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.