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 wrotecontrary 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.
