This talk will investigate the asymptotic behavior of evolution equations, including non-conservative ones, that are expected to converge to equilibrium at exponential speed. After reviewing some of the most classical methods used to derive this type of convergence, we will introduce an original approach to address this question. The key idea is to transform the problem into an equivalent conservative one, where solutions are probability measures, enabling the use of tools from optimal transport. Two examples in 1D -namely, the inhomogeneous heat equation with growth, and growth-fragmentation- will help us illustrate the method. This work is a collaboration with V. Calvez.