论文标题
对于线性和半线性双旋量热方程的库奇问题解决方案的积极性
Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations
论文作者
论文摘要
本文关注的是，与双旋转操作员作为第四阶椭圆形主部分的线性和非线性抛物线方程的库奇问题的积极性。通常，由于基本解决方案的迹象的变化，第四阶抛物线方程方程的凯奇问题没有积极性。一个人最终具有正面的初始数据的局部积极性，但是在短时间内，人们通常也会有消极的区域。本文的第一个目标是在初始数据上找到足够的条件，以确保在所有时间和整个空间中都有线性双旋量热方程的库奇问题解决方案。第二个目标是将这些结果应用于半线性双向抛物线方程中的库奇问题的存在。
This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.
