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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 &gt; 2 and for ∂ t u - Δ(u m ) + b(x)u q  = 0 with m &gt; 1 where 0 ≤ q &lt; 1. Some criteria involving the asymptotics as h → 0 of the first eigenvalue ofare 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.

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