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

Abstract

In this paper we deal with uniqueness of solutions to the following problem \[ \begin{cases} \begin{split} & u_t-\Delta_p u=H(t,x,\nabla u) &\quad \text{in}\quad Q_T,\\ & u (t,x) =0 &\quad \text{on}\quad(0,T)\times \partial \Omega,\\ & u(0,x)=u_0(x) &\quad \displaystyle\text{in }\quad \Omega \end{split} \end{cases} \] where $Q_T=(0,T)\times \Omega$ is the parabolic cylinder, $\Omega$ is an open subset of $\mathbb{R}^N$, $N\ge2$, $1<p<N$, and the right hand side $\displaystyle H(t,x,\xi):(0,T)\times\Omega \times \mathbb{R}^N\to \mathbb{R}$ exhibits a superlinear growth with respect to the gradient term.