We are concerned with degenerate parabolic equations in nondivergence form of the type ∂ t u = a ( δ ( x ) ) u p ( Δ u + λ g ( u ) ) in Ω × ( 0 , ∞ ) , where Ω ⊂ R N ( N ≥ 1 ) is a smooth bounded domain, δ ( x ) = dist ( x , ∂ Ω ) , λ > 0 , p ≥ 1 , and g is either a nondecreasing function having a sublinear growth or g ( u ) = u . The degenerate character of our problem is also given by the potential a ( δ ( x ) ) which may vanish at the boundary ∂ Ω . Under some suitable assumptions on g , a , and λ , we establish the existence and uniqueness of a classical solution and we determine its asymptotic profile as t → ∞ . If g ( u ) = u and a satisfies ∫ 0 1 s / a ( s ) d s < ∞ , we also provide a blow-up result as λ approaches the first eigenvalue λ 1 of the Laplace operator − Δ .