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

Keywords

• biharmonic heat equations
• global positivity
• primary 35k25
• secondary 35b09
• 35k58