Regularity Theory for Mixed Local and Nonlocal Parabolic p-Laplace Equations

Abstract

We investigate the mixed local and nonlocal parabolic p-Laplace equation $$\begin{aligned} \partial _t u(x,t)-\Delta _p u(x,t)+\mathcal {L}u(x,t)=0, \end{aligned}$$where \(\Delta _p\) is the usual local p-Laplace operator and \(\mathcal {L}\) is the nonlocal p-Laplace type operator. Based on the combination of suitable Caccioppoli-type inequality and Logarithmic Lemma with a De Giorgi–Nash–Moser iteration, we establish the local boundedness and Hölder continuity of weak solutions for such equations.