Gradient Hölder regularity in mixed local and nonlocal linear parabolic problem

Abstract: We prove the local H\"older regularity of weak solutions to the mixed local nonlocal parabolic equation of the form \begin{equation*} u_t-\Delta u+\text{P.V.}\int_{\mathbb{R}^{n}} {\frac{u(x,t)-u(y,t)}{{\left|x-y\right|}^{n+2s}}}dy=0, \end{equation*} where $0<s<1$; for every exponent $\alpha_0\in(0,1)$. Here, $\Delta$ is the usual Laplace operator. Next, we show that the gradients of weak solutions are also $\alpha$-H\"older continuous for some $\alpha\in (0,1)$. Our approach is purely analytic and it is based on perturbation techniques.