Doubly Degenerate Parabolic Equations with Variable Nonlinearity II: Blow-up and Extinction in a Finite Time S. N. Antontsev CMAF, University of Lisbon, Portugal S. I. Shmarev ∗ University of Oviedo, Oviedo, Spain Keywords: Nonlinear parabolic equations, double nonlinearity, nonstandard growth conditions, blow-up, extinction, decay rates, 2010 MSC: 35K55, 35K65, 35K67 1. Introduction The paper is devoted to the study of the qualitative properties of energy solutions of the Dirichlet problem for the doubly nonlinear anisotropic parabolic equation with variable nonlinearity            d dt ( |v| m(z) sign v ) = n  i=1 D i  a i (z)|D i v| p i (z)−2 D i v  + b(z)|v| σ(z)−2 v + g in Q T , v = 0 on Γ T , v(x, 0) = v 0 (x) in Ω, (1.1) where Q T =Ω ×(0,T ), Ω ⊂ R n is a bounded domain with Lipschitz-continuous boundary ∂ Ω, Γ T = ∂ Ω × (0,T ) is the lateral boundary of the cylinder Q T . The exponents of nonlinearity m(z), p i (z) and σ(z) are given functions of the argument z =(x,t) ∈ Q T . The total derivative with respect to t is calculated according to the rule d dt ψ(v,m)= ψ v (v,m) v t + ψ m (v,m) m t , which allows one to rewrite equation (1.1) in the formally equivalent form m|v| m−1 v t = n  i=1 D i ( a i |D i v| p i −2 D i v ) + b|v| σ−2 v −|v| m−1 v ln |v| m t + g. ∗ Corresponding author Email addresses: anton@ptmat.fc.ul.pt, antontsevsn@mail.ru (S. N. Antontsev), shmarev@uniovi.es (S. I. Shmarev) Preprint submitted to Elsevier September 24, 2013 Manuscript Click here to download Manuscript: Antontsev-Shmarev-03-10-2012-resub.tex