I am considering the existence of solutions for the following semilinear parabolic PDE \begin{cases} u_t+Lu=f(t,x,u,u_x), &(t,x)\in [0,T]\times\mathbb{R},\\ u(0,x)=g(x), \end{cases} where $L$ is a 2nd-order (uniformly) elliptic operator.
I am interested in the existence of classic solution $u\in C^{1,2}([0,T]\times \mathbb{R})$ where $f$ is monotone in $u$ with polynomial growth. Other assumptions can also be imposed.
Is there any reference about this topic for an unbounded domain?