Fractional Cahn–Hilliard, Allen–Cahn and porous medium equations

Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains

We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. We establish existence and uniqueness results in a suitable class of solutions using the theory of maximal monotone operators on dual spaces. Then we describe the long-time asymptotics in terms of separate-variables s...

The Mesa Problem for the Fractional Porous Medium Equation

We investigate the behaviour of the solutions um(x, t) of the fractional porous medium equation ut + (−∆)(u) = 0, x ∈ R , t > 0. with initial data u(x, 0) ≥ 0, x ∈ RN , in the limit of as m → ∞ with fixed s ∈ (0, 1). We first identify the limit of the Barenblatt solutions as the solution of a fractional obstacle problem, and we observe that, contrary to the case s = 1, the limit is not compactl...

Nonlinear nonlocal diffusion: A fractional porous medium equation

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...

Hamilton’s gradient estimates and Liouville theorems for porous medium equations

Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative

We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account “memory”. The precise model is D t u− div(u(−∆)−σu) = f, 0 < σ < 1/2. We pose the problem over {t ∈ R+, x ∈ Rn} with nonnegative initial data u(0, x) ≥ 0 as wel...