A heteroclinic orbit connecting traveling waves pertaining to different nonlinearities in a channel with decreasing cross section

In this paper we consider a semilinear parabolic equation in an infinite cylinder with shrinking cross section. The portion where the domain is not a straight half-cylinder is assumed to be compact. The spatially varying nonlinearity is such that it connects two (spatially independent) bistable nonlinearities in a compact set in space. We prove that, given such a setting, a traveling wave obeying the equation with the one bistable nonlinearity and starting at the respective side of the decreasing cylinder, will converge to a traveling wave solution prescribed by the nonlinearity on the other side.