Nonlinear and nonlocal diffusion equations

Zusammenfassung

We consider three different models of nonlinear diffusion equations. The prototype is the classical Porous Medium Equation u_t=\Delta u^m. The first model is the Doubly Nonlinear Diffusion Equation u_t=\Delta_p u^m for which we discuss the asymptotic behavior of the homogenous Dirichlet problem. The second model is The Fisher-KPP Equation with nonlinear fractional diffusion u_t+ (-\Delta)^s u^m=u(1-u) for x \in \mathbb{R}^N, t>0. We prove that the level sets of the solution propagate exponentially fast in time. The third model is the Porous Medium Equation with fractional pressure u_t=\nabla \cdot(u^{m-1}\nabla P), P=(-\Delta)^{-s}u for x\in \mathbb{R}^N, t>0. Our main result concerns the effect of the nonlinearity on the finite speed of propagation of the solutions.