Let $a\in (0,1)$, we consider the below equation in a smooth bounded domain (any dimension) $$ \partial_t u -\partial_x^2 u - \dfrac{1}{|u|^a} \partial_x u =f(x).$$ Does anyone have some articles concerning this kind of equation? Or its elliptic version? What keyword should I search for?

Let $a\in (0,1)$ and $(0,1) \subset \mathbb{R}$, we consider the below equation in $(0,1) \times (0,T)$ $$ \partial_t u -\partial_x^2 u - \dfrac{1}{|u|^a} \partial_x u =f(x).$$ And $u(x,0)=x^{1/a}$ and $u(0,t)=0$.

Does anyone have some articles concerning this kind of equation? Or its elliptic version? What keyword should I search for?