Asymptotic behavior of the solutions to a family of PDE ' s arising from the chemotaxis equations of Keller and

The system ut = uxx − (uvx)x , vt = u − Av is considered where A is a non-negative, self-adjoint operator which commutes with the Laplacian. The operator is considered to have eigenvalues λn = nλ1, and the system is considered on [0, 1]× [0, T ] with homogeneous Neumann boundary conditions. The operators which lead to global solutions and those that lead to solutions which blow up in finite time are considered as a function of ρ, using an application of the methods of Hillen and Potapov [Math. Methods Appl. Sci., 27 (2004), pp. 1783− 1801] to analyze the global case and those of Halverson, Levine, and Renclawowicz [Siam J. Appl. Math., 65 (2004), pp. 336−360; 66 (2005), pp. 361−364] to analyze the finite time blowup case. Some numerical results are provided to back up the analysis. Some questions and directions for future study are posed.

考虑系统$u_t = u_{xx} - (uv_x)_x$，$v_t = u - Av$，其中$A$是一个非负的、自伴算子，它与拉普拉斯算子可交换。该算子被认为具有特征值$\lambda_n = n\lambda_1$，并且在$[0, 1]\times[0, T]$上考虑该系统，具有齐次诺伊曼边界条件。利用Hillen和Potapov的方法（[《数学方法与应用科学》，27（2004），第1783 - 1801页]）来分析整体情况，以及利用Halverson、Levine和Renclawowicz的方法（[《工业与应用数学学会应用数学杂志》，65（2004），第336 - 360页；66（2005），第361 - 364页]）来分析有限时间爆破情况，将导致整体解的算子和导致在有限时间内爆破的解的算子视为$\rho$的函数。提供了一些数值结果来支持该分析。提出了一些未来研究的问题和方向。