Local estimates for parabolic equations with nonlinear gradient terms

TL;DR

This paper establishes local estimates for solutions to certain parabolic equations with nonlinear gradient terms, focusing on equations with absorbing first-order terms in an unbounded domain.

Contribution

It provides new local a priori estimates for parabolic equations with nonlinear gradient terms involving absorption, extending existing results to broader classes of equations.

Findings

Derived local estimates for solutions with nonlinear gradient absorption

Extended analysis to equations with $q$ in (1,2]

Applicable to equations with initial data in local $L^1$ spaces

Abstract

In this paper we deal with local estimates for parabolic problems in $\mathbb{R}^N$ with absorbing first order terms, whose model is $\{ {l} u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{in}\, (0,T) \times \mathbb{R}^N\,, \\[1.5 ex] u(0,x)= u_0 (x) &box{in}\, \mathbb{R}^N.$ where $T>0$, $N\geq 2$, $1<q\leq 2$, $f(t,x)\in L^1( 0,T; L^1_{\rm loc} (\mathbb{R}^N))$ and $u_0\in L^1_{\rm loc} (\mathbb{R}^N)$.