In today’s polarized societies, individuals’ opinions evolve not only through rational deliberation and peer influence, but also as a result of psychological, social, and institutional costs that inhibit belief change. We introduce a novel continuum-trait partial differential equation (PDE) model to describe how opinions evolve across two ideological axes (e.g., economic and social views), capturing both spatial and temporal dynamics in a continuous opinion space.

Our model couples diffusion, drift due to cognitive/social movement costs, and nonlocal peer competition, yielding the evolution equation. We rigorously formulate this system with Neumann boundary conditions and prove well-posedness in the appropriate Sobolev spaces. We show that the system admits a Lyapunov (free-energy) functional, and its minimizers—representing evolutionarily stable opinion states—display spontaneous segregation into polarized clusters.

Our theoretical results demonstrate that the combination of rugged cost landscapes and peer crowding effects leads to persistent opinion fragmentation. This work bridges mathematical biology and social dynamics and offers new tools for analyzing the stability and evolution of ideologically divided societies using continuum PDE methods.