Mathematical analysis of extended Fisher-Kolmogorov equation with Neumann boundary conditions

We investigate the extended Fisher-Kolmogorov equation (EFK) under Neumann boundary conditions in a spatial dimension of up to three ($d\leq 3$). This is done on a bounded convex domain with a boundary that possesses $C^{2}$ smoothness. The EFK equation, which is a fourth-order PDE, is reduced into a system of two second-order PDEs. We prove the global existence, uniqueness, and continuous dependence on initial conditions for both strong and weak solutions using Lions' classic Faedo-Galerkin approach and compactness arguments. Furthermore, we provide results on the regularity of weak forms.