The following exercise on p.283 in the book demonstrates nonuniqueness for semilinear parabolic equations under many definitions of a solution that are currently being used

A generalization of this example in the book leads to a new definition of a solution of semilinear parabolic equations under which one has uniqueness. This example, in more abstract form, was first published by the author in Pacific J. Math. 118(1) 1985, pp. 199-214. In response to the article, Dan Henry sent to the author an example similar to the above one.