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5 of 6
see second paragraph.
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Reference Request: Spatially inhomogeneous solutions to parabolic PDE with homogeneous initial data

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, 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. From the responses below, I think that is still true.

JCM
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