Uniform Lipschitz regularity of a singular perturbation problem (Q1894156)

The purpose of the paper is to prove uniform (with respect to \(\varepsilon\)) local Lipschitz continuity in space variables of solutions \(u_\varepsilon\) of the equation \[ \Delta u-u_t=\beta_\varepsilon(u)\quad\text{on }\mathbb{R}^n\times(0,T),\tag{1} \] where \(\beta_\varepsilon\) is an approximation to Dirac's delta measure. It is proved that if \(u_\varepsilon\) is bounded and its negative part is Lipschitz continuous in space then \(u_\varepsilon\) is locally Lipschitz continuous uniformly in space. The proof is based on interesting explicit estimates of solutions to a parabolic equation. The result is valid not only for the heat equation but also for a general nonlinear (divergence or non-divergence form) parabolic operator on the left-hand side of (1).