EXPONENTIAL DICHOTOMIES FOR SOLITARY-WAVE SOLUTIONS OF SEMILINEAR ELLIPTIC-EQUATIONS ON INFINITE CYLINDERS

In applications, solitary-wave solutions of semilinear elliptic equati ons Delta u + g(u, del u) = 0 (x,y) is an element of R x Omega in infi nite cylinders frequently arise as travelling waves of parabolic equat ions. As such, their bifurcations are an increasing issue. Interpretin g elliptic equations on infinite cylinders as dynamical systems in x h as proved very useful. Still, there are major obstacles in obtaining, for instance, bifurcation results similar to those for ordinary differ ential equations. In this article, persistence and continuation of exp onential dichotomies for linear elliptic equations is proved. With thi s technique at hands, Lyapunov-Schmidt reduction near solitary waves c an be applied. As an example, existence of shift dynamics near solitar y waves is shown if a perturbation mu h(x,u,del u) periodic in x is ad ded. (C) 1997 Academic Press.