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

TL;DR

This paper investigates the uniqueness of solutions to a parabolic PDE with superlinear gradient growth, providing comparison results for unbounded solutions under Dirichlet boundary conditions.

Contribution

It establishes uniqueness criteria for unbounded solutions to a nonlinear parabolic PDE with superlinear gradient dependence, extending previous results to broader solution classes.

Findings

Comparison results for unbounded solutions

Uniqueness criteria for solutions with superlinear gradient growth

Extension of existing theory to unbounded solution frameworks

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.