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