Global solutions to semilinear parabolic equations driven by mixed local-nonlocal operators

We are concerned with the Cauchy problem for the semilinear parabolic equation driven by the mixed local-nonlocal operator $\mathcal{L}=-\mathrm{\Delta }+(-\mathrm{\Delta }{)}^s$, with a power-like source term. We show that the so-called Fujita phenomenon holds, and the critical value is exactly the same as for the fractional Laplacian.