Interior Regularity of Mixed Local-Nonlocal Parabolic Semilinear Equations

In this paper we prove the existence, uniqueness and regularity of classical solutions \break of a semilinear parabolic equation with a mixed local and nonlocal diffusion operator \break $\mathcal{L}=-(-\mathrm{\Delta }{)}^s+\mathrm{\Delta }$ and Dirichlet boundary conditions. Here, $(-\mathrm{\Delta }{)}^s$ is the integral fractional laplacian and $\mathrm{\Delta }$ is the classic local laplacian. We then study the interior regularity of said solutions and conclude that they are Hölder continuous in both space and time, and they are $C_{loc}^{2,\alpha }$ in space for all positive times.