# A General Fractional Porous Medium Equation Articles

### publication date

• September 2012

• 1242

• 1284

• 9

• 65

• 0010-3640

• 1097-0312

### 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