Research On The Solution Of Nonlinear Parabolic Equations

There exist amounts of nonlinear phenomena in areas such as physics, chemistry, as well as biological population dynamics, which usually can be modeled by nonlin-ear parabolic partial differential equations, especially, the type of Keller-Segel equations. Currently, the study of nonlinear partial differential equations is a very important research direction in the field of partial differential equations. This article mainly discusses the nonlinear parabolic partial differential equations which describe the chemotactic move-ment of organisms.1. We study the initial boundary value problem of the quasilinear Keller-Segel equa-tions which model the chemotaxis of organisms. We obtain the global existence and uniform boundedness by energy estimates.2. The initial boundary value problem of the semilinear Keller-Segel equations with repulsive signal and logistic source is considered. The global existence, uniform bound-edness as well as large time behaviour are obtained by energy estimates, which reveal the effects of the chemoattraction, the chemorepulsion and the decay of the logistic source.3. The quasilinear chemotaxis-Stokes equations modeling the chemotactic move-ment of organisms in Stokes-fluids are studied. Invoking energy estimates and the prop-erties of Stokes operator, we obtain the global existence and boundedness of the solutions to the corresponding initial boundary value problem.4. The chemotaxis-Navier-Stokes equations describe the chemotactic movement of organisms in Navier-Stokes-fluids. We study both the Cauchy problem and the initial boundary value problem. By constructing an entropy type inequality and using the prop-erties of Stokes operator, the global existence of weak solutions to the Cauchy problem and the global existence as well as large time behaviour of the classical solution to the initial boundary problem are obtained.