New development of the qualitative theory of nonlinear parabolic and elliptic equations

对非线性抛物型和椭圆型方程解的结构进行了研究。我们的主要成果如下：其次，我们研究了一类非线性抛物型偏微分方程具有运动奇异点解的存在唯一性。我们还证明了存在一个具有突然改变其类型的移动奇点的解。，并明确了收敛于奇异稳态的奇异解的渐近行为。我们还研究了一个趋化系统，并明确了在有限时间内通过集中到一点而爆炸的自相似溶液的结构。对于反应-扩散系统，即Gierer-Meinhardt系统，我们研究了模式形成的数学结构，并明确了时变解的行为。

We carried out the investigation about the structure of solutions of nonlinear parabolic and elliptic equations. Our main results are as follows : Next, we studied the existence and uniqueness of solutions with moving singularities for a nonlinear parabolic partial differential equation. We also showed that there exists a solution with a moving singularity that changes its type suddenly., and made clear the asymptotic behavior of singular solutions that converges to a singular steady state. We also studied a chemotaxis system, and made clear the structure of self-similar solutions that blows up by concentrating to a point in finite time.For a reaction-diffusion system, which is called a Gierer-Meinhardt system, we studied the mathematical structure of pattern formation, and also made clear the behavior of time-dependent solutions.