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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.

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  • $\begingroup$ I would try a change of variables in time to make it a standard wave operator (unless that's where you started and were hoping changing to this form would make things easier). $\endgroup$ Oct 9, 2020 at 0:29
  • $\begingroup$ That’s exactly what happened! $\endgroup$
    – Ali
    Oct 9, 2020 at 1:34
  • $\begingroup$ I apologize for the perhaps silly questions, but is $V$ a function or a constant? And if it is a function then $V=V(x)$ or $V=V(t,x)$? $\endgroup$ Oct 9, 2020 at 17:01
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    $\begingroup$ sorry for not clarifying that... It is a smooth function of $t$ and $x$. But perhaps as a simpler question it can just be assumed to be zero. $\endgroup$
    – Ali
    Oct 9, 2020 at 17:16
  • $\begingroup$ After thinking for some time, the only trick I feel it would help you is to use the Mellin transform respect to the time variable $t$: $${\hat u}(s,x)=\mathscr{M}[u](s,x) =\int\limits_0^\infty t^{s-1} u(t, x) \, \mathrm{d}t. $$ Its application to your PDE transforms the modified time derivative $t \partial_t$ in the complex variable $s$, and then you possibly have only to analyze an elliptic problem with complex coefficient. $\endgroup$ Dec 5, 2020 at 19:28

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