Suppose we are given an initial value problem $$ u_t=u_{xx} +f(u), \quad x\in \mathbb{R}, t>0, $$ with an initial datum $u(x,0)$ which is unbounded.
Does it then make sense to search for (essentially) bounded solutions $u$, when the initial datum already is unbounded, i.e. would that be a well-defined problem?
I guess not. But what could be an appropriate solution space? Maybe some weighted function space that renders the initial datum and the corresponding solution (essentially) bounded?
I am aware that the answer probably depends on the concrete choice of $f$ and on the unbounded initial datum. But maybe, my question can be answered in general, nonetheless.