  We study inelastic decay of bosonic excitations in a Luttinger liquid. In a
model with linear excitation spectrum the decay rate diverges. We show that
this difficulty is resolved when the interaction between constituent particles
is strong, and the excitation spectrum is nonlinear. Although at low energies
the nonlinearity is weak, it regularizes the divergence in the decay rate. We
develop a theoretical description of the approach of the system to thermal
equilibrium. The typical relaxation rate scales as the fifth power of
temperature.
