Gradient estimates for solutions of parabolic differential equations degenerating at infinity. (English) Zbl 1152.35013

Summary: 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 \(\Omega\subset\mathbb{R}^n\), \(n\geq 2\), 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.


35B45 A priori estimates in context of PDEs
35P05 General topics in linear spectral theory for PDEs
35K65 Degenerate parabolic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)