Recent Developments in Partial Differential Equations and Their Applications

Dear Colleagues,

The classical Keller–Segel model describes chemotaxis, a biological process in which cells migrate toward higher concentrations of a chemical signal. Chemotaxis and its variant system have been extensively studied in recent years.  In investigating the long-time behavior of Keller–Segel systems, we focus on models with general nonlinear dependence of diffusion and cross-diffusion rates on population density. To address the challenges posed by nonlinear terms, we construct a proper Lyapunov and employ a tailored Moser iteration, in which the time variable is continuously postponed during the iteration process.  This method can be extended to the attraction–repulsion system, haptotaxis system, and related models.

In addition, I am particularly interested in the interplay between boundary geometric properties and boundary regularity across various classes of parabolic equations, including linear equations, p-Laplace equations, and fractional Laplace equations. This line of inquiry underscores the profound connection between mathematical abstraction and real-world applications, highlighting the diverse manifestations of boundary behavior in these distinct mathematical contexts.

Dr. Mengyao Ding
Guest Editor

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