I am interested in spatially inhomogeneous classical bounded solutions $u:\mathbb{R}^n \times [0,T] \to \mathbb{R}$ to the Cauchy problem for semi-linear parabolic PDE, which have homogeneous initial data, i.e;
$$ u\_t - \bigtriangleup u = f(u) \ \ \ \forall (x,t)\in\mathbb{R}^n\times (0,T] $$
$$ u(x,0)= 0 \ \ \ \forall x\in\mathbb{R}^n .$$
If anyone has any references to similar works on this type of problem (specifically concerning spatially inhomogeneous solutions), I would be most appreciative. 


Note that the question of when solutions will be spatially homogeneous (given conditions on $f$) is not of interest to me as it is besides the point. The reason I obtained this result was simply because it seemed somewhat counter-intuitive to most peoples (and initially my own) understanding of these type of problems. 



Reference Request: Spatially inhomogeneous solutions to parabolic PDE with homogeneous initial data
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](https://mathoverflow.net/questions/137384/spatially-inhomogeneous-solutions-to-parabolic-pde-with-homogeneous-initial-data#comment354552_137384), 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. 


