Focusing on linear and nonlinear non-Fredholm operators, this book explores a novel aspect of mathematical theory with wide-ranging applications. It encourages readers to engage with the material and contribute to this intriguing field, highlighting the potential for further exploration beyond the topics covered. The text serves as both a foundational resource and an invitation for collaborative advancement in mathematical research.
Focusing on mitochondrial swelling, this book introduces innovative mathematical models that incorporate spatial effects and the mobility of mitochondria within cells. It explores the well-posedness and long-term dynamics of solutions, tailored to various boundary conditions relevant to experimental settings. The findings have motivated new experimental designs at the Helmholtz Center Munich, bridging theory and practical application. Aimed at graduate students and researchers, it requires a solid mathematical foundation alongside an interest in cell biology.
Focusing on the investigation of nonlinear elliptic problems, this book employs a dynamical system approach to explore symmetrization and stabilization properties of nonnegative solutions in asymptotically symmetric unbounded domains. It introduces innovative methods for analyzing infinite-dimensional systems and finite-dimensional reductions, utilizing trajectory dynamical systems and new Liouville-type results. Additionally, it highlights symmetry and monotonicity in solutions to characterize asymptotic profiles, offering valuable insights for mathematical biologists.
This book deals with the modeling, analysis and simulation of problems arising in the life sciences, and especially in biological processes. The models and findings presented result from intensive discussions with microbiologists, doctors and medical staff, physicists, chemists and industrial engineers and are based on experimental data. They lead to a new class of degenerate density-dependent nonlinear reaction-diffusion convective equations that simultaneously comprise two kinds of degeneracy: porous-medium and fast-diffusion type degeneracy. To date, this class is still not clearly understood in the mathematical literature and thus especially interesting. The author both derives realistic life science models and their above-mentioned governing equations of the degenerate types and systematically studies these classes of equations. In each concrete case well-posedness, the dependence of solutions on boundary conditions reflecting some properties of the environment, and the large-time behavior of solutions are investigated and in some instances also studied numerically.