Asymptotic behavior of solutions for some diffusive equations and its applications

研究了一类具有大扩散的半线性热方程解在齐次Neumann边界条件下在欧氏空间的有界光滑区域中的爆破集的位置.这是与Noriko Mizoguchi教授和Hiroki Yagisita教授共同完成的工作。我们证明了，当扩散系数足够大时，对于几乎所有的初值，解仅在初值投影到第二Neumann特征空间的极大点附近的有限时间内爆破。这是第一个解释特征函数与爆破集位置之间关系的结果。另一方面，我们研究了热方程解的极大值点(热点)的移动。特别地，我们考虑了球外区域内热方程柯西-诺依曼问题和柯西-狄里克莱特问题的解。这个外域很简单，但很难研究热点的运动。利用调和函数，我们得到了当时间趋于无穷大时热点的一些良好的渐近行为。之后，我们与Kabeya教授一起研究了热方程解的导数的衰减率和热点的移动。通过这项研究，我们可以了解如何决定溶液的导数的衰减率和热点的移动的机理。

We studied the location of the blow-up set for the solutions for a semilinear heat equation with large diffusion, under the homogeneous Neumann boundary condition, in a bounded smooth domain of the Euclidean space. This was a joint work with Professors Noriko Mizoguchi and Hiroki Yagisita. We proved that, if the diffusive coefficient is sufficiently large, for almost all initial data, the solution blows-up in a finite time only near the maximum points of the projection of the initial data onto the second Neumann eigenspace. This is the first result that explains the relation between the eigenfunctions and the location of the blow-up set.On the other hand, we studied the movement of the maximum points (hot spots) of the solutions of the heat equations. In particular, we considered the solution for the Cauchy-Neumann problem and the Cauchy-Dirichlet problem to the heat equation in the exterior domain of a ball. This exterior domain is very simple, but it is difficult to study the movement of hot spots. By using harmonic functions, we obtained some good asymptotic behavior of the hot spots as the time tends to infinity. After that, we studied the decay rate of derivatives of the solution and the movement of hot spots for the solution of the heat equation, with Professor Yoshitsugu Kabeya. By this study, we can understand the mechanism how to decide the decay rate of the derivatives of the solutions and the movement of hot spots.