Special issue "Advances in Partial Differential Equations: Theory, Methods and Applications"

Partial differential equations (PDEs) constitute one of the most powerful and versatile frameworks in mathematical analysis, providing essential tools for describing physical phenomena, engineering systems, and complex geometric structures.This Special Issue, Advances in Partial Differential Equations: Theory, Methods and Applications, was conceived as a platform for presenting original contributions that advance both the theoretical foundations and practical methodologies of PDE analysis, whilst demonstrating their applicability to concrete problems in science and engineering.The twenty papers published in this Special Issue reflect the breadth and depth of contemporary research in partial differential equations.They encompass a wide range of topics including existence and uniqueness theory, regularity of solutions, numerical methods, qualitative behaviour, boundary value problems, and applications to mathematical physics.Beyond their methodological diversity, these contributions share a common commitment to mathematical rigour and to advancing our understanding of the fundamental structures underlying differential equations.The collection addresses several core themes in modern PDE theory.A significant group of papers focuses on elliptic and parabolic equations, examining regularity properties, variational formulations, and the interplay between analytic and geometric approaches.These works contribute to our understanding of how solutions behave under various structural assumptions and boundary conditions, with implications for both pure analysis and applications to physical models.Another substantial theme concerns hyperbolic equations and evolution problems, where questions of well-posedness, stability, and long-time behaviour are addressed through sophisticated