**Kaj Nyström** ^{[1]}

*A backward in time Harnack inequality for non-negative solutions to fully non-linear parabolic equations*

**Pages:** 1-14

**Received:** 1 March 2013

**Accepted:** 21 June 2013

**Mathematics Subject Classification (2010):** 35K55.

**Keywords:** Fully non-linear parabolic equations, Lipschitz domain, Harnack inequality,
backward Harnack inequality.

**Author address:**

[1] : Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden

**Abstract:**
We consider fully non-linear
parabolic equations of the form
\[
Hu =F(D^2u(x,t),Du(x,t),x,t)-\partial_tu = 0
\]
in bounded space-time domains \(D\subset\mathbb R^{n+1}\), assuming
only \(F(0,0,x,t)=0\) and a uniform parbolicity condition on \(F\). For
domains of the form \(\Omega_T=\Omega\times (0,T)\), where
\(\Omega\subset\mathbb R^n\) is a bounded Lipschitz and \(T>0\), we
establish a scale-invariant backward in time Harnack inequality for
non-negative solutions vanishing on the lateral boundary. Our
argument rests on the comparison principle, the Harnack inequality
and local Hölder continuity estimates.

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