Starting from the seminal work of H. Jia and V. Šverák (2015), it is now well understood that the existence of linearly unstable forward self-similar solutions can be used to establish non-uniqueness of Leray–Hopf solutions to the unforced incompressible 3D Navier–Stokes equations. Although the existence of such unstable forward self-similar solutions remains an open problem, several works have since built upon the ideas of Jia and Šverák to demonstrate non-uniqueness for various fluid dynamics models with non-zero forcing terms. In this talk, we consider the unforced heat equation with a focusing power-type nonlinearity, and rigorously implement the Jia–Šverák method to establish non-uniqueness of local solutions in the full range of supercritical Lebesgue spaces. In particular, we provide a rigorous verification of the (analogue of the) spectral assumption made by Jia and Šverák for the Navier–Stokes equations. This is joint work with M. Hofmanová (Bielefeld), I. Glogić (Bielefeld), and T. Lange (Pisa).