Global existence of submanifolds of solutions of nonlinear second order differential systems (Q1891576)

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).