Available online at www.sciencedirect.com Mathematics and Computers in Simulation 81 (2011) 2018–2032 Parabolic equations with double variable nonlinearities S. Antontsev a,1 , S. Shmarev b,∗,2 a CMAF, University of Lisbon, Portugal b Department of Mathematics, University of Oviedo, Spain Received 15 November 2009; received in revised form 26 May 2010; accepted 7 December 2010 Available online 21 December 2010 Abstract The paper is devoted to the study of the homogeneous Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: u t = div ( a(x, t, u)|u| α(x,t) |∇u| p(x,t)−2 ∇u ) + f (x, t ) with given variable exponents α(x, t) and p(x, t). We establish conditions on the data which guarantee the existence of bounded weak solutions in suitable Sobolev–Orlicz spaces. © 2011 IMACS. Published by Elsevier B.V. All rights reserved. MSC: 35K57; 35K65; 35B40 Keywords: Parabolic equation; Double nonlinearity; Variable nonlinearity; Nonstandard growth conditions 1. Introduction We study the Dirichlet problem for the doubly nonlinear parabolic equation  u t = div ( a(z, u)|u| α(z) |∇u| p(z)−2 ∇u ) + f (z) z = (x, t ) ∈ Q = Ω × (0,T ], u(x, 0) = u 0 (x) in Ω, u = 0 on Γ = ∂Ω × [0,T ]. (1.1) Eq. (1.1) is formally parabolic, but it may degenerate or become singular at the points where u = 0 or | ∇ u| = 0. Introducing the functions γ (z) = α(z) p(z) − 1 , v(z) =  u 0 |s| γ (z) ds = u|u| γ (z) γ (z) + 1 , u(z) = Φ 0 (z, v) = |v| −γ 1+γ v (1 + γ ) 1 1+γ , (1.2) ∗ Corresponding author. E-mail address: shmarev@orion.ciencias.uniovi.es (S. Shmarev). 1 The author was partially supported by FCT, Financiamento Base 2008-ISFL-1-209, Portugal, and by the research project MTM2008-06208 of the Ministerio de Ciencia e Innovacion, Spain. 2 The author acknowledges the support of the research project MTM2007-65088, Spain. 0378-4754/$36.00 © 2011 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2010.12.015