Gradient estimate and a Liouville theorem for a $p$-Laplacian evolution equation with a gradient nonlinearity

Abstract

In this paper, we establish a local gradient estimate for a $p$-Laplacian equation with a fast growing gradient nonlinearity. This estimate allows us to prove a parabolic Liouville theorem for ancient solutions (i.e., defined for $t < 0$) satisfying some growth restriction near infinity.