Complete quenching for a quasilinear parabolic equation

•Quasilinear parabolic equations with singular absorption terms.•Existence of weak bounded solutions.•Sufficient conditions of the complete quenching in a finite time.•Blow-up in a finite time.

We study the homogeneous Dirichlet problem for the quasilinear parabolic equation with the singular absorption term∂tu−Δpu+1{u>0}u−β=f(x,u)in QT=(0,T)×Ω. Here Ω⊂RdΩ⊂Rd, d⩾1d⩾1, is a bounded domain, Δpu=div(|∇u|p−2∇u)Δpu=div(|∇u|p−2∇u) is the p  -Laplace operator and β∈(0,1)β∈(0,1) is a given parameter. It is assumed that the initial datum satisfies the conditionsu0∈W01,p(Ω)∩L∞(Ω),u0⩾0 a.e. in Ω. The right-hand side f:Ω×R→[0,∞)f:Ω×R→[0,∞) is a Carathéodory function satisfying the power growth conditions: 0⩽f(x,s)⩽α|s|q−1+Cα0⩽f(x,s)⩽α|s|q−1+Cα with positive constants α  , CαCα and q⩾1q⩾1. We establish conditions of local and global in time existence of nonnegative solutions and show that if q⩽pq⩽p and α   and CαCα are sufficiently small, then every global solution vanishes in a finite time a.e. in Ω.