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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.

I imagine all of the above points appear in some form in the paper

Van Moerbeke, Pierre. "On optimal stopping and free boundary problems." Archive for Rational Mechanics and Analysis 60.2 (1976): 101-148.

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.

Existence

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. 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.

I imagine all of the above points appear in some form in the paper

Van Moerbeke, Pierre. "On optimal stopping and free boundary problems." Archive for Rational Mechanics and Analysis 60.2 (1976): 101-148.

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.

I imagine all of the above points appear in some form in the paper

Van Moerbeke, Pierre. "On optimal stopping and free boundary problems." Archive for Rational Mechanics and Analysis 60.2 (1976): 101-148.

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. 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 can be handled in the space of viscosity solutions using a comparison principle argument. I wrote about this in an expository post. The setting there is slightly more general, but handles your case as well.

Existence

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. 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.

I imagine all of the above points appear in some form in the paper

Van Moerbeke, Pierre. "On optimal stopping and free boundary problems." Archive for Rational Mechanics and Analysis 60.2 (1976): 101-148.