Nonlinear nonlocal diffusion: A fractional porous medium equation
نویسنده
چکیده
We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, ∂u ∂t + (−∆)σ/2(|u|m−1u) = 0, x ∈ RN , t > 0, u(x, 0) = f(x), x ∈ RN , with data f ∈ L1(RN ) and exponents 0 < σ < 2, m > m∗ = (N − σ)+/N . An L1-contraction semigroup is constructed. Nonnegative solutions are proved to be continuous and strictly positive for all x ∈ RN , t > 0. We explore the range 0 < m ≤ m∗ by requiring f ∈ L1(RN ) ∩ Lp(RN ) for some p(m) > 1.
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