Decay and local eventual positivity for biharmonic parabolic equations

Alberto Ferrero; Filippo Gazzola; Hans-Christoph Grunau

doi:10.3934/dcds.2008.21.1129

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2008, Volume 21, Issue 4: 1129-1157. Doi: 10.3934/dcds.2008.21.1129

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Decay and local eventual positivity for biharmonic parabolic equations

1.
Dipartimento di Matematica dell’Università di Milano-Bicocca, Via Cozzi 53, Milano, 20125
2.
Dipartimento di Matematica Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano
3.
Fakultät für Mathematik, Otto-von-Guericke-Universität, Postfach 4120, 39016 Magdeburg

Received: July 31, 2007

Revised: November 30, 2007

Published: September 2008

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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.

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Mathematics Subject Classification: Primary: 35K25, 35K55; Secondary: 35B05.

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