Fujita exponent and blow-up rate for a mixed local and nonlocal heat equation

In this paper we consider the blow-up problem for a mixed local-nonlocal diffusion operator, $$u_t=a\mathrm{\Delta }u-b(-\mathrm{\Delta }{)}^{s}u+u^p\mathrm{.}$$ We show that the Fujita exponent is given by the nonlocal part, $p_F=1+2s\mathrm{/}N$. We also determinate, in some cases, the blow-up rate.