Existence to nonlinear parabolic problems with unbounded weights

We consider the weighted parabolic problem of the type ⎧⎨ ⎩ ut − div(ω2(x)|∇u|p−2∇u) = λω1(x)|u|p−2u, x ∈ , u(x, 0) = f (x), x ∈ , u(x, t) = 0, x ∈ ∂, t > 0, for quite a general class of possibly unbounded weights ω1, ω2 satisfying the Hardy-type inequality. We prove existence of a global weak solution in the weighted Sobolev spaces provided that λ > 0 is smaller than the optimal constant in the inequality. The domain is assumed to be bounded or quasibounded. The obtained solution is proven to belong to L p(R+;W1,p (ω1,ω2),0()) ∩ L∞ (R+; L2()).