A general fractional porous medium equation

Abstract

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, $$ \{ll} \dfrac{\partial u}{\partial t} + (-\Delta)^{\sigma/2} (|u|^{m-1}u)=0, & \qquad x\in\mathbb{R}^N,\; t>0, [8pt] u(x,0) = f(x), & \qquad x\in\mathbb{R}^N.%. $$ We consider data $f\in L^1(\mathbb{R}^N)$ and all exponents $0<\sigma<2$ and $m>0$. Existence and uniqueness of a weak solution is established for $m> m_*=(N-\sigma)_+ /N$, giving rise to an $L^1$-contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range $0<m\le m_*$ existence and uniqueness of solutions with good properties happen under some restrictions, and the properties are different from the case above $m_*$. We also study the dependence of solutions on $f,m$ and $\sigma$. Moreover, we consider the above questions for the problem posed in a bounded domain.