  In this article, we study a semi-linear heat equation with the nonlinearity
which is the product of polynomial and logarithmic functions. Using the
invariance of the potential well(s), we have established the global existence
and exponential decay estimates of solutions in $L^2$ - norm without having any
restriction on the exponent in the source term under suitable conditions on the
initial data. Moreover, finite time blow up of solutions at subcritical,
critical and supercritical initial energy levels is also discussed.
