### 2007-01

#### Decay and eventual local positivity for biharmonic parabolic equations

Series: 2007-01, Preprints

MSC:
35K30 Initial value problems for higher-order parabolic equations
35B50 Maximum principles

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 $\partial_t+\Delta^2$ is a sign changing function and the solution of the evolution problem
with a positive initial datum may display almost instantaneous 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.

Keywords:
biharmonic heat kernel, evetual local positivity, semilinear problems, Fujita exponent

This paper was published in:
Discrete Cont. Dynam. Systems 21, 1129 - 1157 (2008).