Examines a nonlinear system of parabolic PDEs arising in mathematical biology and statistical mechanics. The work describes the mathematical and physical principles: derivation of a series of equations, biological modeling based on biased random walks, the study of equilibrium states via the variational structure derived from the free energy, and the quantized blowup mechanism based on several PDE techniques. Suitable for researchers and graduate students of math and applied math interested in nonlinear PDEs in stochastic processes, cellular automatons, variational methods, and their applications to natural sciences.

Autorentext

Takashi Suzuki is a Vice-Director of the Center for Mathematical Modeling and Data Science at Osaka Univeristy.

Inhalt
Summary.- Background.- Fundamental Theorem.- Trudinger-Moser Inequality.- The Green's Function.- Equilibrium States.- Blowup Analysis for Stationary Solutions.- Multiple Existence.- Dynamical Equivalence.- Formation of Collapses.- Finiteness of Blowup Points.- Concentration Lemma.- Weak Solution.- Hyperparabolicity.- Quantized Blowup Mechanism.- Theory of Dual Variation.