A general fractional porous medium equation

Abstract

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, $$ \{ll} \dfrac{\partial u}{\partial t} + (-Δ)^{σ/2} (|u|^{m-1}u)=0, & \qquad x\in\mathbb{R}^N,\; t>0, [8pt] u(x,0) = f(x), & \qquad x\in\mathbb{R}^N.%. $$ 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 $m>m_{\ast }=(N-\sigma {)}_{+}/N$, giving rise to an $L^1$-contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range $0<m\le m_{\ast }$ existence and uniqueness of solutions with good properties happen under some restrictions, and the properties are different from the case above $m_{\ast }$. 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.

43 pages, 2 figures

References in corpus (1)

• Hydrodynamic limit of particle systems with long jumps

Cited by in corpus (1)

• Weak Nonmild Solution of Stochastic Fractional Porous Medium Equation