具有对数敏感度和混合边界的一维趋化模型解的整体存在性和收敛性

Global Existence and Convergence of Solutions to a Chemotactic Model with Logarithmic Sensitivity and Mixed Boundary Conditions

该文主要研究一维有界区间中具有对数敏感度的趋化模型{■tu=Duxx+(u(ln v)x)x,x∈(0,1),t>0.■tu=∈Uxx+uv-μu,x∈(0,1),t>0.根据Cole-Hopf变换将上述带奇性的排斥趋化模型变换为如下的非奇异方程组{pt=pxx+(pq)x,x∈(0,1),t>0,qt=∈qxx+∈(q^2)x+px,x∈(0,1)mt>0,并在混合边界条件下得到对应的初边值问题解的整体存在性和指数收敛性.

This paper investigates the following chemotactic model with logarithmic sensitivity in a one-dimensional bounded domain:{■tu=Duxx+(u(ln v)x)x,x∈(0,1),t>0.■tu=∈Uxx+uv-μu,x∈(0,1),t>0.By using a Cole-Hopf type transformation,we transform the above singular repulsive chemotaxis model into a non-singular system of the form {pt=pxx+(pq)x,x∈(0,1),t>0,qt=∈qxx+∈(q^2)x+px,x∈(0,1)mt>0,Then under some mixed boundary conditions,we prove the global existence and exponential convergence of solutions to the initial-boundary value problem of the above system with regular initial data.