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

This issue Previous Article Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping Next Article Sliding mode control of the Hodgkin–Huxley mathematical model

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: October 31, 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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