Consider equation $$\partial_t u = \partial_x u + \partial_{xx} u - c u + f, \hbox{ on } (t, x) \in (0, \infty) \times \mathbb R$$ with initial condition $u(0, x) = g(x).$

Suppose that $c(t, x)$ and $f(t,x)$ are continuous in $(t, x)$ and $\phi (\cdot) = c(t, \cdot), f(t, \cdot), g(\cdot)$ satisfy $$|\phi|_0 + |\partial_x \phi|_0 + |\partial_{xx} \phi|_0 <K$$ for some $K>0$. [Question.] Is there unique classical solution for the equation with the above conditions?

Remark: I have seen that some conditions for the existence requires at least Holder continuity in $t$ for $c$ and $f$. I want to know if it is still true by dropping Holder $t$-continuity?