Abstract 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.

中文翻译：

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线性和半线性双调和热方程柯西问题解的正性

摘要 本文研究了以双调和算子为四阶椭圆主成分的线性和非线性抛物线方程柯西问题解的正解性。一般情况下，四阶抛物线方程的柯西问题由于其基本解的符号变化，不具有保正性。一个人对积极的初始数据最终具有局部积极性，但在短时间尺度上，一个人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充分条件，以确保线性双调和热方程的柯西问题的解在所有时间和整个空间中都是正的。