Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

TL;DR

This paper investigates conditions under which solutions to linear and semilinear biharmonic heat equations remain positive over time, addressing the challenge posed by the fundamental solution's sign-changing nature.

Contribution

It provides sufficient conditions on initial data to ensure solutions are globally positive for both linear and semilinear biharmonic heat equations.

Findings

Identified conditions for initial data guaranteeing positivity of solutions.

Established existence of globally positive solutions for semilinear biharmonic equations.

Clarified the positivity behavior in the short and long term for these equations.

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.