In this chapter, we study the p-Laplacian evolution equation for p > 1, that is, the gradient flow of the p-Cheeger energy. Our goal is to use the Hilbertian theory of gradient flows to provide a notion of weak solution to the gradient flow associated to the p-Cheeger energy and prove their existence and uniqueness for initial data in L2. We assume that the metric space is complete and separable and that the associated measure is finite on bounded sets (we keep this assumption throughout the whole chapter). Then, Ambrosio, Gigli, and Savaré proved that the p-Cheeger energy is convex and lower semicontinuous; since it’s the domain of definition is dense in L2 as it contains Lipschitz functions with bounded support, by the Brezis–Komura theorem, for every initial data in L2, there exists a unique strong solution to the abstract Cauchy problem. Our goal is to introduce a notion of weak solutions, improving upon the above definition; we will express the subdifferential of the p-Cheeger energy in terms of the differential structure due to Gigli presented in Chapter 1.