论文标题
关键的Hardy-Sobolev抛物线方程的适应性和全球动态
Well-posedness and global dynamics for the critical Hardy-Sobolev parabolic equation
论文作者
论文摘要
我们在能量空间中研究了具有奇异电位的半线性热方程的库奇问题，称为Hardy-Sobolev抛物线方程。本文的目的是确定下面或基态下的初始数据必要且充分的条件，在该数据下，解决方案的行为被完全二分。更确切地说，该解决方案在全球范围内存在，其能量衰减的时间衰减为零，或者在有限或无限的时间内炸毁。通过比较原理，对相应的Dirichlet问题的二分法的结果也显示为副产品。
We study the Cauchy problem for the semilinear heat equation with the singular potential, called the Hardy-Sobolev parabolic equation, in the energy space. The aim of this paper is to determine a necessary and sufficient condition on initial data below or at the ground state, under which the behavior of solutions is completely dichotomized. More precisely, the solution exists globally in time and its energy decays to zero in time, or it blows up in finite or infinite time. The result on the dichotomy for the corresponding Dirichlet problem is also shown as a by-product via comparison principle.
