Here is a theorem that Heinrich Freistuhler & I proved in 1998. There is essentially no assumption.

Let $\phi:{\mathbb R}\rightarrow{\mathbb R}$ be a heteroclinic solution of $\phi'=f(\phi)-q$, where $q$ is some constant. By heteroclinic, we mean that the limits $u_\pm=\phi(\pm\infty)$ exist and are finite. Such functions are viscous standing shocks, that is time-independent solutions of the convection-diffusion equation $$(1)\qquad\partial_tu+\partial_xf(u)=\partial_{xx}^2u.$$ Consider now a function $u_0\in\phi+L^1({\mathbb R})$. Let us define $$h:=\int_{\mathbb R}(u_0-\phi)\,dx.$$ Then the (unique) solution $(x,t)$ of (1) with initial data $u_0$ satisfies (unconditional stability of $\phi$) $$\lim_{t\rightarrow+\infty}\|u(\cdot,t)-\phi(\cdot-h)\|_1=0.$$

This statement assumes neither genuine nonlinearity ($f''$ may vanish arbitrarily), nor decay of $u_0-\phi$ at infinity. The drawback is that the convergence can be arbitrarily slow as $t\rightarrow+\infty$.