Gradient estimates for solutions of parabolic differential equations degenerating at infinity

Abstract

For $p\in (1,+\infty)$ we derive a weighted $L^p$ estimate for the (spatial) gradient of the solution $u$ of a degenerate parabolic differential equation. Here the underlying domain $\Omega\subset\mathbf{R}^n$, $n\ge 2$, is unbounded and the equation may degenerate only at infinity along some unbounded branch of $\Omega$. Our estimate is strictly related with the still-open problem of giving a concrete characterization of the interpolation space between $W^{2,p}(\Omega)$ and $L^p(\Omega)$ to which the (spatial) gradient of $u$ belongs.