Regularity of weak solutions for mixed local and nonlocal double phase parabolic equations

Abstract

We study the mixed local and nonlocal double phase parabolic equation \begin{align*} \partial_t u(x,t)-\mathrm{div}(a(x,t)|\nabla u|^{q-2}\nabla u) +\mathcal{L}u(x,t)=0 \end{align*} in $Q_T=\Omega\times(0,T)$, where $\mathcal{L}$ is the nonlocal $p$-Laplace type operator. The local boundedness of weak solutions is proved by means of the De Giorgi-Nash-Moser iteration with the nonnegative coefficient function $a(x,t)$ being bounded. In addition, when $a(x,t)$ is H\"{o}lder continuous, we discuss the pointwise behavior and lower semicontinuity of weak supersolutions based on energy estimates and De Giorgi type lemma. Analogously, the corresponding results are also valid for weak subsolutions.