Abstract
Location of the discrete and continuous spectrum of the operator corresponding to a boundary value problem for an elliptic system of equations in an unbounded cylinder is studied. Stability of multi‐dimensional travelling waves with respect to small perturbations is proved. These results allow us to prove global stability of travelling waves, i.e., that they describe large time asymptotic of solutions of the initial‐boundary value problem for a class of initial conditions, and to obtain a minimax representation for the wave velocity.