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.

Ferrero, A., Gazzola, F., Grunau, H. (2008). Decay and eventual local positivity for biharmonic parabolic equations. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES S, 21(4), 1129-1157.

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

#### 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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Articolo in rivista - Articolo scientifico
Fujita exponent; global existence; decay estimates
English
1129
1157
Ferrero, A., Gazzola, F., Grunau, H. (2008). Decay and eventual local positivity for biharmonic parabolic equations. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES S, 21(4), 1129-1157.
Ferrero, A; Gazzola, F; Grunau, H
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/7701
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