American Institute of Mathematical Sciences

$u_{t}+u_x+u^pu_{x}-\nu u_{x x}-u_{x xt}=0$ ($*$)

is considered for $\nu > 0$ and $p \geq 1.$ Complementing recent studies which determined sharp decay rates for these kind of nonlinear, dispersive, dissipative wave equations, the present study concentrates on the more detailed aspects of the long-term structure of solutions. Scattering results are obtained which show enhanced decay of the difference between a solution of ($*$) and an associated linear problem. This in turn leads to explicit expressions for the large-time asymptotics of various norms of solutions of these equations for general initial data for $p > 1$ as well as for suitably restricted data for $p \geq 1.$ Higher-order temporal asymptotics of solutions are also obtained. Our techniques may also be applied to the generalized Korteweg-de Vries-Burgers equation

$u_{t}+u_x+u^pu_{x}-\nu u_{x x}+u_{x x x}=0,$ ($**$)

and in this case our results overlap with those of Dix. The decay of solutions in the spatial variable $x$ for both ($*$) and ($**$) is also considered.

$ x''+\partial_{x}V(x,y)=0,\ x\in R^{n};\qquad \ -y''+\partial_{y}V(x,y)=0,\ y\in R^{m}, $

where $V\in C^{1}(R^{n+m},R)$. We are interested in the existence of a brake periodic orbit of prescribed Hamiltonian. This problem may be considered as a generalization of the classical case $m=0$, for which are known many existence results (Seifert theorem and its developments), and also a generalization of the case $n=1$, whose study has been initiated by Hofer and Toland and which is still under investigation. Here we assume at least a quadratic growth on $V$ in order to find a brake orbit via a linking variational principle.

$ y'(t)=f(y(t),u(t))\,\quad y(t) \in \mathbb{R}^n,\ u(t)\in U \subset \mathbb{R}^d. $

We assume $f(x,U)$ to be a convex set with $C^1$ boundary for all $x\in\mathbb{R}^n$ and the target $\kappa$ to satisfy an interior sphere condition. For such problems we prove necessary and sufficient optimality conditions using the properties of the minimum time function $T(x)$. Moreover, we give a local description of the singular set of $T$.