Consider a function $a:[0,1]\to[0,\infty)$, e.g., $a(x)=x$. Are there existence theorems for equations of the form: \begin{align*} u-\big(a(x)u_x\big)_x=f(x) ? \end{align*} what are the boundary conditions to be assumed, if for example $a$ vanishes at $0$? What if $a$ vanishes on an open subset of $(0,1)$ or on, let's say $[0,1/2)$. If $a\in C^1$ and $f\in L^2$, what is the regularity of $u$? here its is not assumed the ellipticity condition for $a$.
What about parabolic equations, \begin{align*} u_t-\big(a(x)u_x\big)_x=f(x,t) ? \end{align*}
Is there any standard reference for this? What about $n$-dimensional case, where $a$ semi-positive definite matrix, but not necessarily a positive-definite one?