Renormalized Solutions for a Class of Nonlinear Parabolic Equation with a Lower Order Term and Variable Exponents

We consider a class of nonlinear parabolic equations $${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}}b(u)-\mathrm{\nabla }\cdot (A(x,t,u,\mathrm{\nabla }u))+H(x,t,\mathrm{\nabla }u)=f,$$ where $H$ is a nonlinear lower order term satisfied the Carath$\acute{e}$odory condition and $$\mid H(x,t,\mathrm{\nabla }u)\mid ⩽g(x,t){\mid \mathrm{\nabla }u\mid }^{\delta (x)}$$ with $$\delta (x)=\frac{p(x)(N+1)-N}{(N+2)(p(x)-1)}(p^{-}-1)\text{and}{p}^{-}=p(x)\mathrm{.}$$ By virtue of truncation metheod,the monotone operator theory and a gradient estimate we prove existence of renormalized solutions without coercivity condition on lower order term in the framework of variable exponents.