Global Solvability And Stabilization In A Three-Dimensional Cross-Diffusion System

Partial differential equations play a very important role as an analytical tool for solving key problems in geometry,physics and biology.Describing complex social systems by means of continuous models based on evolutionary partial differential equations has received considerable interest in the past decades.For instance,the following crossdiffusion system was proposed by Short et al.(?) to describe the spatial-temporal dynamics in urban criminal activity,where u = u(x,t)and v = v(x,t)are unknown functions which represent the density of criminal agents and the attractiveness value,respectively.The given functions B1(x,t)and B2(x,t)denote the density of additional criminal agents who enters into the system and an assumed base attractiveness value which gives rise to the growth function,respectively.Within a suitable generalized framework,the thesis considers the initial-boundary value problem of the cross-diffusion system(*)with no-flux boundary conditions in a three-dimensional setting.As ? ?(0,3^1/2)and B1,B2 are radially symmetric,the local existence and extensibility of the classical solution of the regularized system are obtained by applying the standard cross-diffusion theory.Secondly,both the boundedness of the solution obtained by the Neumann heat semigroup and extensibility yield the global existence of the classical solution to the regularized system.Thirdly,the uniform a priori estimates of the solution to the regularized system are obtained by employing the subtle energy estimates.Finally,with help of Arzelà-Ascoli theorem and Fatou's lemma,the global existence of solution to the cross-diffusion system(*)can be established by taking limits.Moreover,if ? ?(0,6^1/2/2))and B1(x,t)? 0,it is proved that the solution obtained above will converge to a nontrivial steady state as t ? ? by applying the anisotropic LP interpolation inequalities and the Arzelà-Ascoli theorem.