Asymptotic behavior of an aggregating pattern of the reaction diffusion equation with the advection term

1.对于趋化生长模型，我们通过数值模拟给出了在条纹域中具有三重结的行波解。为了证明该解的存在性，我们首先用与原模型方程对应的界面方程构造了该解的近似解。此外，还可以显示出行进溶液的速度与趋化强度之间的成模关系。对于吸附剂诱导相变模型，我们在二维有界区域上证明了周期边界条件下非负整体解和指数吸引子的存在性。由于函数空间的适当选择，我们证明了由方程得到的动力系统的压缩性是成立的。对于平面上的吸附诱导相变模型，我们证明了在边界条件下非负整体解的存在性，使得解在平面边缘趋向于恒定平衡解。在这种情况下，我们不能证明指数吸引子的存在性。对于平面有界区域内吸附诱导相变模型，在Newman边界条件下证明了整体解和指数吸引子的存在性。通过数值模拟，给出了常平衡解分岔点附近参数的六角形和条形等形式。对于趋化性生长模型，需要从生物学的角度考虑具有原点奇点的敏感函数。对于平面上的有界定义域，我们证明了当初始数据相对于某个泛函范数较小时，解趋向于平凡解。此外，在另一种情况下存在非空的极限集。

1.For the chemotaxis growth model, we show the traveling wave solution with a triple junction in the stripe domain by the numerical simulations. In order to show the existence of the solution, we first construct the approximate solution of this solution by the interface equation corresponding to the original model equation. Moreover, it can be shown die relation of the velocity of the traveling solution and the intensity of the chemotaxis.2.For the adsorbate-induced phase transition model, we show the existence of the nonnegative global solution and exponential attractor under the periodic boundary condition in two dimensional bounded domain. Due to the appropriate choice of the functional space, we prove that the sqeezing property is hold for the dynamical system obtained from the equation.3.For the adsorbate-induced phase transition model in the plane, we show the existence of the nonnegative global solutions under the boundary condition such that the solution tends to the constant equilibrium solution at the edge of the plane. In this situation, we can not prove the existence of the exponential attractor.4.For the adsorbate-induced phase transition model in the bounded domain of the plane, we prove the existence of the global solutions and exponential attractor under the Newman boundary condition. By the numerical simulations, we show the hexagonal and stripe patterns and so on for the parameters in the neighborhood of the bifurcation point of constant equilibrium solution.5.For the chemotaxis growth model, we need consider the sensitive function with the singularity at the origin from the biological view points. For the bounded domain in the plane, we prove that the solution tends to the trivial solution if the initial data is small with respect to some functional norm. Moreover, there is a nonempty omega limit set in another case.