Asymptotic Profiles of Solutions to Nonlinear Convection-Diffusion Equations

研究非线性对流扩散方程解的大时间行为和衰减解的渐近分布，并证明整体解的存在性.特别地，我们研究了与趋化性有关的对流扩散方程，得到了如下结果. R^n（n [大于或等于]2）中对流扩散方程柯西问题的每一个有界解随着t [互变异构体]∞衰减到零，并且表现为热核.在初始函数的L^1-范数较小的条件下，得到了上述柯西问题有界解的存在性.考虑了一类在有界区间上不含化学扩散的对流扩散方程，在Neumann边界条件下，给出了解的长时间行为。

The purpose of this research is to study the large time behavior of solutions and asymptotic profiles of decaying solutions to nonlinear convection-diffusion equations, and also to show the existence of global solutions. Especially, we focus on convection-diffusion diffusion equations related to chemotaxis and obtained the following results.1. Every bounded solution of the Cauchy problem to a convection-diffusion equation in R^n (n 【greater than or equal】2) decays to zero as t 【tautomer】∞ and behaves like the heat kernel.2. The existence of bounded solutions of the Cauchy problem mentioned just above is obtained under the smallness of L^1-norm of initial functions.3. We consider a convection-diffusion equation without the chemical diffusion in a bounded interval under Neumann boundary conditions, and show the long time behaviour of solutions.