The initial boundary value problem of the nonlinear parabolic conical degenerate \(p-\) Laplacian operator is considered in this paper. Our first step was to prove the generalized conical Hardy inequality and the embedding result for conical Sobolev spaces on a manifold with conical singularity. We then showed the global existence of a weak solution. We investigated its asymptotic behavior using potential well theory. Finally, we obtained the blow-up property in finite time for the solution of the nonlinear parabolic conical degenerate Laplacian problem with singular potential and source-weighted functions.
