Asymmetric solution of heat equation with heat source

Question

As I guess, the solution $u(x,t)$ of heat equation on $S^n$ $$ \partial_tu=\Delta_g u $$ must have some symmetry, I mean that , $\exists T_0>0$ st $\forall t>T_0$ $u(x,t)=u(-x,t)$, no matter any smooth initial conditions. But I don't know whether the heat source could break this symmetry. For example, for some positive contant $p$, consider $$ \partial_tu=\Delta_g u+u^p $$ it is obvious that for enough small $p$, the solution have the above symmetry. So, what is the maximal $p$ such that the solution has above symmetry ?