Complete quenching for a quasilinear parabolic equation

We study the homogeneous Dirichlet problem for the quasilinear parabolic equation with the singular absorption term ∂ t u − Δ p u + 1 { u > 0 } u − β = f ( x , u ) in Q T = ( 0 , T ) × Ω . Here Ω ⊂ R d , d ⩾ 1 , is a bounded domain, Δ p u = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplace operator and β ∈ ( 0 , 1 ) is a given parameter. It is assumed that the initial datum satisfies the conditions u 0 ∈ W 0 1 , p ( Ω ) ∩ L ∞ ( Ω ) , u 0 ⩾ 0 a.e. in Ω . The right-hand side f : Ω × R → [ 0 , ∞ ) is a Carathéodory function satisfying the power growth conditions: 0 ⩽ f ( x , s ) ⩽ α | s | q − 1 + C α with positive constants α, C α and q ⩾ 1 . We establish conditions of local and global in time existence of nonnegative solutions and show that if q ⩽ p and α and C α are sufficiently small, then every global solution vanishes in a finite time a.e. in Ω.