Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth

In this paper we deal with uniqueness of solutions to the following problem \[ \begincases \beginsplit & u_t-\Delta_p u=H(t,x,\nabla u) &\quad \textin\quad Q_T,\\ & u (t,x) =0 &\quad \texton\quad(0,T)\times \partial \Omega,\\ & u(0,x)=u_0(x) &\quad \displaystyle\textin \quad \Omega \endsplit \endcases \] where $Q_T=(0,T)\times \Omega$ is the parabolic cylinder, $\Omega$ is an open subset of $\mathbbR^N$, $N\ge2$, $1<p<N$, and the right hand side $\displaystyle H(t,x,\xi):(0,T)\times\Omega \times \mathbbR^N\to \mathbbR$ exhibits a superlinear growth with respect to the gradient term.