;:::; ∇= @p1 @pN @u1 @uN Throughout this paper, we assume that (H1 )–(H3 ) hold, for additional assumption we will state when it is needed. Now let us de ne the Legendre transform H (u; ·) of G(u; ·), namely, H (u; p) = (∇G(u; p); p) − G(u; p):

(2.1)

By the strict convexity of G(t; u) and (H1 ), we have H (u; p)¿0 for all p 6= 0

and

H (u; 0) = 0

for all u ∈ RN :

If both u and p are functions of t, then (H (u; p))0 = ((∇G(u; p))0 ; p) − (∇u G(u; p); u0 ):

(2.2)

According to (H2 ) we get (f(t; u); u)¿0

for all t ∈ J; u 6= 0;

and F(t; u) is a potential function of the vector eld f(t; u) for xed t. For the case N = 1, it is obvious Z u f(t; s)ds: F(t; u) = 0

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X. Zhu / Nonlinear Analysis 34 (1998) 701 – 717

It is easy to check that F(t; 0) = 0, and F(t; u) ¿ 0 for all t ∈ J with u 6= 0. We use H (u; p) + F(t; u) as a Lyapunov function for the system (1.1). The condition (H3 ) is essential to the discussion in this paper. If  is in L1 (J ), the rest state is stable but not asymptotically stable. It is delicate when  6∈ L1 (J ). We will consider this case in Section 4. According to (H1 )– (H3 ) we get f(t; 0) = Q(t; u; 0) = 0

and ∇G(u; 0) = ∇u G(t; 0) = 0:

Hence the rest state, u ≡ 0, is a solution of (1.1). Suppose that u is a solution of (1.1). The following identity holds for u and a pair of scalar functions  and ! ∈ C 1 (J ; R), 0

{![H (u; u0 ) + F(t; u)] + (∇G(u; u0 ); u)} = !Ft (t; u) + !0 [H (u; u0 ) + F(t; u)]

−(f(t; u); u) + [(∇G(u; u0 ); u0 ) + (∇u G(u; u0 ); u)] +0 (∇G(u; u0 ); u) + !(Q(t; u; u0 ); u0 ) + (Q(t; u; u0 ); u):

(2.3)

The proof of this identity is essentially in [3]. If  = 0 and ! ≡ 1, we have an important special case of (2.3), that is, {H (u; u0 ) + F(t; u)}0 = (Q(t; u; u0 ); u0 ) + Ft (t; u):

(2.4)

Our proof of Theorem 1.2 mostly depends on choosing proper  and ! in (2.3). We end this section with a lemma. Lemma 2.1. Suppose that u is a solution of (1.1) and H (u; u0 ) + F(t; u) is uniformly bounded for all t ∈ J . We assume in addition |Ft (t; u)| ≤ (t)

for all

t ∈J

and ∈ L1 (J ):

Then H (u; u0 ) + F(t; u) → l ≥ 0

as t → ∞:

Proof. Applying the identity (2.3) with Z t (s)ds and (t) ≡ 0; !(t) = 1 + 0

we get {![H (u; u0 ) + F(t; u)]}0 = [H (u; u0 ) + F(t; u)] + !(Q(t; u; u0 ); u0 ) + !Ft (t; u):

(2.5)

By the fact  ∈ L1 (J ), we get that ! is positive and bounded. For all t ∈ J we obtain !(Q(t; u; u0 ); u0 ) ≤ C1 : 1 + 0 (t)(H (u; u ) + F(t; u))

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It is obvious (t)(H (u; u0 ) + F(t; u)) + !(Q(t; u; u0 ); u0 ) = (t)(H (u; u0 ) + F(t; u))[1 + !(Q(t; u; u0 ); u0 )=((t)(H (u; u0 ) + F(t; u)))]: Since H (u; u0 ) + F(t; u) is uniformly bounded, then (t)(H (u; u0 ) + F(t; u)) + !(t)(Q(t; u; u0 ); u0 ) ∈ L1 (J ): Integrating (2.5) with respect to t yields Z t !(s)(Q(s; u; u0 ); u0 )ds !(t)(H (u; u0 ) + F(t; u)) − ZR t (s)(H (u; u0 ) + F(s; u))ds − ZR t !(s)Ft (s; u)ds = C2 : − R

According to (H2 ) we have |!(t)Ft (t; u)| ≤ const. !(t) (t): Therefore, !(t)(H (u; u0 ) + F(t; u)) goes to a nonnegative constant as t → 0. Since ! has a limit as t → ∞, then there is a constant l such that H (u; u0 ) + F(t; u) → l ≥ 0

as

t → 0:

3. Stability theorems In this section, rst we prove a stability theorem for the following initial value problem ( (∇G(u; u0 ))0 − ∇u G(u; u0 ) + f(t; u) = Q(t; u; u0 ) (3.1) u(R) = u0 ; u0 (R) = u1 : Second, we carry over a similar method to prove stability theorems for the following inhomogeneous system ( (∇G(u; u0 ))0 − ∇u G(u; u0 ) + f(t; u) = Q(t; u; u0 ) + e(t) (3.2) u(R) = u0 ; u0 (R) = u1 : Proposition 3.1. Suppose that H (u; p) + F(t; u) → 0 as t → ∞. Then |u| + |p| → 0: Proof. By the strict convexity of G(u; ·) we have inf {H (u; p): |p| ≥ 1}¿0:

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X. Zhu / Nonlinear Analysis 34 (1998) 701 – 717

It follows easily that |p| → 0 if H (u; p) + F(t; u) → 0. According to (H2 ), there is a positive constant  such that (∇u F(t; u); u) ≥  for all t ∈ J , |u| ≥ u0 , which implies that |u| → 0. The proof is completed. In order to prove Theorem 1.2, we need a modi ed version of the variational identity (2.3). Applying (2.3) with the choice  = 0 gives the following identity, that is, {ln(![H (u; u0 ) + F(t; u)])}0 =

(Q(t; u; u0 ); u0 ) !0 + ! H (u; u0 ) + F(t; u) +

Ft (t; u) H (u; u0 ) + F(t; u)

for H (u; u0 ) + F(t; u)¿0. Proof of Theorem 1.2. By (2.4), (H3 ), and (I) we have (H (u; u0 ) + F(t; u))0 + ((t) − ’− (t))(H (u; u0 ) + F(t; u)) ≥ 0: Integrating both sides with respect to t yields Z t  0 − (H (u; u ) + F(t; u)) exp ( − ’ ) ≥ H (u0 ; u1 ) + F(R; u0 ): 0

Since ; ’− ∈ L1 (J ), then there is a constant m¿0 such that H (u; u0 ) + F(t; u) ≥ m

for all t ∈ J: Rt Applying (3.3) with !(t) = 1 + 0 (s)ds gives {ln(![H (u; u0 ) + F(t; u)])}0 =

(Q(t; u; u0 ); u0 ) Ft (t; u)  + + : ! H (u; u0 ) + F(t; u) H (u; u0 ) + F(t; u)

Integrating the above identity with respect to t yields Z t Z t (s) (Q(s; u; u0 ); u0 ) ln(![H (u; u0 ) + F(t; u)]) − ds − ds 0 0 !(s) 0 H (u; u ) + F(s; u) Z t Ft (s; u) ds = const. − 0 ) + F(s; u) H (u; u 0 Since =! is in L1 (J ) and (Q(t; u; u0 ); u0 ) H (u; u0 ) + F(t; u) ≤ (t); then (Q(t; u; u0 ); u0 ) ∈ L1 (J ): H (u; u0 ) + F(t; u)

(3.3)

X. Zhu / Nonlinear Analysis 34 (1998) 701 – 717

The assumption (I) implies Z t Z t Ft (s; u) ds ≤ 0 0 H (u; u ) + F(s; u) 0

709

+

(s) ds: m

Thus ln(![H (u; u0 ) + F(t; u)]) ≤ C

for all t ∈ J;

which implies that there is positive constant M such that H (u; u0 ) + F(t; u) ≤ M

for all t ∈ J:

Now the assumption (I ) can be replaced by M− ≤ Ft (t; u) ≤ M

+

:

This is the exact condition imposed on Ft (t; u) in Lemma 2.1. According to Lemma 2.1, there is a constant l ≥ m¿0 such that H (u; u0 ) + F(t; u) → l

as t → ∞:

Remark 3.1. Here we do not assume that u is bounded. In fact, we have proved that every solution of (3.1) exists globally and is bounded. If we replace (I) in Theorem 1.2 by the following condition ’(t)F(t; u) ≤ Ft (t; u) ≤ (t)F(t; u); the same conclusion still holds. According to the proof of Theorem 1.2, the following corollary is a straight forward conclusion. Corollary 3.2. With the same assumptions as Theorem 1.2, the rest state for (3.1) is stable but not asymptotically stable. Proof. According to the proof of Theorem 1.2, we have ln(![H (u; u0 ) + F(t; u)]) − ln[H (u0 ; u1 ) + F(R; u0 )] Z t Z t Z t (s) (Q(s; u; u0 ); u0 ) Ft (t; u) ds + ds + ds: = 0 ) + F(s; u) 0 ) + F(s; u) !(s) H (u; u H (u; u 0 0 0 Since every integrand on the right-hand side is in L1 (J ), then ![H (u; u0 ) + F(t; u)] = [H (u0 ; u1 ) + F(R; u0 )](t);

(3.4)

where 0 ≤ (t) ≤const. The above identity implies that H +F is small when H (u0 ; u1 )+ F(R; u0 ) is small. According to Proposition 3.1, H (u0 ; u1 ) + F(R; u0 ) is small provided that |u0 | + |u1 | small enough. If we assume that u is asymptotically stable, namely |u(t)| + |u0 (t)| → 0

as t → ∞;

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then the only solution of system (3.1) is u ≡ 0. Otherwise H (u; u0 ) + F(t; u) → 0 as t → ∞, which contradicts to the fact H (u; u0 ) + F(t; u) → l ¿ 0. Corollary 3.3. Suppose that (H3 ) holds only for |u|; |p| ≤ U in Theorem 1.2. Then the conclusion of Theorem 1.2 holds. Proof. Consider solutions of (3.1) with small initial data, i.e. ( (∇G(u; u0 ))0 − ∇u G(u; u0 ) + f(t; u) = Q(t; u; u0 ) u0 (R) = u1 :

u(R) = u0 ;

We claim that there is a constant C ≤ U such that |u|; |u0 | ≤ C for all t ∈ J , provided that  is small enough. This claim follows immediately from Lemma 2.1 and Proposition 3.1. Applying Theorem 1.2, we complete the proof. Considering the inhomogeneous system (3.2), we are able to show that H + F goes to a nonnegative limit as t → ∞ by a similar method. Theorem 3.4. Suppose that u is a bounded solution of (3.2) with |u| ≤ U . We assume additionally F(t; u) ≥ (e; u); e0 ∈ L1 (J );

e(t) → 0

as t → ∞;

and (I) holds. Then H (u; u0 ) + F(t; u) → k ≥ 0

as t → ∞:

Proof. For the system (3.2), the following identities hold {H (u; u0 ) + F(t; u)}0 = (Q(t; u; u0 ); u0 ) + Ft (t; u) + (e; u0 ); and (e; u0 ) = (e; u)0 − (e0 ; u): Thus {H (u; u0 ) + F(t; u) − (e; u)}0 = (Q(t; u; u0 ); u0 ) + Ft (t; u) − (e0 ; u): By (H2 ) and (H3 ) we get (Q(t; u; u0 ); u0 ) ≤ (t)(H (u; u0 ) + F(t; u)); and

 Z H (u; u0 ) + F(t; u) − (e; u) +

t

≤ ((t) +

+

∞

Ft (t; u) ≤

+U |e0 (s)|ds

(t))(H (u; u0 ) + F(t; u)):

0

+

(t)(H (u; u0 ) + F(t; u))

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Since F(t; u) ≥ (e; u) and |u| ≤ U , then  Z t  Z + 0 exp ((s) + (s))ds H (u; u ) + F(t; u) − (e; u) + 0

∞

t

0

0

U |e (s)|ds

≤ 0: Hence there is a nonnegative constant such that Z ∞ 0 |e0 (s)| ds → const. as t → ∞: H (u; u ) + F(t; u) − (e; u) + U t

According to the fact e(t) → 0 as t → ∞, we complete the proof. With a slight change, we prove the following theorem without the bounded assumption on solutions of (3.2). Theorem 3.5. Suppose that u is a solution of (3.2). We assume additionally (I) holds; and e ∈ L1 (J );

|p| ≤ C(1 + H (u; p)):

Then H (u; u0 ) + F(t; u) → k ≥ 0

as t → ∞:

Proof. It is easy to see {H (u; u0 ) + F(t; u)}0 ≤ ((t) +

+

(t))(H (u; u0 ) + F(t; u)) + C|e(t)|(1 + H (u; u0 )):

By Gronwall’s inequality H (u; u0 ) + F(t; u) ≤ M

for all t ∈ J:

Thus {H (u; u0 ) + F(t; u)}0 ≤ M ((t) +

+

(t)) + M1 |e(t)|;

with M1 = C(1 + M ). Every term on the right hand side is in L1 (J ), then  0 Z ∞ H (u; u0 ) + F(t; u) + (M ((s) + + (s)) + M1 |e(s)|) ds ≤ 0: t

Hence there is a nonnegative constant k such that H (u; u0 ) + F(t; u) → k as t → ∞. 4. An example This section is devoted to study stability of a special case of (1.1): a nonautonomous nonlinear harmonic oscillator. In [4], Kosecki studied stability properties of

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X. Zhu / Nonlinear Analysis 34 (1998) 701 – 717

the following equation with small initial data ( u00 + u = G(u; u0 ) u(0) = u0 ;

u0 (0) = u1 ;

(4.1)

where G(x; y) =

∞ X X

cij xi y j :

n=2 i+j=n

Here we use the idea of [4] to study the stability of a nonautonomous harmonic oscillator, that is ( u00 + u = (t)G(u; u0 ) (4.2) u(0) = u0 ; u0 (0) = u1 ; with  ∈ C 1 (J ). We focus on in uence of  and G(u; u0 ) on stability and asymptotic stability. It is obvious, for (4.2) we choose, H (u; u0 ) = u0 2 =2;

F(t; u) = u2 =2

and (Q(t; u; u0 ); u0 ) = (t)G(u; u0 )u0 :

We now consider the following two cases:  ∈ L1 (J )

or

0¡m ≤  ≤ M:

Case 1.  ∈ L1 (J ): In this case, we have G(u; u0 )u0 ≤ (u0 2 + u2 )

for |u|; |u0 | ≤ U;

provided that U is small enough. This is the exact condition assumed in Corollary 3.3. According to Corollary 3.3 the following theorem holds. Theorem 4.1. If  is small enough; then the rest state of (4:2) is stable but not asymptotic stable. The above theorem is an obvious conclusion of Corollary 3.3. In fact, we have a better description of solutions around the rest state. We actually show an oscillatory behavior of solutions around the rest state. Theorem 4.2. If  is small enough; then solutions of system (4:2) are oscillatory. Proof. According to Corollary 3:3; there is a positive constant L() such that u2 + u0 2 → L()

as t → ∞;

and L() = M2 + o(2 );

X. Zhu / Nonlinear Analysis 34 (1998) 701 – 717

713

where M only depends on ; u0 ; u1 . Hence there exists a positive number T such that u2 + u02 = L() + o(2 )

as t ≥ T:

In order to prove that u is oscillatory, we need to show that |u| doespnot tend p to a limit. We claim that |u| cannot always stay inside or outside the domain [ L()=4; L()=2]. Otherwise, suppose that there is a positive number T1 ¿T such that p |u|¿ L()=4 for all t ≥ T1 : By u00 = − u + O(2 ) we have p p u00 ≤ − L()=2 or u00 ≥ L()=2

for all t¿T1 :

It implies that |u0 (t)| → ∞ as t → ∞, which is impossible. Suppose that there is a positive number T2 ¿T such that p |u|¡ L()=2 for all t ≥ T2 : Then we have p |u0 |¿ L()=4

for all t ≥ T2 ;

which implies that |u| → ∞ as t → ∞. It is a contradiction. Above all, the proof is completed. Case 2. 0¡m ≤  ≤ M: Let us start with de ning an energy function E = E1 = − E2 − E3 − E4 ; where E1 =

u2 + u 0 2 ; 2

E2 = S − Z S=

E3 =

0

c11 03 2c02 3 c12 04 u + c02 uu0 2 + u + c03 u03 u − u ; 3 3 4

∞ uX

cn0 sn ds;

n=2

2 2 −(c11 − 2c02 ) 04 c02 c20 4 u − u ; 4 2

E4 = M2 uu03 − M1 u0 u3 ;   2 2 0 (c11 − 2c02 ) M1 = sup c11 c02 + 4 t≥0

 and

M2 = sup t≥0

0 c02 c20 2

 :

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X. Zhu / Nonlinear Analysis 34 (1998) 701 – 717

Next set A1 = c21 + 3c03 + (c11 c20 − 2c02 c11 ) + 3M1 + 3M2 ; A2 = c11 c02 +

2 2 0 (c11 − 2c02 ) − M2 ; 4

0 c02 c20 − M1 ; 2 and it is obvious that A2 and A3 are nonpositive. By calculation we have A3 =

0 E1 + A1 u2 u0 2 + A2 u04 + A3 u4 + O(5); 2

E0 = −

(4.3)

where O(5) is a term of u, u0 with degree at least 5. If u is a solution of (4.2), then u 0 2 u2 =

(u2 + u0 2 )2 − (uu03 − u0 u3 )0 + O(5): 8

(4.4)

Theorem 4.3. If (0 )− is in L1 (R+ ) and sup A1 = A¡0: t≥0

Then the rest state is asymptotically stable. Proof. First, we show that E¡C2 , where Z ∞  2(u02 + u12 ) exp C= −2M (0 )− (s)=2 (s) ds + 1: m 0 Suppose that there exists a positive number T3 such that E(T3 ) = C2 , and E¡C2 for t ≤ T3 . Set A 03 (uu − u0 u3 ): 8 Combine (4.3) with (4.4) to get W =E +

W0 ≤ −

0 A E1 + E12 + A2 u04 + A3 u4 + O(5): 2  8

For   1 we have E1 ≤ 2MW: By the de nitions of A, A2 and A3 we get A 2 E + A2 u04 + A3 u4 + O(5)¡0: 8 1 Thus W0 ≤ −

2M (0 )− W: 2

(4.5)

X. Zhu / Nonlinear Analysis 34 (1998) 701 – 717

715

Integrating (4.5) over [0; T3 ] gives, Z T3  W (T3 ) ≤ W (0) exp −2M (0 )− (s)=2 (s) ds : 0

Since  is bounded away from 0 and (0 )− ∈ L1 (R+ ), then Z T3  0 − 2 exp −2M ( ) (s)= (s) ds ≤ const. 0

Now, it is easy to see that W (T3 ) ≤

2(u02 + u12 ) exp m

Z 0

∞

 −2M (0 )− (s)=2 (s) ds ;

2

which implies that E(T3 )¡C . It is a contradiction. Second, we show that E → 0. Without loss of generality we assume that for all t ∈ R+ :

W (t)¿0

Since E ≤ C2 for all t ∈ R+ and   1, then A −2M (0 )− W0 + ≤ 2 W 2 W 64M 2 and

Z t  −A 0 − 2 exp −2M ( ) (s)= (s) ds 64M 2 0 Z t 0  1 exp −2M (0 )− (s)=2 (s) ds : ≤ W 0

Since (0 )− ∈ L1 (R+ ) and  ≥ m¿0, then Z t  −A 0 − 2 exp −2M ( ) (s)= (s) ds ≥ const. ¿ 0: 64M 2 0 Hence 1 exp W

Z 0

t

0 −

2

−2M ( ) (s)= (s) ds

 →∞

as t → ∞:

Therefore, 1=W (t) → ∞ as t → ∞, which implies E1 (t) → 0 as t → ∞. The proof is completed. Taking  ≡ 1, we get A1 = c11 (c20 + c02 ) + 3c03 + c21 , which is the exact condition in [4]. It is clear that the asymptotic stability of the rest state can be totally destroyed by changing cij , even  is xed. Next we consider the case when the rest state is not asymptotically stable. Let us de ne another energy function. E = E1 = − E2 − E3 − E4 ;

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X. Zhu / Nonlinear Analysis 34 (1998) 701 – 717

where E1 =

u2 + u 0 2 ; 2

E2 = G − Z G= E3 =

c11 03 2c02 3 c12 04 u + c02 uu0 2 + u + c03 u03 u − u ; 3 3 4

∞ uX

0 n=2

cn0 sn ds;

2 2 −(c11 − 2c02 ) 04 c02 c20 4 u − u ; 4 2

E4 = M2 uu03 − M1 u0 u3 ;   2 2 0 (c11 − 2c02 ) M1 = inf c11 c02 + t≥0 4

 and

M2 = inf

t≥0

0 c02 c20 2

 :

Let A1 = c21 + 3c03 + (c11 c20 − 2c02 c11 ) + 3M1 + 3M2 ; A2 = c11 c02 + A3 =

2 2 0 (c11 − 2c02 ) − M2 ; 4

0 c02 c20 − M1 : 2

By calculation we get E0 = −

0 E1 + A1 u2 u0 2 + A2 u04 + A3 u4 + O(5): 2

(4.6)

The above identity is almost the same as the identity (4.3) except A1 , A2 and A3 . It is obvious that both A2 and A3 are non-negative now. The following theorem holds. Theorem 4.4. If (0 )+ is in L1 (R+ ) and sup A1 = A¿0: t¿0

Then the rest state is not asymptotically stable. Proof. We only need to show that if E(t0 ) ≤ 2 and E(t0 ) 6= 0, then there exists t1 ≥ t0 such that E(t1 ) ≥ 2 . Suppose that such a number t1 does not exist. Thus E(t) ≤ 2 for all t ≥ t0 . Set $=E +

A 03 (uu − u0 u3 ): 8

X. Zhu / Nonlinear Analysis 34 (1998) 701 – 717

717

By (4.4) and (4.6) we have $0 ≥

−(0 )+ A E1 + E12 2  16

for all t ≥ t0 :

Since E ≤ 2 and   1, then E=$ ≥

1 m 2

and E=$ ≤ 2M:

Thus $0 +

2M (0 )+ Am 2 $ ≥ 0; $≥ 2  64

which implies that $¿0 for all t ≥ 0. By integration we obtain Z t  −Am2 0 + 2 exp −2M ( ) (s)= (s) ds 64 0 Z t 0  1 0 + 2 exp −2M ( ) (s)= (s) ds : ≥ $ 0 Since (0 )+ ∈ L1 (R+ ) and  ≥ m, then Z t  −Am2 0 + 2 exp −2M ( ) (s)= (s) ds ≤ const.¡0: 64 0 Thus 1 exp $

Z 0

t

 −2M (0 )+ (s)=2 (s) ds −

1 exp $(t0 )

Z 0

t0

−2M (0 )+ (s)=2 (s) ds



≤ c(t − t0 ); for some positive constant c, and all t ≥ t0 . Letting t → ∞ gives 1=$ → − ∞, which is a contradiction. The proof is completed. References [1] P. Pucci, J. Serrin, Continuation and limit properties for solution of strongly nonlinear second order dierential equations, Asymptotic Anal., 4 (1993) 97–160. [2] P. Pucci, J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order system, Acta Math., 170 (1993) 275 –307. [3] P. Pucci, J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986) 681–703. [4] R. Kosecki, The stability properties of a nonlinear harmonic oscillator, J. Math. Anal. Appl., 187(3) (1994) 716 –722.