Prove comparison principle for $u_t + f(u)_x = k u_{xx} + g(t,x)$ with $g \in L^\infty(0,T; L^2(\mathbb R))$

Question

Let us consider the PDE $$u_t + f(u)_x = k u_{xx} + g(t,x),$$ with $k>0$, $g \in L^\infty(0,T; L^2(\mathbb R))$ and $f\in C^2(\mathbb R$) and $f$ strictly convex.

Assume $u(0,\cdot) = u_0(\cdot) \in L^\infty(\mathbb R) \cap L^1(\mathbb R)$.

Assume that there exists a smooth solution $u$ of this PDE. How can I prove that $$u(t,\cdot) \in L^\infty(\mathbb R), \quad t >0 \ ?$$ Or, for a fixed $T>0$, that $$\Vert u \Vert_{L^\infty((0,T)\times\mathbb R)} <\infty$$ holds?