Existence to nonlinear parabolic problems with unbounded weights

We consider the weighted parabolic problem of the type \begin{equation*} \begin{split} \left\{\begin{array}{ll} u_t-\mathrm{div}(\omega_2(x)|\nabla u|^{p-2} \nabla u )= \lambda \omega_1(x) |u|^{p-2}u,& x\in\Omega, u(x,0)=f(x),& x\in\Omega, u(x,t)=0,& x\in\partial\Omega,\ t>0, \end{array}\right. \end{split} \end{equation*} for quite a general class of possibly unbounded weights $\omega_1,\omega_2$ satisfying the Hardy-type inequality. We prove existence of a global weak solution in the weighted Sobolev spaces provided that $\lambda$ is smaller than the optimal constant in the inequality.