Renormalized solutions for nonlinear parabolic systems in the Lebesgue–Sobolev spaces with variable exponents

The existence result of renormalized solutions for a class of nonlinear parabolic systems with variable exponents of the type
\begin{align*} \partial_{t} e^{\lambda u_i(x,t)}& -\mathop{\mathrm{div}} (|u_i(x,t)|^{p(x)-2}u_i(x,t))\\ & + \mathop{\mathrm{div}}(c(x,t)|u_i(x,t)|^{\gamma(x)-2}u_i(x,t))=f_{i}(x,u_{1},u_{2})-\mathop{\mathrm{div}}(F_{i}), \end{align*}
for $i=1,2,$ is given. The nonlinearity structure changes from one point to other in the domain $\Omega$. The source term is less regular (bounded Radon measure) and no coercivity is in the nondivergent lower order term $\mathop{\mathrm{div}}(c(x,t)|u(x,t)|^{\gamma(x)-2}u(x,t))$. 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.