While not exactly the same problem that you solved, there has been previous results considering nonuniqueness of solutions (with zero initial data) for power-law type semilinear term. Interestingly, contrary to what you wrote, Lipschitz may not be enough (depending on the function spaces in consideration) for uniqueness.
Some relevant papers: In the case where the nonlinearity is Lipschitz and the function spaces used are $L^p$ type spaces, we have
- Haraux and Weissler. "Nonuniqueness for a semilinear initial value problem". http://www.ams.org/mathscinet-getitem?mr=648169
- Ni and Sacks. "Singular behavior in nonlinear parabolic equations". http://www.ams.org/mathscinet-getitem?mr=768731
- Baras. "Non-unicité des solutions d'une équation d'évolution non-linéaire". http://www.numdam.org/item?id=AFST_1983_5_5_3-4_287_0
In the case where the nonlinearity is not Lipschitz, we have
- Fujita and Watanabe, "On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations". http://www.ams.org/mathscinet-getitem?mr=234129
This should be enough to get you started with the literature search on MathSciNet.
