Existence of solutions to degenerate parabolic problems with two weights via the Hardy inequality

The paper concentrates on the application of the following Hardy inequality \begin{equation*} \int_\Omega \ |\xi(x)|^p \omega_{1 }(x)dx\le \int_\Omega |\nabla \xi(x)|^p\omega_{2 }(x)dx, \end{equation*} to the proof of existence of weak solutions to degenerate parabolic problems of the type \begin{equation*} \left{\begin{array}{ll} u_t-div(\omega_2(x)|\nabla u|^{p-2} \nabla u )= \lambda W(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{equation*} on an open subset $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, not necessarily bounded, where $$W(x)\le \min \{m,{\omega }_1(x)\},m\in {\mathbb{R}}_{+}\mathrm{.}$$