Complete quenching for singular parabolic problems

We prove finite time extinction of the solution of the equation u t − Δ u + χ { u > 0 } ( u − β − λ f ( u ) ) = 0 in Ω × ( 0 , ∞ ) with boundary data u ( x , t ) = 0 on ∂ Ω × ( 0 , ∞ ) and initial condition u ( x , 0 ) = u 0 ( x ) in Ω, where Ω ⊂ R N is a bounded smooth domain, 0 < β < 1 and λ > 0 is a parameter. For every small enough λ > 0 there exists a time t 0 > 0 such that the solution is identically equal to zero.