Local Continuity and Harnack’s Inequality for Double-Phase Parabolic Equations

Abstract

We consider parabolic equations of the form $$ u_{t}-\text{div} \left( |\nabla u|^{p-2}\nabla u+ a(x,t)|\nabla u|^{q-2}\nabla u\right)= 0, a(x,t)\geq 0. $$ In the range $\frac {2n}{n+1}<p< q$ we establish local properties of bounded solutions u to equation as local continuity and Harnack’s type inequality. These properties in a neighborhood of a point (x0, t0) depend on the value of the function a(x0, t0).