J. Nonl. Evol. Equ. Appl. 2015 (5), pp. 67-90, published on February 16, 2016:

Nonlinear parabolic inequalities in Sobolev space with variable exponent

Y. Akdim, N. El Gorch, M. Mekkour

Sidi Mohamed Ben Abdellah University, Laboratory LSI; Poly-Disciplinary Faculty of Taza P.O. Box 1223, Taza Gare, Morocco

Received on September 9, 2014, revised version on January 15, 2015
Accepted on December 19, 2014 (with modifications), revised version on January 16, 2015

Communicated by Ti-Jun Xiao

Abstract.  We give an existence result of the obstacle parabolic associated to the problem
u ≥ ψ a.e. in Ω × (0, T),
∂u/∂t − div(a(x, t, u, ∇u)) + H(x, t, u, ∇u) = f in Q = Ω × (0, T).
The main contribution of this work is to prove the existence of an entropy solution without the sign and the coercivity conditions on H(x, t, u, ∇u), the critical growth condition on H is with respect to ∇u, no growth condition with respect to u. The second term f belongs to L1(Q) and u0 ∈ L1(Ω). The main methods are the so called ”penalization methods”.
Keywords:Entropy solutions. Penalized equations. Nonlinear parabolic problem.
2010 Mathematics Subject Classification:   35J15, 35J70, 35J85.

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