Renormalized Solutions for Nonlinear Parabolic Systems in the Lebesgue-Sobolev Spaces with Variable Exponents

Abstract

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 ),

Onpz–Laplacian System Involving Critical Nonlinearities

“…For the systems with singular source data, we refer to Saoudi [5] and Papageorgiou et al [6]. For more results, we refer to [7][8][9][10][11], as well as to [12], and the references therein. Before explaining the novelty of this paper, we give an overview of the literature on this kind of system in W 1,p ðQ Þ. Adriouch and El Hamidi in [13] proved the existence and multiplicity of solutions to the following system:…”