  We study the long time behavior of solutions of the Cauchy problem for
nonlinear reaction-diffusion equations in one space dimension with the
nonlinearity of bistable, ignition or monostable type. We prove a one-to-one
relation between the long time behavior of the solution and the limit value of
its energy for symmetric decreasing initial data in $L^2$ under minimal
assumptions on the nonlinearities. The obtained relation allows to establish
sharp threshold results between propagation and extinction for monotone
families of initial data in the considered general setting.
