There is a well established theory for the uniformly parabolic equations, i.e. for equations of the form $u_t=a(x,t)u_{xx}, x \in D, t\in (0,T], u(x,0)=u_0$ when $a(x,t)\geq a_0 >0$. In fact, if write the fundamental solution for that (i.e. the Green's function) in the special case of $a(x,t)\equiv a_0$, I can see that $a_0$ and $t$ both are present in the denominator(or in several dimension it is a determinant). However, if $a(x,t)$ takes value $0$ for some $t>0$, for example, if $a(x,t)=x(x-1)$, then it doesn't fall into the class of uniformly parabolic p.d.e. Thus, I have two questions I am confused about:
Does the fundamental solution in fact in this case has a similar form as in the case of constant coefficients and similar to heat density with minimum of $a(x,t)$ being in the denominator? Because if it is the case, then the fundamental solution becomes a delta function for any $t>0$. Is there a problem with a fundamental solution then? Or existence of it?
Might be a bit of a general question but can someone provide an example or simple explanation why "degenerate" equations of high importance. Is that because the tools used for proving existence and uniqueness in the case of uniformly parabolic equations are not valid or what's the reason behind a separate theory for those equations? What are the tipical issues with those equations?