带分数阶拉普拉斯算子的发展方程

In recent years there has been a surge of activity focused on the use of so-called fractional diffusion operators to replace the standard Laplace operator. And there are many applications of these equation in physics, biology, chemistry. The often used method and technique for parabolic equations will not apply to the case of fractional diffusion equation. Thus give us a new difficulty. Then we must develop new method. To study the exact dynamic behavior the fractional diffusion equation is meaningful project. In this project, we mainly study two kinds of fractional diffusion equation, including the Fujita critical exponent, the blow-up time, the self-similar solutions, the long time behavior, and so on.

近年来许多数学工作者将标准的Laplace算子用分数阶Laplace算子代替。带分数阶Laplace算子的发展方程具有丰富的物理背景和内涵，在物理、生物、化学等多个学科有着广泛的应用。传统的研究抛物方程的技巧对带分数阶扩散项的发展方程基本不适用，这就迫使研究者引进新的思想或方法技巧才有可能取得新的突破。深入研究带分数阶Laplace算子的发展方程的动力学行为是一个有着重要意义和丰富内容的课题。本项目主要研究两类带分数阶Laplace算子的发展方程，所涉及的问题包括Fujita临界指标的存在、爆破时间的上下界估计、自相似解、长时间渐进行为等。