This dissertation studies the existence and stability of solutions for a class of non-autonomous systems of differential equations in multi-dimensional time, within the framework of Banach space. Focusing on both hyperbolic and parabolic cases, it presents a comprehensive range of results on existence and stability over finite and infinite time intervals. By leveraging the theory of evolution families, this work uncovers the conditions under which these systems have solutions and provides an in-depth analysis of how these solutions evolve, offering fresh perspectives on their dynamic behavior over time
