Decay and eventual local positivity for biharmonic parabolic equations

Abstract

(Communicated by Peter Polacik) Abstract. We study existence and positivity properties for solutions of Cauchy problems for both linear and semilinear parabolic equations with the biharmonic operator as elliptic principal part. The self-similar kernel of the parabolic operator ∂t + ∆2 is a sign changing function and the solution of the evolution problem with a positive initial datum may display almost in-stantaneous change of sign. We determine conditions on the initial datum for which the corresponding solution exhibits some kind of positivity behaviour. We prove eventual local positivity properties both in the linear and semilinear case. At the same time, we show that negativity of the solution may occur also for arbitrarily large given time, provided the initial datum is suitably constructed. 1. Introduction and results