Time-vanishing properties of solutions of some degenerate parabolic equations with strong absorption

Let Ω be an open bounded subset of ℝ N and b a measurable nonnegative function in ℝ. We study the time compact support property for ∂ t u - Δ p u + b(x)u q = 0 with p > 2 and for ∂ t u - Δ(u m ) + b(x)u q = 0 with m > 1 where 0 ≤ q < 1. Some criteria involving the asymptotics as h → 0 of the first eigenvalue of are given which imply such a phenomenon. Applying these criteria we deduce that the time vanishing property occurs when 1/b belongs to L s (Ω) for some s depending on q, and p or m.