I came across the following eigen-function problem $$ \begin{align} (a(x,t)\nabla + b(t)\Delta)f =& \lambda f\\ f(x)=&0 (\forall x \in \partial \Omega),\\ f(x)>&0 (\forall x \in \Omega), \end{align} $$ $\Omega$ is a convex, bounded, open domain in $\mathbb{R}^d$, $d\geq 1$, with continuously differentiable boundary. Here $a$ is an affine function in $x$ and continuously differentiable in $t$, and $b$ is a matrix.
Under what conditions does the minimal eigen-value exist? When it exists, what is the corresponding eigenfunction?
I came across this problem in my readings and it is not directly my field, so I thought I should as here for some guidance.