Rigorous solution to Ricci Flow on dumbbell $S^3$
The neckpinch solution on a dumbell has been constructed by Angenant and Knopff. A whole chapter of the book "The Ricci Flow: An Introduction" by Chow and Knopff is devoted to the construction of such a solution.
For the "weak solution" side, it has not been settled yet. Angenant and Knopff has some result in the rotationnaly symmetric case in a preprint that was put on the arXiv something like two years ago.