Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth
Tommaso Leonori, Martina Magliocca
TL;DR
The paper advances the theory of uniqueness for nonlinear parabolic Cauchy--Dirichlet problems with p-Laplacian-type operators and superlinear gradient growth by developing a renormalized framework capable of handling unbounded solutions and irregular data. It combines a linearization approach (valid for $1<p\le 2$) with a convexity-based technique (for $p\ge 2$), deriving precise data and function-space conditions under which comparison principles hold. The results identify a sharp energy-space uniqueness class determined by a power $\gamma(q,p,N)$, and they provide counterexamples in the $p=2$ case to illustrate the necessity of this class. The work thus generalizes elliptic unbounded-solution results to the parabolic setting, offering detailed regimes and regularity assumptions that ensure well-posedness in the presence of gradient-driven, superlinear lower-order terms.
Abstract
In this paper we deal with uniqueness of solutions to the following problem \[ \begin{cases} \begin{split} & u_t-Δ_p u=H(t,x,\nabla u) &\quad \text{in}\quad Q_T,\\ & u (t,x) =0 &\quad \text{on}\quad(0,T)\times \partial Ω,\\ & u(0,x)=u_0(x) &\quad \displaystyle\text{in }\quad Ω\end{split} \end{cases} \] where $Q_T=(0,T)\times Ω$ is the parabolic cylinder, $Ω$ is an open subset of $\mathbb{R}^N$, $N\ge2$, $1<p<N$, and the right hand side $\displaystyle H(t,x,ξ):(0,T)\timesΩ\times \mathbb{R}^N\to \mathbb{R}$ exhibits a superlinear growth with respect to the gradient term.