Approximate identities in variable $L^p$ spaces

We give conditions for the convergence of approximate identities, both pointwise and in norm, in variable $L^p$ spaces. We unify and extend results due to Diening, Samko and Sharapudinov. As applications, we give criteria for smooth functions to be dense in the variable Sobolev spaces, and we give solutions of the Laplace equation and the heat equation with boundary values in the variable $L^p$ spaces.