Local boundedness for forward-backward parabolic De Giorgi classes without assuming higher regularity

Abstract

We define a homogeneous De Giorgi class of order $p \geqslant 2$ that contains the solutions of two evolution equations of elliptic-parabolic and forward-backward parabolic type like $\rho (x,t) u_t + A u = 0$ and $(\rho (x,t) u)_t + A u = 0$, where $\rho$, for simplicity, takes values in the set $\{ -1, 0, 1 \}$, and $A$ a suitable monotone operator. For functions belonging to this class, we prove an unusual local boundedness result.