Harnack's estimate for a mixed local-nonlocal doubly nonlinear parabolic equation

We establish Harnack's estimates for positive weak solutions to a mixed local and nonlocal doubly nonlinear parabolic equation, of the type & part;t (|u|(p-2)u) - div A (x, t, u, Du) + Lu(x, t) = 0, where the vector field A satisfies the p-ellipticity and growth conditions and the integrodifferential operator 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.