Alternative probabilistic representations of Barenblatt-type solutions

Abstract

A general class of probability density functions \[u(x,t)=Ct^{-αd}\left (1-\left (\frac{\|x\|}{ct^α}\right )^β\right )_+^γ,\quad x\in \mathbb{R}^d,t>0,\] is considered, containing as particular case the Barenblatt solutions arising, for instance, in the study of nonlinear heat equations. Alternative probabilistic representations of the Barenblatt-type solutions $u(x,t)$ are proposed. In the one-dimensional case, by means of this approach, $u(x,t)$ can be connected with the wave propagation.

arXiv commentPublished at https://doi.org/10.15559/20-VMSTA151 in the Modern Stochastics: Theory and Applications (https://vmsta.org/) by VTeX (http://www.vtex.lt/)

References in corpus (8)

• Fractional Cauchy problems on bounded domains
• Fractional diffusion equations and processes with randomly varying time
• Space-time fractional diffusion on bounded domains
• Nonlocal porous medium equation: Barenblatt profiles and other weak solutions
• Space-time duality for fractional diffusion
• Numerical method for the time-fractional porous medium equation
• Random flights related to the Euler-Poisson-Darboux equation
• Stochastic models associated to a Nonlocal Porous Medium Equation