混合机制对半线性热方程解爆破的抑制作用

Suppression of blow up by mixing mechanism in semilinear heat equations

本文考虑带有混合机制的半线性热方程,混合机制是通过加入含不可压流的对流项实现的.在没有混合机制时,方程的解在有限时间会发生爆破.在对流项合适的混合条件下,本文研究方程的大初值整体解的适定性.对于带有混合机制的经典半线性热方程,本文通过能量方法得到了整体的Lp(p>d/2)估计,并且获得了经典解的整体存在性.然而,对于带有混合机制的分数阶半线性热方程,由于技术困难,通过能量方法无法得到整体的Lp (p> 2)估计,这里利用非线性极值原理得到了整体的L∞估计,并且获得了经典解的整体存在性.

In this paper, we consider the semilinear heat equation with additional mixing mechanism of advection by an incompressible flow. In the absence of mixing mechanism, the solution of the equation is blow up in finite time. Under a suitable mixing condition of the advection, we study global well-posedness of the solution with large initial data. For the classical semilinear heat equation with mixing mechanism, we establish the global Lp(p >d/2) estimate of the solution by the energy method, and obtain the global existence of the classical solution.However, for the fractional semilinear heat equation with mixing mechanism, due to technical difficulties, we cannot get the global Lp(p > 2) estimate by the energy method;we establish the global L∞ estimate of the solution through the nonlinear maximum principle, and obtain the global existence of the classical solution.