Combining robust local construction of solutions to PDE with appropriate global approximation theorems, we can obtain solutions to PDE with prescribed geometric or topological properties. This technique has been used in many contexts involving elliptic equations. In this talk I will focus on showing global approximation theorems for parabolic equations and their application to prove the existence of solutions of the heat equation with local hot spots with prescribed behavior. This is a joint work with A. Enciso and D. Peralta-Salas.