The boundary value problem with the second order elliptic operator \(A(x,D_x)\): \[ D_t[m(x)u(x,t)] + A(x,D_x)u(x.t)=f(x,t), \quad (x,t)\in\Omega\times[0,\tau], \;\Omega\subset {\mathbb R}^n, \]
\[ u(x,t)=0, \;(x,t)\in\partial\Omega\times[0,\tau], \quad m(x)u(x,t) \to_{t\to0+} m(x)u_0(x), \;\text{ for \;a.e.} \;x\in \Omega, \] is considered. Since a nonnegative function \(m\in L^\infty(\Omega)\) is not bounded away from zero, the parabolic equation considered is singular. The estimate for ‘the resolvent’ in \(X=L^p(\Omega)\) is obtained: \[ \| L(\lambda M+L)^{-1}\| _{{\mathcal L}(X)}\leq C(1+| \lambda| )^{1-\beta}, \quad \operatorname{Re} \lambda\geq -c(1+| Im\lambda| \})^\alpha, \] \(c>0\), \(0<\beta\leq\alpha\leq1\), \(\alpha+\beta>1. \) The existence and uniqueness theorem for the boundary value problem is derived. An application to an inverse problem associated to the considered one is announced.