Autonomous Formation of Spatial Structures in Solutions of Parabolic Partial Differential Equations

Takagi与明尼苏达大学的倪卫明和东北大学的铃木Kanako Suzuki合作，研究了Gierer和Meinhardt提出的一类激活剂-抑制物类型的反应扩散系统的解的性态，得到了以下结果：(I)在初始数据为常数函数的情况下，如果激活剂激活其产物比抑制物的产物更强，则存在在有限时间内爆破的解。有两种类型的爆炸解决方案。或者(A)只有激活剂爆炸，或者(B)激活剂和抑制剂都爆炸。在前一种情况下，我们可以选择初始值，使抑制器收敛到任何指定的正数。(Ii)在激活剂的方程不包含源项的情况下，如果激活剂产生的抑制剂多于其自身，则没有解在有限时间内爆炸。此外，在这种情况下，可能会发生模式崩溃。在这里，我们所说的模式崩溃是指当时间变量趋于无穷大时，解收敛到原点。西村研究了脉冲解和光斑解的散射现象。他发现，根据不稳定稳态或周期解邻域内的局部动力学以及解轨道的位置，可以形成各种输入/输出关系。Yanagida考虑了一类拟线性抛物型方程，证明了该解要么是(A)全局增加的，(B)是行波的，要么是在有限时间内消失的，具体取决于初始数据。Kozono证明了在三维外域上可以构造出满足所有平方可积初值的强能量不等式的Navier-Stokes方程的弱解。

This objective of this project is to pursue the behavior of solutions of nonlinear partial differential equations of parabolic type.In collaboration with Wei-Ming Ni (University of Minnesota) and Kanako Suzuki (Tohoku University), Takagi studied the behavior of solutions of a reaction-diffusion system of activator-inhibitor type proposed by Gierer and Meinhardt and obtained the following results : (i)In the case where the initial data are constant functions, there exist solutions blowing up in finite time if the activator activates the its production stronger than that of the inhibitor. There are two types of blow-up solutions. Either (a) only the activator blows up or (b) both the activator and the inhibitor blow up. In the former case, we can choose the initial value so that the inhibitor converges to any specified positive number. (ii)In the case where the equation for the activator does not contain the source term, no solution blows up in finite time if the activator produces the inhibitor more than itself. Moreover, in this case the collapse of patterns can occur. Here, by the collapse of patterns we mean that the solution converges to the origin as the time variable tends to infinity.Nishiura studied scattering phenomena of pulse solutions and spot solutions. He found that various input/output relationships can be formed depending on the local dynamics in the neighborhood of unstable steady-states or periodic solutions and on the location of solution orbits.Yanagida considered a certain quasilinear parabolic equation and showed that the solution is either (a) globally increasing, (b) a traveling wave, or (c) extinct in finite time, depending on the initial data. The asymptotic behavior of the solution is also investigated.Kozono proved that in three dimensional exterior domains one can construct weak solutions of the Navier-Stokes equations which satisfy the strong energy inequality for all square-integrable initial data.