Harnack’s estimate for a mixed local–nonlocal doubly nonlinear parabolic equation

Abstract

We establish Harnack’s estimates for positive weak solutions to a mixed local and nonlocal doubly nonlinear parabolic equation, of the type $$\begin{aligned} \partial _t\left( |u|^{p-2}u\right) -{\textrm{div}}\,\textbf{A}\left( x,t,u,Du\right) +\mathcal {L}u(x,t)=0, \end{aligned}$$where the vector field \(\textbf{A}\) satisfies the p-ellipticity and growth conditions and the integro-differential operator \(\mathcal {L}\) whose model is the fractional p-Laplacian. All results presented in this paper are provided by using sharp tools in the doubly nonlinear theory together with quantitative estimates.