  We consider a blow-up solution for a strongly perturbed semilinear heat
equation with Sobolev subcritical power nonlinearity. Working in the framework
of similarity variables, we find a Lyapunov functional for the problem. Using
this Lyapunov functional, we derive the blow-up rate and the blow-up limit of
the solution. We also classify all asymptotic behaviors of the solution at the
singularity and give precisely blow-up profiles corresponding to these
behaviors. Finally, we attain the blow-up profile numerically, thanks to a new
mesh-refinement algorithm inspired by the rescaling method of Berger and Kohn
in 1988. Note that our method is applicable to more general equations, in
particular those with no scaling invariance.
