Effect of decay rates of initial data on the sign of solutions to Cauchy problems of polyharmonic heat equations
Effect of decay rates of initial data on the sign of solutions to Cauchy problems of polyharmonic heat equations
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DOI：
10.1007/s00208-022-02466-w
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发表时间：
2022-09
影响因子：
1.4
通讯作者：
Nobuhito Miyake
中科院分区：
文献类型：
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作者：
Nobuhito Miyake
在本文中，我们考虑线性和半线性多调和热方程柯西问题解的符号。高阶抛物线方程的柯西问题一般不具有正性保持性质，但是，如果初始数据衰减得足够慢，则预计这些柯西问题的解最终是全局正的。我们首先证明了初始数据衰减率阈值的存在性，该阈值区分了线性多调和热方程柯西问题的相应解最终是否全局为正。应用这一结果，我们最终构造了超藤田条件下半线性多调和热方程柯西问题的全局正解。
In this paper, we consider the sign of solutions to Cauchy problems of linear and semilinear polyharmonic heat equations. Cauchy problems for higher order parabolic equations have no positivity preserving property in general, however, it is expected that solutions to these Cauchy problems areeventually globally positiveif initial data decay slowly enough. We first show the existence of the threshold of the decay rate of initial datum which separates whether the corresponding solution to the Cauchy problem of the linear polyharmonic heat equation is eventually globally positive or not. Applying this result, we construct eventually globally positive solutions to the Cauchy problem of the semilinear polyharmonic heat equation under the super-Fujita condition.