Qualitative theory and asymptotic analysis of nonlinear partial differential equations

研究了幂非线性和指数非线性非线性热方程爆破时间后解的行为。结果表明，爆破时出现的奇异点消失，解立即变得光滑（Matano, SIAM j . math . al.）。（待印）。我们还研究了具有幂非线性的非线性热方程的爆破率，并证明了只要幂在中超临界范围内，爆破总是2型的（Matano, Comm.Pure Appl.Math）。, 2004)。Yamamoto研究了一个确定二维椭圆方程中两个未知对流项的反问题，并证明了这些项可以通过所谓的Dirichlet-Neumann映射来确定（inverse Problems, 2004）。Weiss研究了适用于双障碍问题的抛物方程的奇异极限问题。利用新的单调性公式，他成功地估计了自由边界的豪斯多夫维数（Calc.Var）。PDE, 2003)。Ei研究了二维圆柱体上的反应扩散系统，并分析了具有脉冲剖面的溶液的行为。他推导了一个控制慢脉冲运动的方程，并证明了行进脉冲是相互排斥的(DCDS, Ser。A, in press)。Taniguchi考虑了所谓的“奇异极限特征值问题方法”（SLEP方法），它是分析奇异极限问题平稳解稳定性的有力工具。他推广了这种方法，使之适用于无界域的问题。利用这一结果，他证明了平面行波在双稳态反应扩散体系(DCDS, Ser。B, 2003)。

We have considered the behavior of solutions after the blow-up time for nonlinear heat equations with a power nonlinearity and those with an exponential nonlinearity. It is shown that singularities that apper at the blow-up time disappear and the solutions become smooth immediately (Matano, SIAM J.Math.Anal., in press). We have also studied the blow-up rate for nonlinear heat equations with a power nonlinearity and proved that the blow-up is always type 2 so far as the power is in the intermediate supercritical range (Matano, Comm.Pure Appl.Math., 2004).Yamamoto has studied an inverse problem of determining two unknown convection terms in a two-dimensional elliptic equation, and proved that those terms can be determined by the so-called Dirichlet-Neumann map (Inverse Problems, 2004).Weiss has considered a singular limit problem for parabolic equations that are applicable to the double obstacle problem. Using a new monotonicity formula, he has succeeded in exstimating the Hausdorff dimension of the free boundary (Calc.Var.PDE, 2003).Ei has studied a reaction-diffusion system on a two-dimensional cylinder and analysed the behavior of solutions having a pulse-like profile. He derived an equation governing the motion of slow pulses and proved that traveling pulses are mutually repelling (DCDS, Ser.A, in press).Taniguchi has considered the so-called "singular limit eigenvalue problem method" (SLEP method), which is a powerful tool in analyzing the stability of stationary solution of the singular limit problem. He has generalized this method so that it applies to problems in unbounded domains. Using this result, he has proved the stability of planar traveling waves in a bistable reaction-diffusion system (DCDS, Ser.B, 2003).