Guiding potentials and bounded solutions of differential equations on finite-dimensional non-compact manifolds
Let \((M^n,g)\) be an \(n\)-dimensional compact manifold, \(X\) be a vector field on \(M^n\), that is, a \(TM^n\)-valued map on \(M^n\). We denote by \(Xf\) the derivative of a real-valued function \(f\) on \(M^n\) along \(X\), and defined as \(Xf=df(X)=\langle\nabla_g f,X\rangle\). We denote by \(X(t,x)\) a smooth set of variables \(t\in\mathbb R\) and \(x\in M^n\) vector field on \(M^n\). The author assumes that \(X(t,x)\) is just a locally Lipschitz vector field on \((M^n,g)\), \(\Omega\) is a domain in \(M^n\) which is homeomorphic to an open ball in \(\mathbb R^n\), \(\mathcal{O}\) is a relative compact domain containing the closure of \(\Omega\), and \(\varphi\) be a real-valued smooth proper function on \(M^n\), i.e., the preimage of a bounded subset in \(\mathbb R\) is relatively compact in \(M^n\). Then he states that the following first-order equation: \[ \frac{dx(t)}{dt}=X(t,x(t))\tag{\(*\)} \] has at least one uniformly bounded solution for all positive \(t\) whenever the following conditions are satisfied. \begin{itemize} \item For the derivative \(\varphi\) along direction \(X(t,x)\) on the \(M^n\) for all \(t>0\), the inequality \(X\varphi<C\) holds for some positive constant \(C\). \item On the boundary of \(\Omega\) for all \(t>0\), the inequality \(X\varphi>0\) holds on the boundary of \(\mathcal{O}\) for all \(t>0\). \item We have \(X\varphi<0\), all points on the boundary \(\partial\Omega\) are no return for solutions to \((*)\) with initial conditions on \(\partial\Omega\). \item For the covector field \(d\varphi\), we have that the index of type \(\widehat{\gamma}(d\varphi,\partial\Omega)\) does not vanish. \end{itemize}
scientific article; zbMATH DE number 7374475