Large time behavior of solutions to degenerate parabolic equations with weights

We study the degenerate parabolic equation ∂ t u = a ( δ ( x ) ) u p Δ u − g ( u ) in Ω × ( 0 , ∞ ) , where Ω ⊂ R N ( N ⩾ 1 ) is a smooth bounded domain, p ⩾ 1 , δ ( x ) = dist ( x , ∂ Ω ) and a is a continuous nondecreasing function such that a ( 0 ) = 0 . Under some suitable assumptions on a and g we prove the existence and the uniqueness of a classical solution and we study its asymptotic behavior as t → ∞ .