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 Generalized p(x)-elliptic system with non physical data Farah Balaadich,Elhoussine Azroul Keywords:p(x)-Laplacian systems, Variable exponents, Weak solutions, Young measures Abstract: This paper is concerned with the following Dirichlet problem of the form $-\text{div}\,\big(\Phi(Du-\Theta(u)\big)=v(x)+f(x,u)+\text{div}\,\big(g(x,u)\big),$ which corresponds to a diffusion problem with a source $v$ in moving and dissolving substance, the motion is described by $g$ and the dissolution by $f$. By the theory of Young measure we will prove the existence result in variable exponent Sobolev spaces $W^{1,p(x)}_0(\Omega;\R^m)$.