Cauchy problem for parabolic equation

Question

Let $\mathcal{L}=a \partial_{xx}+b\partial_{x}+c$, where $a>0,c\in\mathbb{R},b\in \mathbb{R}$, I want to know if the following parabolic equation gets a smooth and unique solution \begin{align} &v_t-\mathcal{L}v=f,\quad (t,x)\in(0,T]\times\mathbb{R},\\ &v(0,x)=g(x).\quad x\in \mathbb{R}. \end{align} where $f$ and $g$ are smooth and satisfy $|f|+|g|\leq A\exp(Ax^2)$ for some $A>0$.