Existence and extinction in finite time for Stratonovich gradient noise porous media equations

Mattia Turra

doi:10.3934/eect.2019042

Article Contents

2019, Volume 8, Issue 4: 867-882. Doi: 10.3934/eect.2019042

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Existence and extinction in finite time for Stratonovich gradient noise porous media equations

Mattia Turra^,

Dipartimento di Informatica, Università degli Studi di Verona, Strada Le Grazie 15, I–73134, Verona, Italy

^* Corresponding author: Mattia Turra
^* Corresponding author: Mattia Turra

Received: November 2018

Early access: June 2019

Published: June 2019

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Abstract

We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $ dX - \bigl( \nu \Delta X + \Delta \psi (X) \bigr) dt = \sum_{i = 1}^N \langle b_i, \nabla X \rangle \circ d\beta_i $ in $ ]0,T[ \times \mathcal{O} $, with $ X(0) = x(\xi) $ in $ \mathcal{O} $ and $ X = 0 $ on $ ]0,T[ \times \partial \mathcal{O} $. Moreover, we prove extinction in finite time of the solutions in the special case of fast diffusion model and of self-organized criticality model.

Keywords:

Stochastic partial differential equations,
porous media equations,
multiplicative Stratonovich gradient noise,
fast diffusion,
self-organized criticality.

Mathematics Subject Classification: Primary: 60H15; Secondary: 35K55, 76S05.

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