I will get you started, but there are lots of blanks to fill. We are
interested in the PDE
\begin{align*}
\min\left\{ -u_{t}+\Delta u,u-\varphi\right\} & =0 & \text{in }(0,T]\times\mathbb{R}^{n}\\
u(0,\cdot)-\varphi(0,\cdot) & =0 & \text{in }\mathbb{R}^{n}
\end{align*}
where $\varphi$ is a smooth function of polynomial growth.
Existence
Let $u^{0}$ be a classical solution of
\begin{align*}
-u_{t}^{0}+\Delta u^{0} & =0 & \text{in }(0,T]\times\mathbb{R}^{n}\\
u^{0}(0,\cdot)-\varphi(0,\cdot) & =0 & \text{in }\mathbb{R}^{n}
\end{align*}
that is unique in an appropriately picked space of functions of polynomial
growth. Define inductively $u^{k}$ as a classical solution of
\begin{align*}
-u_{t}^{k}+\Delta u^{k}+k\min\left\{ u^{k-1}-\varphi,0\right\} & =0 & \text{in }(0,T]\times\mathbb{R}^{n}\\
u^{k}(0,\cdot)-\varphi(0,\cdot) & =0 & \text{in }\mathbb{R}^{n}
\end{align*}
that is unique in an appropriately picked space of functions of polynomial
growth. Now you would have to show that you can pass to limits ($k\rightarrow\infty$)
to obtain a solution of the original PDE.
The limiting solution is not, in general, twice differentiable in space. But, you should probably be able to establish sufficient conditions for it to be a once differentiable viscosity solution.
This technique is called a penalty method and is at least useful for establishing existence.
Uniqueness
Uniqueness can be handled in the space of viscosity solutions using a comparison principle argument.
I wrote about this in an expository post, but I'm sure this is available in different forms elsewhere. The setting of the post proves uniqueness in the space of bounded functions, but you should be able to generalize the arguments.
Heat Kernel
You can use smooth pasting to write the solution with the heat kernel, but you will probably not be able to find a "nice expression" for the free boundary, as this is believed to be hard.
References
Van Moerbeke, Pierre. "On optimal stopping and free boundary problems." Archive for Rational Mechanics and Analysis 60.2 (1976): 101-148.
Pham, Huyên. "Optimal stopping, free boundary, and American option in a jump-diffusion model." Applied mathematics & optimization 35.2 (1997): 145-164.
