Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders
The authors investigate the elliptic equation \[ u_{xx}+ \Delta_yu+ g(y,u,u_x,\nabla_y u)= 0,\quad (x,y)\in \mathbb{R}\times\Omega\tag{1} \] in the infinite cylinders \(\mathbb{R}\times \Omega\), where \(\Omega\) is an open and bounded subset in \(\mathbb{R}^n\) with boundary conditions prescribed on \(\mathbb{R}\times \partial\Omega\). The main idea of the paper is to consider problem (1) as a dynamical system with respect to \(x\). In terms of this idea, results on stability and continuity of the exponential dichotomy for problem (1) are obtained. The presentation is supplied with an example.
dynamical system treatment of elliptic equations
scientific article; zbMATH DE number 1109521
scientific article