A general fractional porous medium equation

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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 xinmathbb{R}^N,; t>0, [8pt] u(x,0) = f(x), & qquad xinmathbb{R}^N.%.

We consider data

f i n L^{1} (m a t h b b R^{N})

and all exponents

0 < s i g m a < 2

and

m > 0

. Existence and uniqueness of a weak solution is established for

m > m_{*} = (N - s i g m a)_{+} / 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 l e 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

s i g m a

. Moreover, we consider the above questions for the problem posed in a bounded domain.

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