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The existence result of renormalized solutions for a class of nonlinear parabolic systems with variable exponents of the typefor i = 1, 2, is given. The nonlinearity structure changes from one point to other in the domain Ω. The source term is less regular (bounded Radon measure) and no coercivity is in the nondivergent lower order term div(c(x, t)|u(x, t)| γ(x)−2 u(x, t)). The main contribution of our work is the proof of the existence of renormalized solutions without the coercivity condition on nonlinearities which allows us to use the Gagliardo-Nirenberg theorem in the proof.where v is the velocity, [∇v]v is the convective term, π denotes the pressure, S denotes the extra stress tensor, g is the external body force, E is the electric field, and P is the electric polarization. The extra stress tensor is given by S = α 2 1((1 + |D| 2 ) p−1 2 − 1)E ⊗ E + (α 2 1 + α 2 1|E| 2 )(1 + |D| 2 ) p−2 2 ),

AbstractWe prove the existence of renormalized solutions to a class of nonlinear evolution equations, supplemented with initial and Dirichlet condition in the framework of generalized Sobolev spaces. The data are assumed merely integrable.

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