Weak Nonmild Solution of Stochastic Fractional Porous Medium Equation

TL;DR

This paper investigates a stochastic fractional porous medium equation with non-mild solutions, establishing existence and uniqueness using RKHS, and explores how noise and anomalous diffusion affect energy growth.

Contribution

It introduces a novel approach to define solutions for a stochastic fractional PDE lacking a heat kernel representation, using RKHS methods.

Findings

Existence and uniqueness of solutions established.

Effect of space-time white noise on fractional operators analyzed.

Energy moment growth behavior studied under anomalous diffusion parameters.

Abstract

Consider the non-linear stochastic fractional-diffusion equation \begin{eqnarray*} \left \{\begin{array}{lll} \frac{\partial}{\partial t}u(x,t)= -( \Delta)^{\alpha/2} u^m(x,t) + \sigma(u(x,t)) \dot{W}(x,t),\, x\in \mathbb{R}^d,t>0, u(x,0)= u_0(x),\,\,\, x\in\mathbb{R}^d \end{array}\right. \end{eqnarray*} with initial data $u_0(x)$ an $L^1(\mathbb{R}^d)$ function, $0<\alpha<2$, and $m>0$. There is no mild solution defined for the above equation because its corresponding heat kernel representation does not exist. We attempt to make sense of the above equation by establishing the existence and uniqueness result via the reproducing kernel Hilbert space (RKHS) of the space-time noise. Our result shows the effect of a space-time white noise on the interaction of fractional operators with porous medium type propagation and consequently studies how the anomalous diffusion parameters influence the energy moment growth behaviour of the system.