Weak Nonmild Solution of Stochastic Fractional Porous Medium Equation

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.