A short proof of increased parabolic regularity

We present a short proof of the increased regularity obtained by solutions to uniformly parabolic partial differential equations.Though this setting is fairly introductory, our new method of proof, which uses a priori estimates Microfiber Towel and an inductive method, can be extended to prove analogous results for problems with time-dependent coefficients, advection-diffusion or reaction diffusion equations, and nonlinear PDEs even when other tools, such Walker as semigroup methods or the use of explicit fundamental solutions, are unavailable.