Continuous in time bubbling and Soliton Resolution for Non-negative Solutions of the Energy-Critical Heat Flow

We show that any finite energy solution of the energy-critical nonlinear heat flow in dimensions $d\ge 3$ asymptotically resolves into a sum of possibly time-dependent solitons, a radiation term, and an error term that vanishes in the energy space. As a consequence, when the initial data has finite energy and is non-negative, we settle the Soliton Resolution Conjecture for all dimensions $d\ge 3.$