Let $\Omega$ be a bounded domain with smooth boundary and consider the following initial boundary value problem: \begin{equation}\label{pf0} \begin{aligned} \begin{cases} t\partial_t(t\partial_t u)-\Delta u+Vu=0\,\quad &\text{on $(0,T)\times \Omega$}, \\ u=f\,\quad &\text{on $\Sigma=(0,T)\times \partial \Omega$,}\\ u(0,x)=\partial_t u(0,x)=0 \,\quad &\text{on $\Omega$,} \end{cases} \end{aligned} \end{equation}
Is there a good theory for solving these kind of problems? The issue of course is the degeneracy at $t=0$. Any answers or references are greatly appreciated.