Long time behaviour for the reinitialization of the distance function

In this article we study the long-time behaviour of a class of non-coercive Hamilton-Jacobi equations, that includes, as a notable example, the so called reinitialization of the distance function. In particular we prove that its viscosity solution converges uniformly as $t\to +\mathrm{\infty }$ to the signed distance function from the zero level set of the initial data.