Consider a reaction-diffusion equation $$u_t=u_{xx}+f(u)$$ on $(x,t)\in\mathbb{R}\times [0,\infty)$ with non-linearity $f(u)$.
Suppose, we have a continuous initial datum $u_0(x):=u(x,0)$ such that the limits $\lim_{x\to\pm\infty}u_0(x)$ exist.
I have read that, assuming the solution has no blow-up, standard parabolic estimates imply that for the associated solution $u(x,t)$, the limits $\lim_{x\to\pm\infty}u(x,t)$ exist for all times $t>0$.
To be honest, I have no idea which standard parabolic estimates are meant.
I googled a lot but did not find any estimates which give me this implication. Does anybody know what might be meant?
Without additional assumptions this statement is false, take e.g. $$ u_0(x)=1\\ f(u)=u^2\\ \Rightarrow u(x,t)=\frac{1}{1-t} $$
Please share all assumptions.