  In this work, we investigate the qualitative properties as uniqueness,
regularity and stabilization of the weak solution to the nonlinear parabolic
problem involving general $p(x)$-homogeneous operators: \begin{equation*}
  \left\{
  \begin{alignedat}{2}
  {} \frac{q}{2q-1}\partial_t(u^{2q-1}) -\nabla.\, a(x, \nabla u)
  & {}= f(x,u) + h(t,x) u^{q-1}
  && \quad\mbox{ in } \, (0,T) \times \Omega;
  u & {}> 0
  && \quad\mbox{ in }\, (0,T) \times \Omega ;
  u & {}= 0
  && \quad\mbox{ on }\, (0,T) \times \partial\Omega;
  u(0,.)&{}= u_0
  && \quad\ \mbox{in}\, \ \Omega.
  \end{alignedat}
  \right. \end{equation*} Thanks to the Picone's identity obtained in [10], we
prove new results about comparison principles which yield a priori estimates,
positivity and uniqueness of weak solutions.
