I know that this question may result rater vague and somehow out of context, still I am hopping you could help me. Assume we have the following equation

\begin{align} \boxed{\partial_t u(t,x,z)=\mathcal L(t,x) u(t,x,z)+h(t,x)\partial_z u(t,x,z)} \end{align}

where \begin{align} \mathcal L(t,x)=\sum_{i=1}^{d}\sum_{j=1}^{d} a_{ij}(t,x)\frac{\partial^2}{\partial x_j \partial x_i}+\sum_{i=1}^{d} b_i(t,x)\frac{\partial}{\partial x_i}+c(t,x). \end{align}

where the $a$'s an the $b$'s are very well behaved functions (we can assume they are $C^{\infty}$, bounded, and with bounded derivatives) and we can assume that $a_{ij}$ is uniformly elliptic.

The fact that there's no second order derivative with respect to $z$ makes me think that we are dealing with the prototipe of "hypoelliptic" PDEs.

The problem is that I am not really familiar with Hörmander's results, and I find particularly difficult to read and understand his papers. I was wondering under which assumptions regarding $h$ can we conclude that the latter equations has a solution? Is is possible to show existence even though we don't assume the necessary conditions for hypoellipticity?

Thanks in advance and please let me know if something is not clear.

I know that this question may result rater vague and somehow out of context, still I am hoping you could help me. Assume we have the following equation

\begin{align} \boxed{\partial_t u(t,x,z)=\mathcal L(t,x) u(t,x,z)+h(t,x)\partial_z u(t,x,z)} \end{align}

where \begin{align} \mathcal L(t,x)=\sum_{i=1}^{d}\sum_{j=1}^{d} a_{ij}(t,x)\frac{\partial^2}{\partial x_j \partial x_i}+\sum_{i=1}^{d} b_i(t,x)\frac{\partial}{\partial x_i}+c(t,x). \end{align}

where the $a$'s an the $b$'s are very well behaved functions (we can assume they are $C^{\infty}$, bounded, and with bounded derivatives) and we can assume that $a_{ij}$ is uniformly elliptic.

The fact that there's no second order derivative with respect to $z$ makes me think that we are dealing with the prototipe of "hypoelliptic" PDEs.

The problem is that I am not really familiar with Hörmander's results, and I find particularly difficult to read and understand his papers. I was wondering under which assumptions regarding $h$ can we conclude that the latter equations has a solution? Is is possible to show existence even though we don't assume the necessary conditions for hypoellipticity?

Thanks in advance and please let me know if something is not clear.