In Partial Differential Equations book by Evans they treat a nonlinear system of reaction-diffusion equations. The nonlinearity comes from the reaction term $f$
\begin{align*} & \partial_t u - \Delta u = f(u) \ \ \ \ \text{ in } U & u =0, \ \ \text{ on } \partial U \times [0,T], \ \ \ \ u=g \ \ \text{ on } U \times \{0\} \end{align*}
It is assumed that the reaction term $f: \mathbb{R}^m \to \mathbb{R}^m$ is Lipschitz so it satisfies the growth assumption
$$|f(z)| \leq C(1 + |z|)$$
My question is what is the physical interpretation of the growth assumption?
I am assuming that it puts a restriction on the values of $f$ so it does not grow too fast, or it does not pass a certain value maybe. However, how is this different from assuming that $f$ is bounded?
Are there any other growth assumptions used in the study of PDEs?
EDIT:
In this paper for example they assume that $F$ is continuous and $sF(s) \geq 0$. I do not think this has something to do with the growth of $F$, however.