Iovanna, Francesco (2025) From reaction-diffusion to porous media and complex fluids: stability and pattern formation. [Tesi di dottorato]

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Abstract

This doctoral thesis investigates the mathematical mechanisms leading to pattern formation and instabilities in spatially extended systems governed by nonlinear partial differential equations. The work focuses on two main modeling settings: reaction–diffusion systems, with applications to ecological dynamics, and fluid flow models in porous and complex media, relevant to geophysical and industrial processes. The first part establishes the theoretical foundation by reviewing functional analysis tools, weak formulations of PDEs, and methods in linear and nonlinear stability theory, including spectral analysis and Lyapunov techniques. The second part concerns spatial pattern formation in ecological communities. A three-species intraguild predation model with cross-diffusion is studied, identifying conditions under which diffusion drives Turing and Turing–Hopf instabilities. Analytical results are supported by numerical simulations illustrating complex spatiotemporal dynamics. The third part addresses penetrative convection in porous media with variable gravity. Critical thresholds for instability are derived through linear and nonlinear analyses, and simulations highlight the destabilizing role of gravity variation. The fourth part investigates oscillatory instabilities in non-isothermal Jeffreys fluids under vertical throughflow. Memory effects and transport enhance the richness of the dynamical behavior, and global stability results are obtained through energy methods. Overall, the thesis contributes new insights into the onset and development of spatial organization in nonlinear PDE models, bridging ecological and fluid-mechanical applications.

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