On Chemotaxis Model with Linear and Porous Medium Diffusion, Logistic Source and Consumption on \(\R^N\)

Abstract

This dissertation is devoted to the study of chemotaxis systems with both linear diffusion and porous-medium-type diffusion, logistic source terms, and consumption of a chemical substance on $\mathbb{R}^N$. Chemotaxis systems are mathematical models describing the aggregation of cells driven by their directed movement in response to gradients of chemicals in their environment, which may act as attractants or repellents.

In the first part of this dissertation, we investigate a chemotaxis model with linear diffusion. We study fundamental problems such as the local and global existence of classical solutions with nonnegative initial data, which may be integrable or non-integrable. Under suitable smallness assumptions on the product of the initial chemical concentration and the chemotactic sensitivity, we prove the existence of a unique global classical solution. For non-integrable initial data, we develop a novel weighted energy method to establish global existence and boundedness. By introducing carefully chosen cut-off functions, we localize $L^p$-estimates uniformly in space. This approach extends known results for bounded domains and is applicable to other chemotaxis systems. We also study the stability of strictly positive solutions and the spreading behavior of solutions with compactly supported initial data. We show that the chemical does not, in general, hinder the spreading of the species, and it does not accelerate the spreading speed when the initial chemical concentration decays spatially or in the chemorepellent case with small sensitivity. Numerical simulations further reveal a phase transition in the sensitivity $\chi$: when the chemical is initially uniformly distributed in space, acceleration occurs only when $\chi$ exceeds a critical positive value.

In the second part, we study the local and global existence of weak solutions for the porous-medium diffusion case. For general bounded, possibly non-integrable initial data, we prove the existence of global weak solutions that remain uniformly bounded for all time. The proof is based on local $L^p$ estimates, uniform in time, obtained through a new continuity-type argument combined with Moser iteration to derive $L^\infty$ bounds. We also investigate regularity and prove uniqueness of weak solutions for sufficiently smooth initial data under suitable conditions on the diffusion exponent.