On a singular anisotropic parabolic equation related to the $\vec{p}(x)$-Laplacian

This paper investigates a singular anisotropic parabolic equation associated with the $p_{i}(x)$-Laplacian operator. By employing the anisotropic Gagliardo-Sobolev-Nirenberg inequality and a modified Di Giorgi iteration technique, we derive a uniform $L^{\infty}$-estimate for the weak solution $u_{n}$ of the approximate problem. Using the Hölder inequality and embedding theorems in anisotropic Sobolev spaces, we successfully eliminate the effects of both the damping and source terms. Furthermore, using Egoroff’s theorem, we prove the strong convergence of $u_{nx_{i}}\rightarrow u$ in the $L^{p_{i}(x)}(Q_{T})$ space. The existence of a weak solution is finally confirmed through the renormalized solution method.