In this article, we prove two versions of the Lyapunov center theorem for symmetric potentials. We consider a second order autonomous system $$ \ddot q(t)=-\nabla U(q(t)) $$ in the presence of symmetries of a compact Lie group $\Gamma.$ We look for non-stationary periodic solutions of this system in a neighborhood of a $\Gamma$-orbit of critical points of the $\Gamma$-invariant potential $U.$ Our results generalize that of [13, 14]. As a topological tool, we use an infinite-dimensional generalization of the equivariant Conley index due to Izydorek, see [9].

We consider the Cauchy problem for the fractional nonlinear Schrödinger equation \[ \left\{ \begin{array} [c]{c}% i\partial_{t}u+\frac{1}{\alpha} \left| \partial_{x} \right| ^{\alpha }u=\lambda\left| u \right| ^{2}u,\text{ }t>0, \quad x\in\mathbb{R}% \mathbf{,}\\ u \left( 0,x \right) =u_{0} \left( x \right) , \quad x\in\mathbb{R}% \mathbf{,}% \end{array} \right. \] where $\lambda\in\mathbb{R},$ the order of the fractional derivative $\alpha\in\left( 1,\frac{3}{2} \right) .$ We obtain the large time asymptotic behavior of solutions which has a logarithmic phase modifications for a large time comparing with the linear problem.

We study quasi-positive dissipative reaction diffusion systems with polynomial growth and show several criteria for global-in-time existence of the classical solution uniformly bounded.