Stochastic Nonlinear Diffusion Equations with Singular Diffusivity

2009 ◽  
Vol 41 (3) ◽  
pp. 1106-1120 ◽  
Author(s):  
Viorel Barbu ◽  
Giuseppe Da Prato ◽  
Michael Röckner
Keyword(s):  
Nonlinear Diffusion ◽  
Diffusion Equations ◽  
Nonlinear Diffusion Equations ◽  
Singular Diffusivity

Total variation flow perturbed by gradient linear multiplicative noise

2018 ◽  
Vol 21 (01) ◽  
pp. 1850003 ◽  
Author(s):  
Ionuţ Munteanu ◽  
Michael Röckner
Keyword(s):  
Nonlinear Diffusion ◽  
Existence And Uniqueness ◽  
Multiplicative Noise ◽  
Diffusion Equations ◽  
Stochastic Variational Inequalities ◽  
Nonlinear Diffusion Equations ◽  
Variational Solutions ◽  
Total Variation Flow ◽  
Gradient Type ◽  
Singular Diffusivity

We consider stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative gradient-type noise. We study existence and uniqueness of non-negative variational solutions in terms of stochastic variational inequalities. We also show the positivity preserving property of the solutions and extinction in finite time with probability one. These kinds of equations arise, e.g., in the use for simulation of image restoring techniques or for modeling turbulence.

scholarly journals The Neumann problem for nonlocal nonlinear diffusion equations

2007 ◽  
Vol 8 (1) ◽  
pp. 189-215 ◽  
Author(s):  
Fuensanta Andreu ◽  
José M. Mazón ◽  
Julio D. Rossi ◽  
Julián Toledo
Keyword(s):  
Neumann Problem ◽  
Nonlinear Diffusion ◽  
Diffusion Equations ◽  
Nonlinear Diffusion Equations ◽  
The Neumann Problem ◽  
Nonlocal Nonlinear

scholarly journals Conditional Lie-Bäcklund Symmetries and Reductions of the Nonlinear Diffusion Equations with Source

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Junquan Song ◽  
Yujian Ye ◽  
Danda Zhang ◽  
Jun Zhang
Keyword(s):  
Invariant Subspace ◽  
Dynamic Systems ◽  
Nonlinear Diffusion ◽  
Invariant Subspaces ◽  
Higher Order ◽  
Diffusion Equations ◽  
Canonical Forms ◽  
Nonlinear Diffusion Equations ◽  
Symmetry Approach ◽  
Finite Dimensional

Conditional Lie-Bäcklund symmetry approach is used to study the invariant subspace of the nonlinear diffusion equations with sourceut=e−qx(epxP(u)uxm)x+Q(x,u),m≠1. We obtain a complete list of canonical forms for such equations admit multidimensional invariant subspaces determined by higher order conditional Lie-Bäcklund symmetries. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamic systems.

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