  We consider equivariant wave maps from a wormhole spacetime into the
three-sphere. This toy-model is designed for gaining insight into the
dissipation-by-dispersion phenomena, in particular the soliton resolution
conjecture. We first prove that for each topological degree of the map there
exists a unique static solution (harmonic map) which is linearly stable. Then,
using the hyperboloidal formulation of the initial value problem, we give
numerical evidence that every solution starting from smooth initial data of any
topological degree evolves asymptotically to the harmonic map of the same
degree. The late-time asymptotics of this relaxation process is described in
detail.
