Extinction properties of solutions for a class of fast diffusive [formula omitted]-Laplacian equations

Abstract

We consider the extinction properties of solutions for the homogeneous Dirichlet boundary value problem for the p -Laplacian equation u t − div ( ∣ ∇ u ∣ p − 2 ∇ u ) + β u q = λ u r with 1 < p < 2 , q ≤ 1 and r , λ , β > 0 . For β = 0 , it is known that r = p − 1 is the critical extinction exponent for the weak solution. For β > 0 , we show that r = p − 1 is still the critical extinction exponent when q = 1 . Moreover, the precise decay estimates of solutions before the occurrence of the extinction are derived. However, extinction can always occur when 0 < q ≤ r < 1 .