Main Research Line

My primary research focus lies in the study of differential equations, both ordinary (ODEs) and partial (PDEs), with an emphasis on the interplay between nonlinear functional analysis and variational methods. My work explores the existence, uniqueness, and qualitative behavior of solutions.

Ongoing Research Projects

Project: Transition Solutions for Allen-Cahn Type Equations

Duration: 2024–Present

Description: The stationary solutions of Allen-Cahn-type equations have been the subject of extensive research in recent years due to their complexity and the variety of structures they exhibit. These equations, characterized by the formulation

−Δu+A(x)V′(u)=0,

where V(u) is the classical Ginzburg-Landau double-well potential and A(x) represents an oscillatory factor, allow for the emergence of various families of transition-type solutions, such as periodic, heteroclinic, homoclinic, stationary layered, saddle, and multi-bump solutions. These structures play a fundamental role in modeling phenomena that involve transitions between equilibrium states in complex systems. The study of these solutions is essential for understanding the dynamics of mathematical systems, with applications that include pattern formation in materials and the propagation of excitations in neuroscience. This project aims to investigate the existence, multiplicity, and qualitative properties of transition solutions for Allen-Cahn-type equations and systems, addressing various scenarios, including cases where the Laplacian operator is replaced by quasilinear and nonlocal operators. Additionally, we plan to study these equations in more sophisticated contexts, such as structured domains like differentiable manifolds and situations where the potential V exhibits jump discontinuities. These transition dynamics are particularly relevant as they provide mathematical explanations for natural patterns and stationary states in various dynamical systems.

Other Research Interests

In addition to my main research project, I am also interested in several mathematical fields, particularly in problems where functional analysis, PDEs, and geometric methods intersect:

🔹 Geometric Analysis – Investigating partial differential equations (PDEs) on Riemannian manifolds, focusing on their analytical and geometric properties, including curvature effects and their influence on solutions.

🔹 Schrödinger Equations – Studying the existence and qualitative properties of normalized solutions in nonlinear Schrödinger-type equations, with emphasis on variational methods and stability analysis.

🔹 Quantum Metric Graphs – Exploring the geometric structure and spectral theory of metric graphs, with applications to differential equations, quantum systems, and wave propagation.

🔹 Ordinary Differential Equations & Applications – Investigating generalized models of pendulum-type equations, extending their applicability to nonlinear oscillations, damping effects, and external forces.

🔹 Evolutionary PDEs – Applying the Galerkin method to wave and heat equations in generalized settings, addressing existence, uniqueness, and long-term behavior of solutions.

Secondary Research Topics