Effect of decay rates of initial data on the sign of solutions to Cauchy problems of polyharmonic heat equations

Abstract

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 are eventually globally positive if 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.