My research interests lie primarily in the analysis and application of partial differential equations, or PDEs. On one hand, the analysis of PDEs is often a challenging and nuanced venture with a functionally infinite amount of existing results to learn and discover. On the other hand, the application of PDEs to real world phenomena can act as a focusing lens to motivate the analytical tools one might develop.
On the analysis side of PDEs, I have spent a portion of time studying the existence and regularity of solutions to parabolic and elliptic equations, or systems of equations. Some problems of interest include equations with singular nonlinearities and the expected regularity, particularly near the boundary, that these solutions might have.
On the application side of PDEs, I have spent a portion of time studying time dependent problems modelling the population densities of various interacting species. This includes classical competition models, such as a Lotka-Volterra competition models with diffusion, but also includes more exotic equations, such as impulsive reaction-diffusion models.
My goal is to combine these two aspects of study in order to understand the properties of solutions (existence, stability, long term behaviour) in order to make predictions in the real world.