In this article, we study the existence of a renormalized solution for the nonlinear $p(x)$-parabolic problem associated to the equation: $$\frac{\partial b(x,u)}{\partial t} - \mbox{div} (a(x,t,u,\nabla u)) + H(x,t,u,\nabla u) = f - \mbox{div}F \;\mbox{in }\;Q= \Omega\times(0,T)$$with $ f $ $ \in L^{1} (Q),$\; $b(x,u_{0}) \in L^{1} (\Omega)$ and $ F \in (L^{P'(.)}(Q))^{N}. $The main contribution of our work is to prove the existence of a renormalized solution in the Sobolev space with variable exponent. The critical growth condition on $ H(x,t,u,\nabla u)$\; is with respect to$ \nabla u$, no growth with respect to $u$ and no sign condition or the coercivity condition.
@article{23667,
title = {Existence of Renormalized Solutions for p(x)-Parabolic Equation with three unbounded nonlinearities},
journal = {Boletim da Sociedade Paranaense de Matem\'atica},
volume = {34},
year = {2015},
doi = {10.5269/bspm.v34i1.23667},
language = {EN},
url = {http://dml.mathdoc.fr/item/23667}
}
Akdim, Youssef; Gorch, Nezha El; Mekkour, Mounir. Existence of Renormalized Solutions for p(x)-Parabolic Equation with three unbounded nonlinearities. Boletim da Sociedade Paranaense de Matemática, Tome 34 (2015) . doi : 10.5269/bspm.v34i1.23667. http://gdmltest.u-ga.fr/item/23667/