Global existence of submanifolds of solutions of nonlinear second order differential systems. (English) Zbl 0824.34004

Let \(G= G(u,p)\) be a \(C^ 1(\mathbb{R}^ n\times \mathbb{R}^ n)\) function, strictly convex with respect to \(p\) for every \(u\) and \(G(u, 0)= 0\), \(G_ p(u, 0)= 0\). Moreover, let \(F= F(t, u)\) be of class \(C^ 1(\mathbb{R}^ n\times \mathbb{R}^ n)\) and \(Q= Q(t,u,p)\) be continuous on \(\mathbb{R}_+\times \mathbb{R}^ n\times \mathbb{R}^ n\) with values in \(\mathbb{R}^ n\). Consider the second order nonlinear differential system \[ (G_ p(u,u'))'- G_ u(u,u')+ F_ u(t,u)= Q(t,u,u').\tag{1} \] Denote by \(H= H(u,p)\) the partial Legendre transform of the function \(G(u,p)\) defined as \(H(u,p)= (G_ p(u,p),p)- G(u,p)\) (\((G_ p,u)\) denotes the scalar product of vectors \(G_ p\), \(p\in \mathbb{R}^ n\)).
Using the construction of Lyapunov functions based on the perturbation of the total energy \(H+F\) and applying some comparison methods the author obtains sufficient conditions for global existence in the future of submanifolds of solutions of the equation (1).


34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34C30 Manifolds of solutions of ODE (MSC2000)
34A34 Nonlinear ordinary differential equations and systems
70H03 Lagrange’s equations