You do not need the PDE to implement the LSM algorithm.
The $T$ maturity American call price on time $t$ is
$$v_t = \max_{\tau} E_t\left[e^{-\int_t^\tau r(u) du} (S_\tau - K)^+\right]$$ where the max is over all the stopping times $t \leq \tau \leq T$.
After discretizing time along a discrete time line $t_k$, this leads to the recursion
$$
v_{t_k} = \max\left\{(S_{t_k} - K)^+, E_{t_k}\left[v_{t_{k+1}} e^{-\int_{t_k}^{t_{k+1}} r(u) du}\right] \right\}
$$
where $E_{t_k}\left[v_{t_{k+1}} e^{-\int_{t_k}^{t_{k+1}} r(u) du}\right]$ is the option continuation value (the value if you choose not to exercise on $t_k$).
The Longstaff and Schwartz LSM algorithm is a "trick" to compute the continuation value at any point in the Monte Carlo simulation without having to resort to a new Monte Carlo simulation that would originate at that point.
As you can see you do not need a PDE for $v$ to implement the algorithm. The only thing you require is the SDE for $S_t$ for generating the Monte Carlo paths, as well as appropriate projection functions for the LS algorithm itself (there are many good references on the latter, for instance Monte Carlo Methods in Finance - Peter Jäckel).
