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

TL;DR

This paper investigates the regularity and pointwise behavior of weak solutions to a mixed local and nonlocal double phase parabolic equation, establishing boundedness, lower semicontinuity, and pointwise properties under certain conditions.

Contribution

It provides new regularity results for weak solutions of mixed local and nonlocal parabolic equations, including boundedness and semicontinuity, using De Giorgi-Nash-Moser techniques.

Findings

Weak solutions are locally bounded via De Giorgi-Nash-Moser iteration.

Pointwise behavior and lower semicontinuity are established for solutions with Hölder continuous coefficients.

Results apply to both weak subsolutions and supersolutions.

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.