Differential and Integral Equations, Volume 8, Number 6, July 1995, pp. 1385 – 1394.
GLOBAL EXISTENCE OF SUBMANIFOLDS OF SOLUTIONS
OF NONLINEAR SECOND ORDER DIFFERENTIAL SYSTEMS
Giancarlo Cantarelli1
Dipartimento di Matematica, Università degli studi di Parma, Via M. D’Azeglio 85, 43100 Parma, Italy
(Submitted by: James Serrin)
Abstract. Several works, quoted in the introduction, provide criteria which establish global
existence in the future of all solutions of nonlinear second order differential systems. The
aim of the present paper is to establish further su⇥cient conditions for global existence in
the future for submanifolds of solutions.
1. Introduction. We consider scalar functions G, F and an N -vector Q (N ⇧ 1)
defined as follows: G = G(u, p) is of class C 1 (RN ⇥ RN ), strictly convex in the
variable p for every u RN , and satisfies the conditions G(u, 0) = 0, Gp (u, 0) = 0
on RN ; F = F (t, u) is of class C 1 (RN ⇥ RN ); Q = Q(t, u, p) is continuous on
R+ ⇥ RN ⇥ RN . In the present paper we establish su⇤cient conditions for global
existence in the future of solutions u = u(t) of the second order nonlinear di⇥erential
system (see, e.g., [9])
(Gp (u, u⌅ ))⌅
Gu (u, u⌅ ) + Fu (t, u) = Q(t, u, u⌅ ),
(1.1)
where ( · )⌅ denotes di⇥erentiation with respect to the independent variable t R+ .
A function u(t) defined on an interval I (⌃ R+ ) is called a solution of (1.1) if u(t)
and Gp (u(t), u⌅ (t)) are functions of class C 1 (I) such that
(Gp (u(t), u⌅ (t)))⌅
Gu (u(t), u⌅ (t)) + Fu (t, u(t)) = Q(t, u(t), u⌅ (t)) on I.
We denote by H = H(u, p) the partial Legendre transform of the function G(u, p),
defined
H(u, p) = (Gp (u, p), p) G(u, p) on RN ⇥ RN ,
(1.2)
where (Gp , p) denotes the scalar product of vectors Gp , p RN . For every fixed
u RN the function H(u, p) is positive definite with respect to p, and along any
solution u(t) of (1.1) the following identity holds [12, Theorem 8]:
[H(u(t), u⌅ (t)) + F (t, u(t))]⌅ = (Q(t, u(t), u⌅ (t)), u⌅ (t)) + Ft (t, u(t)).
(1.3)
Su⇤cient conditions for global existence in the future of solutions of the system
(1.1) were first established in [4] and [5], and later in [11], in the particular case
Received March 1994.
This work was reported on at the International Conference on Differential Equations in August,
1993.
1 Partially supported by the Italian Ministero dell’Università e della Ricerca Scientifica e Tecnologica, fondi 40%.
AMS Subject Classification: 34A15, 34A34, 70H35.
1385