In this talk, I will discuss a class of elliptic and parabolic partial differential equations characterized by anisotropic $\vec{\textbf{p}}(u)$-Laplace operator, where the vector-valued exponent $\vec{\textbf{p}} = (p_1, \ldots, p_N)$ depends on the unknown function $u$ and a non-local function of $u$, respectively. This dependence necessitates the use of variable exponent Sobolev spaces specifically tailored to the anisotropic framework and destroys the homogeneity and standard convexity properties of the operator, making the analysis substantially more delicate. For the elliptic case, we discuss the existence of a weak solution by employing the theory of pseudomonotone operators in conjunction with suitable approximation techniques. In the parabolic setting, the existence of a weak solution will be discussed via a time discretization scheme and Schauder fixed-point theorem, supported by a priori estimates and compactness arguments.