The present paper is concerned with semilinear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local Hamilton-Jacobi equations. The starting point is a new formula for the operator. It permits to prove the key a priori estimate that stands for the scalar conservation law and the Hamilton-Jacobi equation. The smoothing effect of the operator is also put in light and used to solve both equations. As far as Hamilton-Jacobi equations are concerned, a non-local vanishing viscosity method is used to construct a (viscosity) solution when existence of regular solutions fails, and a rate of convergence is provided. Turning to conservation laws, global-in-time existence and uniqueness are established. We also show that our formula allows to obtain entropy inequalities for the non-local conservation law, and thus to prove the convergence of the solution, as the non-local term vanishes, toward the entropy solution of the pure conservation law.