Well/Ill posedness on Generalized heat equation

Question

Suppose we have the following one dimensional generalized heat equation:

$$u_t(x,t)=g(x,t)\Delta u(x,t) \ \ \ x\in\mathbb{R},t\in(0,\infty)$$

I need to prove that this equation is ill posed, for some initial data and some particular g(x,t). Is there any literature on such equation? I have found loads on equation like $$ u_t= \Delta u(x,t) + f(x,t)$$ which different f(x,t), but no one where the Laplacian is multiplied by another function.

Thanks a lot guys!

Answer 1

If you are free to pick the function $g$ to show ill-posedness, just take $g=-1$. Then, the ill-posedness for that case will follow from the smoothing effect when $g=1$.