Solutions to PDEs with prescribed properties

Combining the construction of suitable local solutions to PDEs with appropriate global approximation theorems, we can obtain global solutions to PDEs 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 utilisation to prove the existence of solutions of the heat equation with local hot spots with prescribed behaviour, among other applications. This is a joint work with A. Enciso and D. Peralta-Salas.