Singularly perturbed solutions of reaction-diffusion systems and concentration phenomena

1. Takagi考虑了一类活化剂-抑制剂型反应扩散方程组定态解的构造和稳定性。在倪伟明和柳田英二的合作下，他在一维区域的情况下证明了：（i）当活化剂扩散缓慢而抑制剂扩散非常快时，存在集中在边界点的定态解。(ii)如果抑制剂反应的弛豫参数很小，那么这些解是稳定的;而如果弛豫参数足够大，那么它们是不稳定的。(iii)从稳态解分支出一个集中在边界点附近的单参数周期解族，并将这些结果推广到高维区域，当抑制剂的扩散速率为无穷大时. Nishiura和Iida研究了产生尖锐过渡层的反应扩散方程组的初边值问题的解的行为。Nishiura建立了一个理论来解释自我复制模式的机制。Iida构造了一个反应扩散方程组，其奇异极限退化为经典Stefan问题. Tsutsumi、Tachizawa和中野运用真实的分析技术研究了薛定谔方程。他们在初值问题的适定性和特征值的渐近分布方面获得了新的结果.增田和长泽主要考虑了非线性扩散方程解的行为。增田证明了弱解的最大值原理。Nagasawa改进了Navier-Stokes方程弱解的能量不等式。

1. Takagi considered the construction and stability of stationary solutions of a reaction-diffusion system of activator-inhibitor type. With the cooperation of Wei-Ming Ni and Eiji Yanagida, he proved the following in the case of one dimensional domains : (i) The existence of stationary solutions concentrating at the boundary point when the activator diffuses slowly and the inhibitor diffuses very fast. (ii) If the relaxation parameter of the inhibitor reaction is small then these solutions are stable ; while they are unstable if the relaxation parameter is sufficiently large. (iii) A one-parameter family of periodic solutions concentrating around the boundary point bifurcates from the stationary solution.Moreover, these results are generalized to higher dimensional domains in the case where the diffusion rate of the inhibitor is infinite.2. Nishiura and Iida studied the behavior of solutions to the initial-boundary value problem for reaction-diffusion systems which generate sharp transition layers. Nishiura established a theory to explain the mechanism of self-replicating patterns. Iida constructed a reaction-diffusion system whose singular limit reduces to the classical Stefan problem.3. Tsutsumi, Tachizawa and Nakano studied Schroedinger equations by applying techniques in real analysis. They obtained new results on the well-posedness of the initial value problem, and on the asymptotic distribution of eigenvalues.4. Masuda and Nagasawa considered mainly the behavior of solutions to nonlinear diffusion equations. Masuda proved the maximum principle for weak solutions. Nagasawa refined the energy inequality for weak solutions to the Navier-Stokes equations.